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Stability Analysis and Numerical Simulation of Non-Newtonian Fluids of Oldroyd Kind Den Naturwissenschaftlichen Fakult¨ aten der Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg zur Erlangung des Doktorgrades vorgelegt von Nicoleta Dana Scurtu aus Bra¸ sov

Stability Analysis and Numerical Simulation of Non-Newtonian

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Page 1: Stability Analysis and Numerical Simulation of Non-Newtonian

Stability Analysis and NumericalSimulation of Non-Newtonian Fluids

of Oldroyd Kind

Den Naturwissenschaftlichen Fakultaten

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

zur

Erlangung des Doktorgrades

vorgelegt von

Nicoleta Dana Scurtu

aus Brasov

Page 2: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 3: Stability Analysis and Numerical Simulation of Non-Newtonian

Als Dissertation genehmigt von den Naturwissen-

schaftlichen Fakultaten der Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 24. Oktober 2005

Vorsitzender der

Promotionskommision: Prof. Dr. D.-P. Hader

Erstberichterstatter: Prof. Dr. Eberhard Bansch

Zweitberichterstatter: Prof. Dr. Wolfgang Borchers

Page 4: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 5: Stability Analysis and Numerical Simulation of Non-Newtonian

To the man who pleases him,

God gives wisdom, knowledge and happiness...

(the Bible)

Page 6: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 7: Stability Analysis and Numerical Simulation of Non-Newtonian

Acknowledgment

First of all, I am especially thankful to Prof. Dr. Eberhard Bansch who gave me the oppor-

tunity to work in his research group in the fascinating field of computational fluid dynamics

and for the time he spent to supervise my research.

I also want to thank Prof. Dr. Kunibert G. Siebert who kindly placed his program package

Albert at my disposal which was the starting point and the major tool for my numerical

computations.

Finally, I want to acknowledge my colleagues from Weierstraß Institute Berlin and from

Brandenburg Technical University of Technology Cottbus for their moral encouragement,

for text improvements and for their hints concerning computational issues.

Erlangen , July 2005 Nicoleta D. Scurtu

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Contents

Introduction 1

1 Description of Newtonian and non-Newtonian fluids 5

1.1 Kinematics, deformation and balance laws . . . . . . . . . . . . . . . . . . . . 5

1.2 Frame indifference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Simple incompressible fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Newton’s viscosity law . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.2 Quasi-Newtonian models . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 Differential type fluids (Rivlin-Ericksen) . . . . . . . . . . . . . . . . . 11

1.3.4 Rate type fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.5 Integral type fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Mathematical formulation 19

2.1 The Oldroyd system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 The elastic-viscous-split-stress method . . . . . . . . . . . . . . . . . . 19

2.1.2 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Dimensionless Oldroyd system . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 The high Weißenberg number problem . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Notes on the high Weißenberg number problem . . . . . . . . . . . . . . . . . 24

3 Existence results and finite element formulation 25

3.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 The stationary Oldroyd system . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 The discontinuous Galerkin method . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Finite element formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Time approximation using the fractional θ-scheme 39

4.1 Application of the fractional step θ-scheme to the Oldroyd system . . . . . . 40

4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Well-posedness of the subproblems . . . . . . . . . . . . . . . . . . . . 43

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vi Contents

4.2.2 Fixed-point iteration scheme for the stationary Oldroyd system . . . . 44

5 Stability analysis 47

5.1 Spectral analysis of the linearized Oldroyd system . . . . . . . . . . . . . . . 47

5.1.1 Spectral analysis of the linearized continuous Oldroyd system . . . . . 47

5.1.2 Spectral analysis of the θ-scheme for the linearized Oldroyd system . . 52

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Strong stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Choice of the splitting parameters k and ω . . . . . . . . . . . . . . . 66

Plots of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Contribution of the β-term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.1 Influence of the β-term on the stability of the constitutive equation . 68

Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Stability of the constitutive equation without the stress convective term 70

Stability of the constitutive equation . . . . . . . . . . . . . . . . . . . 71

5.2.2 Influence of the nonlinearity β on the stability of the Oldroyd system 72

Influence of the convective terms on the stability of the Oldroyd system 76

Stability of the time semi-discretized constitutive equation . . . . . . . 76

5.3 A priori stability estimation of the linear Oldroyd problem . . . . . . . . . . . 79

5.3.1 A priori stability estimation of the linear Oldroyd problem in weak form 79

5.3.2 A priori stability estimation of the fractional θ-scheme . . . . . . . . . 80

6 Implementation aspects 89

6.1 Solution of the subproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Solution of the Stokes problem . . . . . . . . . . . . . . . . . . . . . . 89

6.1.2 Solution of the Burgers problem . . . . . . . . . . . . . . . . . . . . . 91

6.1.3 Solution of the stress convective problem . . . . . . . . . . . . . . . . 92

6.1.4 Approximation of the boundary conditions . . . . . . . . . . . . . . . 92

6.2 Assembly of the element matrices . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Implementation of the jump-terms . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3.1 Computational costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4 List of own implementations and program modifications . . . . . . . . . . . . 95

7 Numerical examples 97

7.1 Experimental order of convergence . . . . . . . . . . . . . . . . . . . . . . . . 97

7.1.1 EOC tests for example 1 . . . . . . . . . . . . . . . . . . . . . . . . . 98

Stress equation without convective term for a = 1 . . . . . . . . . . . 99

Stress constitutive equation for a = 1 . . . . . . . . . . . . . . . . . . 107

Oldroyd system without the β-term . . . . . . . . . . . . . . . . . . . 113

Oldroyd-B system without the stress convective term . . . . . . . . . . 115

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Contents vii

Oldroyd-B system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Oldroyd-B system in a domain comprising the stagnation point . . . . 123

Stress transport problem for a = 0 . . . . . . . . . . . . . . . . . . . . 128

Oldroyd system without the convective terms for a = 0 . . . . . . . . 128

Oldroyd system for a = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.1.2 EOC tests for example 2 . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.1.3 EOC tests for example 3 . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2 Benchmark problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.2.1 Lid-driven cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.2.2 Four-to-one planar contraction . . . . . . . . . . . . . . . . . . . . . . 162

Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Geometry and computational meshes . . . . . . . . . . . . . . . . . . . 164

Numerical results for the Oldroyd-B model . . . . . . . . . . . . . . . 165

8 Concluding remarks 173

A Stability analysis 175

A.1 A priori stability estimates of the fractional θ-scheme . . . . . . . . . . . . . 175

A.2 Spectral analysis of the Oldroyd System . . . . . . . . . . . . . . . . . . . . . 175

A.2.1 Spectral analysis of the θ-scheme for the linearized Oldroyd system . . 175

Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

B Implementation aspects 183

C Numerical Examples 187

C.1 Experimental order of convergence . . . . . . . . . . . . . . . . . . . . . . . . 187

C.1.1 EOC test for example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Bibliography 189

Zusammenfassung 195

Curriculum 197

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Introduction

Many fluids appearing in nature do not satisfy the Newtonian law. This is also the case

of many fluids created for industrial purposes. Unfortunately, a general constitutive law

satisfied by all of them is not conceivable. Moreover, it has been observed experimentally

that the constitutive law can change from one fluid to another.

Generally, materials encountered in industry fall outside the classical extremes of the

Newtonian viscous fluid and Hookean elastic solid. When such materials are classified as

fluids, the adjective non-Newtonian is usually employed. Such fluids may or may not possess

a memory of past deformation. If they do, they are called non-Newtonian viscoelastic fluids .

Non-Newtonian fluids appear in chemical and process engineering, glaciology, biomedical

applications, food technology, geology and in modeling boundary layer-type behavior. The

chemical, automobile, aerospace, electronics and plastic processing industries use widely

polymeric materials.

As opposed to their Newtonian counterparts, non-Newtonian fluids present many effects

that can not be predicted by the Navier-Stokes equations. Among them, viscoelastic fluids

are characterized by specific features like the shear rate dependence on the shear viscosity,

the presence of normal stresses in viscometric flows, the high resistance to elongational

deformation and memory effects related to their elastic properties. An additional constitutive

equation is generally introduced to account for such effects either in differential or integral

form. Even though, significant progress has been obtained in the last few decades, properly

simulating the physics of viscoelastic flows remains a major challenge.

The main aim of this thesis is to introduce a numerical method for solving flow problems

for non-Newtonian viscoelastic fluids of Oldroyd type, and to present a stability analysis of

the corresponding system of equations. Assuming constant temperature, the mathematical

formulation comprises the conservation of mass and momentum together with the consti-

tutive law relating the stress field to the motion. The material constitutive equation in

combination with the equations of motion taking into account the elastic and memory ef-

fects, leads to a highly nonlinear system of partial differential equations. The system of

equations governing the motion of an Oldroyd fluid is usually of mixed parabolic-hyperbolic

(in the stationary case of elliptic-hyperbolic) type. The basic idea in our numerical approach

is to first decouple the system into the parabolic and the hyperbolic parts.

Page 14: Stability Analysis and Numerical Simulation of Non-Newtonian

2 Introduction

The work is divided into seven chapters. Simple fluids of Newtonian and non-Newtonian

type are described in chapter 1. At the beginning a short and general overview of the

continuum mechanics will be given. Thereafter, the implications of the frame indifference

principle on material constitutive equations are presented. Special attention is being given to

the non-Newtonian viscoelastic fluids of Oldroyd type. For this fluid model the viscometric

properties will be recalled.

Chapter 2 deals with the mathematical formulation of the Oldroyd type fluid flow. The

conservations of mass, momentum and the material constitutive law form the Oldroyd sys-

tem of equations. Using the elastic-viscous-stress-splitting method [10, 33, 26], the non-

dimensional Oldroyd system is deduced. The non-dimensional Oldroyd equation system

contains four free parameters: the Reynolds number, Re ≥ 0, and the Weißenberg num-

ber, We ≥ 0, the rate of viscoelastic viscosities, α ∈ [0, 1], and the index of the β-term,

a ∈ [−1, 1]. In the limit of α = 0, the system reduces to the Navier-Stokes equations,

corresponding to the purely viscous fluid, whereas the limit case α = 1 corresponds to the

purely elastic Maxwell fluid. After an interpretation of the non-dimensional parameters,

the Reynolds number and the Weißenberg number, the so called high Weißenberg number

problem is reviewed.

Some results concerning the existence of solutions for the Oldroyd problem with boundary

conditions are presented at the beginning of chapter 3. Next, the weak formulation of the

Oldroyd system is clarified. By concentrating first on the stationary case, the two main

difficulties arising from solving the system by a finite element method will be presented. The

hyperbolic character of the stress constitutive equation and the solvability of the system in

the case of We = 0 leads to the choice of the discontinuous Galerkin method for solving the

stress equation. For the velocity and pressure fields the stable Taylor-Hood element [27] has

been chosen, whereas for the stress field linear discontinuous finite elements are used. This

choice of finite elements for stress is justified by an inf-sup condition in correlation with the

velocity finite elements [28, 26].

For the time discretization the fractional step θ-scheme [18] is introduced in chapter 4.

This time approximation is applied to the Oldroyd system as a double operator splitting

method. Here, the splitting idea from [18], where the θ-scheme is applied on the Navier-

Stokes equations, and that from [63], where this scheme is applied to the Oldroyd system for

slow flows, are combined. Based on this time discretization, an algorithm for solving numeri-

cally the full Oldroyd system is presented. Through this algorithm the three major numerical

difficulties of the Oldroyd system are decoupled: the treatment of the nonlinearity in the

momentum equation, given by the velocity transport term, the solenoidal condition, and the

stress transport term in the constitutive equation. By the operator splitting algorithm, one

reduces the Oldroyd system to three considerably simpler subproblems: a linear selfadjoint

Stokes problem, a nonlinear Burgers problem and a transport problem for the stress tensor.

The last one was solved through the discontinuous Galerkin method [43, 36, 28].

Chapter 5 deals with the stability analysis of the instationary linearized Oldroyd system,

fully discretized by the finite element method and the fractional step θ-scheme. Then, a

Page 15: Stability Analysis and Numerical Simulation of Non-Newtonian

Introduction 3

spectral analysis of the linearized Oldroyd system is done, showing good stability properties

and second order accuracy of the time discretization scheme. The last part of this chapter

was dedicated to show the influence of the β-term on the stability of the constitutive stress

equation and of the Oldroyd system. Here, an upper limit of the Weißenberg number, Wecris found, beyond that the Oldroyd system was not longer stable. When neglecting the stress

transport term, this upper limit was sharp. So, beyond Wecr no stability of the continuous

equation system and thus of the numerical scheme was achieved. For the full Oldroyd-B

system, due to the influence of the nonlinear convective term, the stability limit was found

to be greater than Wecr.

The numerical implementation of the Oldroyd system was based on the program package

Albert. Chapter 6 summarizes our own implementations and program modifications.

Finally, in chapter 7 numerical results for some two-dimensional problems will be pre-

sented. To prove the correctness of the algorithm implementation and the theoretical stabil-

ity bounds determined in chapter 5, some experimental order of convergence (EOC) problems

will be tested. Then, two benchmark problems are investigated, the lid driven cavity and

the four-to-one contraction problem.

Page 16: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 17: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 1

Description of Newtonian and

non-Newtonian fluids

In this chapter a short overview of the continuum mechanics and of fluids types will be given.

Only homogeneous and incompressible fluids were considered here, without elaborate on

thermodynamical concepts. The implication of the frame indifference principle on material

constitutive equations is shown, and the objective time derivative is introduced. Special

attention is given to the non-Newtonian viscoelastic fluids of Oldroyd type. The presentation

is based on the monographs of Coleman, Markovitz and Noll [20], Truesdell [69], Truesdell

and Noll [70], Giesekus [30], Bohme [16], Renardy [59], and the dissertation of Videman [71].

1.1 Kinematics, deformation and balance laws

Continuum mechanics deals with the motion and deformation of bodies. A body B is a

smooth three-dimensional manifold, a continuous medium, that consist of material points

X , called particles . A configuration χ of a body B is a smooth one-to-one mapping of B

onto a region of the three-dimensional Euclidian space E

x = χ(X),

here the point x denotes the place in E occupied by the particle X in the configuration χ.

A body is assumed to admit a non-negative scalar measure m, called the mass distribution

of the body. One assumes the existence of a corresponding mass density ρ, satisfying

m(P) =

χ(P)

ρ dv,

for every (Lebesgue) measurable part P of B and where dv denotes the volume measure.

Page 18: Stability Analysis and Numerical Simulation of Non-Newtonian

6 Chapter 1. Description of Newtonian and non-Newtonian fluids

A motion of the body is a one-parameter family of configurations x = χ(X, t) with the

real parameter t denoting the time. More than the motion of the material points themselves,

one finds it useful to study the relative motion (and deformation) of the particles. To this

end, let x denote the position in the Euclidian space E of a particle X at time t and suppose

that the same particle has occupied the position ξ at some time s, say s ≤ t. Then

ξ = χ(X, s) = χ(χ−1(x, t), s) = χt(x, s).

We call χt the relative deformation function or the relative motion of the particle X . The

velocity u = u(x, t) and the acceleration a = a(x, t) of the material point X are vectors

defined through

u = u(x, t) =d

dtχt(x, t),

a = a(x, t) =d

dtu(x, t).

The velocity gradient ∇u(x, t) is a tensor, whose symmetric part D and the skew-symmetric

part W are called the stretching (or the rate of deformation) tensor and the spin tensor ,

they are defined by

D =1

2(∇u+∇uT ), W =

1

2(∇u−∇uT ),

where ∇uT denotes the transpose of ∇u.The material (substantial) time derivative d/dt is given by the classical formula from Euler

(1.1)d

dt=

∂t+ u · ∇,

whereas ∂/∂t denotes the partial time derivative.

The relative deformation gradient Ft(s) is the tensor

Ft(s) =∂ξ

∂x= ∇χt(x, s).

One assumes that Ft(s) is continuous and invertible, considered as a linear transformation

from regions of R3 into R

3, and that detFt(s) > 0. Moreover, one has

Ft(t) = I.

To measure the strain, one defines the right relative Cauchy-Green (strain) tensor Ct(s) by

Ct(s) = F Tt (s)Ft(s).

The Rivlin-Ericksen tensors An are defined through

An =

[

dnCt(s)

dsn

]

s=t

.

Page 19: Stability Analysis and Numerical Simulation of Non-Newtonian

1.1. Kinematics, deformation and balance laws 7

One can show the recurrence formulae (see e.g. [16, 70]).

(1.2)

A1 = 2D = ∇u+∇uT ,

An =

(

∂t+ u · ∇

)

An−1 +An−1∇u+∇uTAn−1.

The operator which applied on An−1 provides An, is called convective (Oldroyd) time-

derivative, see [53, 68].

If φ(t) is any scalar-, vector- or tensor-valued function of time, one defines the history of

φ through

φt(s) = φ(t − s), s ≥ 0.

A motion is called steady if the velocity field u, expressed in Eulerian coordinates, is

independent of t, i.e. u = u(x). In an isochoric motion the volume of the body is preserved,

hence the density ρ(X, t) is constant. If only isochoric motions are possible for a certain

material, then this material is called incompressible and it holds

detCt(s) = 1, ∀s ≥ 0.

In all isochoric motions it holds

tr∇u = trD = 0.

Assume that a body B and a motion of B are given. A system of forces for B is charac-

terized by the following conditions:

(i) At each time t a vector field f(x, t), defined for x in the region occupied by B at time

t, is given. The vector f(x, t) is called the density per unit mass of the external force, or

simply external body force, acting on B.

(ii) At any given time t, to each part P of the body B corresponds a vector field t(x,P),

defined for the points x on the boundary ∂P of P. This vector is referred to as the density

of the contact force, or simply as the stress acting on the part P of B.

(iii) The total resultant force exerted on the part P of B is given by∫

P

ρfdv +

∂P

t dΓ.

(iv) There is a vector-valued function t(x, n), defined for all points in B and for all unit

vectors n, such that the stress can be expressed as

t(x,P) = t(x, n),

where n is the exterior unit normal vector at the point x on ∂P. The vector t(x, n) is called

the stress vector at x acting across the oriented surface element with normal n.

As a consequence of the conservation laws for the linear and angular momentum, one

deduces that the dependence of t on n can be expressed by a symmetric tensor T = T (x),

called the stress tensor , in the form

t = Tn.

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8 Chapter 1. Description of Newtonian and non-Newtonian fluids

A further consequence of the balance of linear momentum is the Cauchy’s first law of motion,

or simply the equation of motion

(1.3) div T + ρf = ρa.

Moreover, the conservation of mass implies that the mass density ρ satisfies the equation of

continuity

(1.4)∂ρ

∂t+ div(ρu) = 0.

In a homogeneous incompressible material, the density is constant everywhere and the equa-

tion (1.4) reduces to

(1.5) div u = 0.

1.2 Frame indifference

One of the main axioms of mechanics is the requirement that the material response is

independent of the observer. This is the principle of frame indifference or material objectivity

(cf. [20]). It states that if a given process is compatible with a constitutive equation, then all

processes obtained from the given process by changes of frame must also be compatible with

the same constitutive equation. An observer O at a fixed reference point x, or synonymously

a frame (of reference), is represented by a basis ek or ej

ej · ek = δkj ,

and by a set of measuring devices for other physical quantities. The measuring devices

generate tensor components of tensor of second order

Σlk → Σ : = elΣlkek.

Denote an arbitrary but fixed observer O∗ as standard frame of reference and another with

respect to O∗ arbitrarily moving frame O as the chosen observer. A change of frame in

classical mechanics is described by a proper orthogonal time dependent rotation (Qjl)(t) ≡

Q(t) and by time dependent translation (cj)(t) ≡ c(t). Consequently, the transformation of

the position coordinates is given by

(1.6) x∗ = Q(t)(x− c(t)).

The components of the material velocity transform themselves in the following way

u∗ = Q(u+ urel).

Here, urel is defined by

urel = −∂c∂t

+QT∂Q

∂t(x− c),

Page 21: Stability Analysis and Numerical Simulation of Non-Newtonian

1.2. Frame indifference 9

and represents the relative velocity between the two frames O∗ and O. In general a scalar,

a vector and a second order tensor transform, respectively, according to

φ∗ = φ+ φrel,

v∗ = Q(v + vrel),

Σ∗ = Q(Σ + Σrel)QT .

By definition, some quantities are called objective, if their relative parts vanish for all ob-

servers. In the non-relativistic physics the time and mass density, for example, are objective,

but the velocity vector is not objective. The velocity gradient tensor ∇u is transformed ac-

cording to

∇u∗ =∂Q

∂tQT +Q∇uQT ,

so, it is not objective. On the other hand, the rate of deformation tensor D is objective,

obeying the transformation law

D∗ = QDQT .

Most of the physical laws are assumed to be form invariant , in other words the equations

are assumed to have exactly the same form when changed to a new reference frame. This

assumption, applies to the equation of continuity (1.4), but it is not true for the dynamical

equation (1.3). In fact, one normally assumes that the equation of motion is valid only in

some preferred reference frame called inertial frame and a time-dependent change of frame

transforms the equation into a form to which some inertial force must be added.

It is obvious that equations (1.3) and (1.4) cannot alone define the material response

of any fluid whatsoever. To complete the mathematical formulation, one needs to have

further information about the relation between the stress and the kinematic variables of

the material. The equation that defines the properties of the material is called constitutive

equation. Accordingly to the frame indifference principle, the constitutive equation has to

be invariant under the change of frame (1.6), and the stress tensor Σ must be objective.

In constitutive relations, there usually appear time derivatives of objective tensors (rate

of deformation tensor D, stress tensor T ). As a matter of fact, it is obvious that neither the

partial time derivative nor the material time derivative (1.1) is objective when applied to

objective tensors. Following the convected coordinates approach of Oldroyd [53], one may

define the following objective time derivative

(1.7)∂aT

∂t=

∂T

∂t+ (u · ∇)T + βa(T,∇u).

The βa-term is defined by

(1.8) βa(T,∇u) = TW −WT − a(DT + TD),

were a is a real-valued constant. For a ∈ −1, 0, 1 one obtains the lower convected,

corotational and upper convected derivative, respectively (see also [45, 50, 68]). Note that,

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10 Chapter 1. Description of Newtonian and non-Newtonian fluids

this derivative is symmetry-preserving. The operator applied on An−1 in the recurrence

formula (1.2), corresponds to the case a = −1.

The slip parameter a ∈ [−1, 1] was interpreted by Johnson and Segalman [37] as a mea-

sure of the non-affinity of polymer deformation, i.e. the fractional stretch of the polymeric

material with respect to the stretch of the flow field.

1.3 Simple incompressible fluids

To have a complete mathematical description of the flow it is necessary to introduce an

additional constitutive law , relating the stress tensor T to the motion. In many processing

operations the temperature variations are considerable, and the dependence of the stress

tensor on temperature can be a very important factor. However, in connection with purely

mechanical problems the temperature can be assumed as constant.

For simple fluids the constitutive law is of the form

(1.9) T = −p(ρ)I + TE(ρ,D),

where p is the hydrostatical pressure and TE is the extra stress tensor. It will be also assumed

that the fluids are incompressible, because in the flow of polymeric liquids compressibility

is seldom of importance. In the following only incompressible fluids will be accounted, for

which (1.9) becomes

T = −pI + TE(D).

1.3.1 Newton’s viscosity law

In a relatively simple but very realistic situation, the constitutive law is the following

(1.10) TE = 2µD.

Here µ is the proportionality coefficient called viscosity . This law shows that the viscosity,

i.e. the friction of particles at the molecular level, is uniquely responsible for the existence of

stresses. At each point (x, t), the stress tensor depends linearly on the spatial derivatives ∂iujat (x, t) (tangential stresses increase when neighboring particles possess different velocities).

Furthermore, the second law of thermodynamics implies µ ≥ 0. If Newton’s law (1.10) holds,

it will be said that the fluid is Newtonian. Since T is an explicit function of D, it is possible

to replace (1.10) in (1.3), which leads in the case of incompressible flow to the Navier-Stokes

equations

ρ

[

∂u

∂t+ (u · ∇)u

]

− µ∆u+∇p = f,

div u = 0,

which are to be solved under adequate boundary and initial conditions.

Page 23: Stability Analysis and Numerical Simulation of Non-Newtonian

1.3. Simple incompressible fluids 11

1.3.2 Quasi-Newtonian models

Sometimes, it is possible to model the fluid by replacing Newton’s law by another explicit ,

nonlinear law. In this case, it is said that the considered fluid is of generalized Newtonian

type. An example is given by the viscoplastic law

(1.11) TE = 2µ(|D|)D,

where the general viscosity is a nonlinear (power) function of the second invariant |D| =√D : D.

Power law models (1.11) have been found to be successful in describing the behavior of rub-

ber solutions, adhesives, biological fluids, colloids, suspensions and a variety of polymeric

liquids.

1.3.3 Differential type fluids (Rivlin-Ericksen)

When the extra-stress tensor is a tensor-valued function in the form

TE = F (A1, A2, ..., Ak) ,

the fluid is said to be of differential type with grade k. Here An are the Rivlin-Ericksen

tensors (1.2). An example of application field of such models are glacier ice in creeping flow,

described by the constitutive law

TE = µ|A1|rA1 + α1A2 + α2A21.

Here, r ≈ − 23 , µ > 0, α1 and α2 are material constants. The model with r = 0 describes

what is called the second grade fluid .

1.3.4 Rate type fluids

The fluids of rate type are described through a differential law of the form

∂jaTE

∂t= F

(

TE,∂1aT

E

∂t, ...,

∂j−1a T

∂t;D,

∂1aD

∂t, ...,

∂kaD

∂t

)

,

with j and k positive integers.

Maxwell constitutive law

A simple way to account for the elastic effects in a non-Newtonian fluid is to consider a

properly invariant version of the one-dimensional Maxwell model of linear viscoelasticity.

Regarding the objective time derivative (1.7) of the stress tensor, the Maxwell fluid model

is defined by the constitutive equation

TE + λ1∂aT

E

∂t= 2µ0D.

Page 24: Stability Analysis and Numerical Simulation of Non-Newtonian

12 Chapter 1. Description of Newtonian and non-Newtonian fluids

The material constant µ0 is the null viscosity coefficient defined as

µ0 = lim|D|→0

TE12(|D|)|D| ,

and λ1 denotes the relaxation time. When a constant strain is imposed on a fluid, it could

happen that the stress needed to maintain the constant strain will decrease continuously

in time (stress relaxation). So, upon application of a step strain the material becomes

instantaneously elastic, and the stress produced relaxes, eventually to zero, according to the

constant λ1 > 0. That means, if the motion is stopped, TE decays as exp(−t/λ1). The

larger λ1 is, the slower is the relaxation. A large relaxation time means that the elastic

response of the fluid is persistent; the fluid can be said to have a long memory.

The case a = 1 corresponds to the upper-convected Maxwell model (UCM), which can be

written also in the integral form

TE(x, t) =µ0

λ1

∫ t

−∞

e−(t−s)/λ1(

C−1t (x, s)− I

)

ds.

Oldroyd constitutive law

The Oldroyd model has a stress that comprises the linear superposition of a Maxwell and a

Newtonian contribution. This provides a good description of the behavior of some fluids that

have in part properties found for elastic solids and, also in part, properties similar to those of

viscous fluids (this is why they are called viscoelastic). For instance, the Oldroyd model can

be used to describe the behavior of a polymer in a Newtonian solvent (see Oldroyd [53]). The

constant viscosity elastic Borger fluid, consisting of a dilute solution of high molecular weight

polymer dissolved in a viscous Newtonian solvent, is generally modeled through Oldroyd’s

constitutive equation. This equation has the differential form

(1.12) TE + λ1∂aT

E

∂t= 2µ0

(

D + λ2∂aD

∂t

)

.

To guarantee that the viscometric properties (1.18) are realistic, the material constant λ2,

called the retardation time, is required to fulfill the restriction

(1.13) 0 ≤ λ2 ≤ λ1.

The retardation time characterizes the time according to which the strain relaxes after the

removal of stress. When a constant stress is imposed on a continuum (solid or fluid), induced

deformations can increase in time (creep). In some cases, such deformations does not vanish

even when the stress is suddenly switched to zero. If stresses are removed, the strain will

decay as exp(−t/λ2). The case λ1 = λ2 = 0 corresponds to a Newtonian fluid which is

purely viscous, while the case λ1 > 0, λ2 = 0, corresponds to the Maxwell fluid which is

purely elastic.

One can split the extra stress tensor in its Newtonian part 2µnD and its elastical part τ

(1.14) TE = 2µnD + τ.

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1.3. Simple incompressible fluids 13

The solvent viscosity µn is defined by

µn = µ0λ2

λ1,

and due to the restriction (1.13) relative to the retardation time, one obtains 0 ≤ µn ≤ µ0.

At this point, the rate of viscoelastic viscosities α is introduced by

(1.15) α = 1− µnµ0

= 1− λ2

λ1, 0 ≤ α ≤ 1

which takes values in the interval [0, 1]. The case α = 0 corresponds to the Newtonian fluid,

whereas the other extreme case, α = 1, corresponds to the Maxwell fluid. The zero viscosity

contains also the solvent viscosity µn and the polymer viscosity µe

µ0 = µn + µe, µe = µ0 (1− α) .

Under these considerations, from (1.12) and (1.14) one deduces that the elastic part of the

stress tensor τ satisfies

(1.16) τ + λ1∂aτ

∂t= 2µeD.

By linearizing relation (1.16) around the rest state, one obtains the following equation for τ

τ + λ1∂τ

∂t= 2µeD.

Observe that this equation can also be written as follows

∂t

(

et/λ1τ)

=2µeλ1

et/λ1D.

Assuming that τ is known in the flow domain Ω at time t = 0, namely

τ|t=0= τ0,

then, by integration, one has

τ(x, t) = e−t/λ1τ0(x) +2µeλ1

∫ t

0

e−(t−s)/λ1D(x, s)ds.

Thus, one can see that the extra stress tensor TE(x, t) is not determined exclusively by the

instantaneous value of the rate of deformation tensor D(x, t), as in the case of a Newtonian

fluid, but by all the values D(x, s), with 0 ≤ s ≤ t, i.e. along the trajectory described by the

particle located at x ∈ Ω at time t.

The particular cases of the Oldroyd fluid with a lower and upper convective derivative, are

referred to as Oldroyd-A (a = −1) and Oldroyd-B (a = 1) fluid, respectively (see [53]). For

Page 26: Stability Analysis and Numerical Simulation of Non-Newtonian

14 Chapter 1. Description of Newtonian and non-Newtonian fluids

these models, the elastic part of the extra stress tensor admits also an integral representation,

namely

τ(x, t) =µeλ1

∫ t

−∞

e−(t−s)/λ1 (I − Ct(x, s)) ds

for an Oldroyd-A fluid and

τ(x, t) =µeλ1

∫ t

−∞

e−(t−s)/λ1(

C−1t (x, s)− I

)

ds

for an Oldroyd-B fluid. The particular case a = 0 corresponds to the Jaumann model.

A simple generalization of the Oldroyd type model is obtained by considering a fluid

having several relaxation times. In this case, the extra stress tensor is given by

TE = 2µnD +

N∑

i=1

τ i,

where each of the modes τ i satisfies an equation similar to (1.16).

Behavior in steady viscometric flows

The first step in evaluating constitutive models is to consider their predictions in a number

of simple flows in which the velocity field is known explicitely and so it is easy to find the

stress predicted by a given constitutive model. In steady simple shear (viscometric) flow,

the flow is two-dimensional and the velocity is uni-directional

u = (v(y), 0, 0).

Consequently, the velocity gradient is the matrix

0 v′(y) 0

0 0 0

0 0 0

.

The quantity κ = v′(y) is known as the shear rate. As consequence of the frame indifference

principle, the corresponding stress tensor has the form

T =

T11(κ) T12(κ) 0

T12(κ) T22(κ) 0

0 0 T33(κ)

.

Because of the presence of an undetermined pressure in an incompressible fluid, the diagonal

components of the extra stress tensor have physical meaning only modulo an arbitrary

constant; hence it is natural to consider their differences. This leads to the viscometric

functions

(1.17)

T12(κ) = µ(κ)κ,

T11(κ)− T22(κ) = N1(κ),

T22(κ)− T33(κ) = N2(κ).

Page 27: Stability Analysis and Numerical Simulation of Non-Newtonian

1.3. Simple incompressible fluids 15

Here, µ(κ) is called the viscosity , T12 is the shear stress and N1 and N2 are called the first

and second normal stress differences .

For a Newtonian fluid (1.10), µ(κ) is constant and N1 and N2 are zero, but in the case of

Oldroyd type fluids (1.12), the viscometric functions (1.17) can be easily deduced as

(1.18)

µ(κ) = µ01 + λ1λ2κ

2(1− a2)

1 + λ21κ

2(1− a2),

N1(κ) = 2µ0(λ1 − λ2)κ

2

1 + λ21κ

2(1− a2),

N2(κ) = −µ0(λ1 − λ2)(1− a)κ2

1 + λ21κ

2(1− a2).

The assumptions −1 ≤ a ≤ 1 and 0 ≤ λ2 < λ1 guarantee that, for all κ ≥ 0, the viscosity

is positive and the normal stress differences are non-zero and opposite in sign with the first

normal stress difference being positive. The viscosity decreases with increasing shear rate,

thus predicting shear-thinning behavior, except for the special cases a = ±1 corresponding

to the Oldroyd A and B models. In real polymeric fluids, the viscosity typically decreases

with increasing shear rate (have shear-thinning behavior), the first normal stress difference

grows quadratically at low shear rates but then grows more slowly as the shear rate increases

further, and the second normal stress difference is negative, but much smaller in magnitude

than the first (a typical value of N2/N1 is −0.1). For Oldroyd-B fluids the viscosity is shear

independent and the second normal stress difference is equal to zero. However, since this

model presents all the main problems, as regarding the mathematical analysis of rate type

fluids, in the sequel the author will mainly concentrate on this fluid.

Behavior in steady elongational flows

Elongational flows are flows in which the fluid undergoes a stretching motion. In the case of

uniaxial extensions the velocity field is of the form

u = (κx,−κy/2,−κz/2),

where κ is a positive parameter. The velocity gradient and thus the stress tensor in such a

flow is diagonal, with T22 = T33, and one is interested in the stress difference T11 − T22 as a

function of κ. The quantity (T11 − T22)/κ is called the elongational viscosity, and the rate

of the elongational viscosity to the shear viscosity is called the Trouton ratio.

For Newtonian fluids, the stress components are

T11 = 2µκ, T22 = −µκ,

and hence the Trouton ratio is 3. In the case of Oldroyd fluids the Trouton ratio is

(1.19) Tr = 21− 2aλ2κ

1− 2aλ1κ+

1 + aλ2κ

1 + aλ1κ.

Page 28: Stability Analysis and Numerical Simulation of Non-Newtonian

16 Chapter 1. Description of Newtonian and non-Newtonian fluids

At low elongation rates, Tr is 3 as in a Newtonian fluid, but the elongational viscosity

increases rapidly with the elongation rate and becomes infinite at κ = (2aλ1)−1. Steady

elongational flows with a large elongation rate are impossible. Although a solution of the

flow equations with a negative T11 − T22 exists, this solution is unattainable, it cannot be

reached if the fluid is deformed starting from equilibrium. Polymeric fluids do indeed show

an increase in elongational viscosity with elongation rate. In polymer solutions, this increase

can well be of three or four orders of magnitude, whereas in polymer melts it seldom reaches

even one order of magnitude. Hence the prediction of a limited elongation rate where the

elongational viscosity becomes infinite is reasonable for some polymeric fluids, but not for

all of them.

The flow of blood

For the flow of blood in tubes Yeleswarapu et al. proposes in [72] a generalized Oldroyd-B

of the form

TE + λ1∂aT

E

∂t= 2µ(D)

(

D + λ2∂aD

∂t

)

,

where the generalized viscosity function µ(D) accommodates the shear thinning phenomena.

The following function has been found in [72] to fit the experimental viscometric data

µ(κ) = µ∞ + (µ0 − µ∞)1 + ln(1 + Λκ)

1 + Λκ, κ =

1

2trD2.

Here, µ0 and µ∞ are the asymptotic apparent viscosities as κ→ 0 and ∞, respectively and

the parameter Λ describes the shear dependence of the apparent viscosity.

Giesekus model

Some phenomena can be easier described by adding a quadratic term in the stress at the

left side of the UCM (or Oldroyd-B) constitutive equation

(

1 +ξλ1

µTE)

TE + λ1∂1T

E

∂t= 2µ0D,

where 0 ≤ ξ ≤ 1 is interpreted as an anisotropic mobility of the structure unity induced by

the flow process. For ξ = 12 , the model is called Leonov . For details regarding the behavior

of this model in viscometric flows the reader is referred to [59].

Phan-Thien-Tanner (PTT) model

This model is used for polymer materials including melts, fiber suspensions, and liquid crys-

tal polymers. It is obtained by adding a proportional term to TE trTE at the left side of

Page 29: Stability Analysis and Numerical Simulation of Non-Newtonian

1.3. Simple incompressible fluids 17

the UCM (or Oldroyd-B) model

linear case:

(

1 +λ1ξ

µtrTE

)

TE + λ1∂1T

E

∂t= 2µ0D,

exponential case: exp

(

ξλ1

µtrTE

)

TE + λ1∂1T

E

∂t= 2µ0D.

The viscometric analysis leads to the possibility to have two different shear rates at the same

shear stress (see e.g. [59]). This has been suggested as an explanation for formation of shear

bans and spurt.

White-Metzner model

The White-Metzner constitutive law is obtained by variation parameters µ0 and λ1 as func-

tion of the second invariant of the rate-of-deformation tensor D.

FENE model

The origin of this model is the consideration of the polymer macromolecules as dumbbells

suspended in a Newtonian solvent of given viscosity µn. The Finitely-Extensible-Non-linear-

Elastic-dumbbell model is described by

τ =µnλ1

(

1

1− tr(A)/L2A− 1

1− 3/L2I

)

,

1

1− tr(A)/L2A+ λ1

∂A

∂t=

1

1− 3/L2I.

where A is the configuration tensor and L is a measure of the extensibility of the dumbbells.

1.3.5 Integral type fluids

In an integral model the stress is given in the form of integrals of the deformation history.

The K-BKZ model is one of the most widely used incompressible integral model, and can

be used in the limit of small deformations. It can be formulated in terms of a stored energy

function W (I1, I2, t− s), where I1 and I2 are the principal invariants of the relative Cauchy

strain

I1 = trC−1t (s), I2 = trCt(s).

The model has the form

TE =

∫ t

∂W (I1, I2, t− s)∂I1

(Ct(s)− I) ds−∫ t

∂W (I1, I2, t− s)∂I2

(Ct(s)− I) ds.

The K-BKZ model is based on the idea that every prior configuration of the material can be

viewed as a temporary equilibrium configuration, and the stress is found by superposition of

the elastic stresses resulting from all deformations relative to these temporary equilibrium

Page 30: Stability Analysis and Numerical Simulation of Non-Newtonian

18 Chapter 1. Description of Newtonian and non-Newtonian fluids

states. Throughout experimental tests one finds that real fluids violate such an hypothesis.

Despite its shortcomings the K-BKZ model is widely used, and it is a resonable first step in

the investigation of integral models. Integral models like K-BKZ have finite memory effects

and predict shear-dependent viscosity as well as non-zero first and second normal stress

differences.

Page 31: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 2

Mathematical formulation

In this chapter the Oldroyd system is introduced, and by using the elastic-viscous-split-stress

method, the dimensionless system of equations is derived. For the non-dimensional param-

eters a short interpretation is given. In the last part of this chapter, the high Weißenberg

number problem is presented. In addition, different numerical methods applied to the flow

of an Oldroyd-B fluid through the four-to-one contraction, together with the reached upper

limit of the Weißenberg number, are reviewed.

2.1 The Oldroyd system

Let Ω be a domain, i.e. an open and connected subset in RN , N ∈ 2, 3, with the boundary

Γ = ∂Ω. The mathematical model of viscoelastic flow for the Oldroyd fluid consists of the

set of three coupled equations which are to be solved simultaneously, namely the continuity

equation (1.5), the momentum equation (1.3) and in addition the constitutive equation

(1.12). These three equations constitute the following system of equations, which will be

called in this work the Oldroyd system

(2.1)

λ1∂aT

E

∂t+ TE = 2µ0

[

λ2∂aDu

∂t+Du

]

,

ρ

[

∂u

∂t+ (u · ∇)u

]

= div TE −∇p+ f,

div u = 0.

2.1.1 The elastic-viscous-split-stress method

The elastic-viscous-split-stress (EVSS) method [10, 26] was used to split the extra stress

tensor TE into its Newtonian part 2µnDu and its elastical part τ , as written in (1.14).

There are two principal features associated with this method, stress-splitting and recovery

Page 32: Stability Analysis and Numerical Simulation of Non-Newtonian

20 Chapter 2. Mathematical formulation

of velocity gradients. Applying the EVSS method the Oldroyd system becomes

(2.2)

λ1∂aτ

∂t+ τ − 2αµeDu = 0,

ρ

[

∂u

∂t+ (u · ∇)u

]

− 2(1− α)µ0 divDu− div τ +∇p = f,

div u = 0.

This system is to be solved in the domain Ω during the time interval [0, T ] under adequate

boundary and initial conditions. The mass density ρ is constant and the tensorial stress

field τ , the vectorial velocity field u and the scalar pressure field p are unknown functions of

x ∈ Ω and t ∈ [0, T ].

In the instationary case the Oldroyd system is a mixed hyperbolic-parabolic system

whereas in the stationary case it is of hyperbolic-elliptic type.

2.1.2 Boundary and initial conditions

The parabolic equation of motion for the Oldroyd fluids requires a condition of e.g. Dirichlet

type for the velocities

u = uΓ on Γ ,

Γ

uΓ · n ds = 0.

In [24] Fernandez-Cara et al. note that in the context of viscoelastic fluids, the role of

boundary conditions of other kinds is not well understood. In general, they lead to open

theoretical problems.

The hyperbolic constitutive equation requires a condition for the stress components

(2.3) τ = τΓ on Γ−

on the upstream boundary section

Γ− = x ∈ Γ; u(x) · n(x) < 0 .

Here n is the outward unit normal vector to Ω at the boundary Γ.

Since the Oldroyd system is transient one needs initial conditions at time t = 0

u(0) = u0, τ(0) = τ0 in Ω.

2.2 Dimensionless Oldroyd system

System (2.2) can be non-dimensionalized in the usual way, by putting

x =x

L, u =

u

U, t =

U

Lt ,

τ =L

µ0Uτ , p =

L

µ0Up , f =

L2

µ0Uf.

Page 33: Stability Analysis and Numerical Simulation of Non-Newtonian

2.2. Dimensionless Oldroyd system 21

Here, L and U are the characteristic values of the length and velocity respectively and

x, u, t, τ , p and f are non-dimensional quantities. Denoting them again by x, t, etc., one

obtains the dimensionless Oldroyd system

(2.4)

We

[

∂τ

∂t+ (u · ∇)τ + βa(τ,∇u)

]

+ τ − 2αDu = 0,

Re

[

∂u

∂t+ (u · ∇)u

]

− 2(1− α)divDu− div τ +∇p = f,

div u = 0.

2.2.1 Non-dimensional parameters

In the Oldroyd system (2.4) there are two non-dimensional parameters, the Reynolds num-

ber (Re) and the Weißenberg number (We), which appear as a consequence of the above

procedure of non-dimensionalization.

The Reynolds number is defined as

Re =ρLU

µ0,

and represents the ratio of the order of magnitude of the inertial forces by the viscous forces.

The Weißenberg number is defined as the ratio of average relaxation time of the polymer

λ1 to an external given time which is a typical length to a typical velocity in the flow

We = λ1U

L.

Small values of the Weißenberg number incorporate both the concept of small characteristic

times (corresponding to slight elastic liquids) and small speed of flow. The behavior of

non-Newtonian fluids depends crucially on how the time of the polymer relates to the flow

relevant time scale. If the flow is on a time scale that is long relative to the memory of the

fluid, We → 0, then memory is unimportant, and the fluid behaves like a Newtonian fluid.

On the other hand, memory effects will be crucial if the relaxation time of the fluid exceeds

the time scale of the flow. In the extreme case where the time scale of the flow is very short,

We→∞, the fluid will behave like an elastic solid.

For shear flows, the Weißenberg number is defined as the average relaxation time of the

polymer λ1 by the inverse shear rate of the base flow. The Weißenberg number represents

the effect of normal stress and it is a measure of the degree of nonlinearity or the degree

to which normal stress differences are exhibited in a flow. In a shear flow, We is used to

characterize the effective strength of the flow. Considering bead-rod chain in elongational

flow, for We < 1 the chain can resist the flow. The conformation of the chain will be a

(deformed) coil. For We > 1 the chain is deformed with a rate larger than the rate of

relaxation, the chain will stretch.

Page 34: Stability Analysis and Numerical Simulation of Non-Newtonian

22 Chapter 2. Mathematical formulation

2.3 The high Weißenberg number problem

Until now, there are some unsolved problems for the numerical simulation of the flow of vis-

coelastic fluids. One of them is that the equations cannot be integrated when the relaxation

time is large. The simulation problems associated with highly elastic viscoelastic fluids is

called the high Weißenberg number problem. It refers to the failure of numerical simulations

when We is large.

One of the reasons for the failure of the numerical simulations first encountered in the early

1980s (e.g. Keunings [39]), is that a straightforward Galerkin discretization of the constitu-

tive law has poor stability properties if the advection term (u · ∇)τ becomes dominant. The

other problem is that the equations for viscoelastic fluid are of combined hyperbolic-parabolic

type, and the behavior of such equations under discretization is not very well understood.

High values of the Weißenberg number for the UCM fluid are connected with a change of

type of the equation system, as was shown in Joseph et al. [38]. They associated the high

Weißenberg number problem in steady flows with a change of type like the transition from

subsonic flow to supersonic flow. Likewise, Renardy in [59] investigated the characteristics

of the Maxwell system of equations considered as quasi-linear system. Linearizing the UCM

system of equations at a state of no motion and constant stress, Rutkevich [60] finds that a

change of type leading to imaginary wave speeds and an instability of Hadamard type occur

if the principal values of τ satisfy certain inequalities.

A limit on the Weißenberg number is found in all published works by applying numerical

techniques, to differential and integral models. Minor changes in the constitutive equation

and/or the algorithm employed can lead to higher limit values of We. For example, using

the Oldroyd-B model in contrast to the upper-convected Maxwell model one can extend the

range of We with respect to the convergence of the numerical solution. As We approaches the

critical value Wecr it is often observed that spurious oscillations appear in the field variables;

the stress components are then more severely affected than the velocity components, yielding

large and erroneous stress gradients. The spurious oscillations have no physical background,

and their wavelength depends upon the mesh used for the discretization. Mesh refinement

and corner strategies affect the critical conditions for breakdown, but it is difficult to discern

an overall consistent trend in published works.

The developments in the field of computational rheology applied to the prediction of

flow of polymeric liquids in complex geometries, are reviewed by Keunings [40]. Here, the

two current avenues towards complex flow simulation are visited, namely the macroscopic

and micro-macro approaches. Progress in macroscopic simulations has been studied along

the path of obtaining numerical solutions of the discrete, non-linear algebraic equations

at significant values of We; assessing their numerical accuracy and assessing their physical

relevance. For a number of benchmark problems for steady flows, in [19], agreements between

a number of different formulations has been demonstrated for increasing values of We.

In the field of viscoelastic flows, the flow through a planar contraction (see Fig. 7.86) is

accepted as a torture test case since 1988. The progress made, in the period 1987-1997, in

Page 35: Stability Analysis and Numerical Simulation of Non-Newtonian

2.3. The high Weißenberg number problem 23

the application of mixed finite element methods to solve viscoelastic flow problems using

differential constitutive equations is reviewed by Baaijens in [10]. Crochet et al. [21] have

used a mixed finite element method and they have obtained solutions to the flow in the

four-to-one contraction up to We = 1.75 before arriving at numerical simulation breakdown.

The upper limit above which the numerical algorithm fails is relatively low and often in

a region where the solution before breakdown are no more than perturbations about the

Newtonian case.

For the flow of an Oldroyd-B fluid through an axisymmetric four-to-one contraction with

4 × 4 streamline-upwind formulation, Marchal and Crochet [46] achieved solutions upto

a Weißenberg number beyond 60. In this case, the Weißenberg number is taken as the

relaxation time times the fully developed wall shear rate at the downstream channel. How-

ever, convergence with mesh refinement has not been demonstrated and in view of Crochet

and Legat [22] the accuracy of these results may be questioned. Similar remarks hold for

the results of low-order constant stress interpolation in conjunction with the discontinuous

Galerkin method reported by Baaijens [8].

As shown by Keunings [39], the maximum attainable Weißenberg number decreases with

increasing the mesh resolution. Baaijens [9] has demonstrated that using the DEVSS/DG

method, stable and accurate results can be obtained upto Weißenberg number beyond 24 as

well, while the limiting Weißenberg number increases with continued mesh refinement.

Saramito [62, 63, 64] has used the θ-scheme, the zero divergence Raviart-Thomas element

for approximating the velocities, and the Lesaint-Raviart element for the stresses for creeping

(slow) flows. He applied the algorithm for the flow in a plane or axisymmetric abrupt

contraction for Oldroyd-B fluids and Phan-Thien-Tanner models. The numerical results

seem to show that no upper limit of the Weissenberg number is encountered also for the

Oldroyd-B model.

Renardy [59] gives a formal asymptotic description for an UCM fluid flow, which is pre-

sumed to exist, in the high Weißenberg limit. Here, an analogy to the Euler equations

for Newtonian fluid flows at high Reynolds number is obtained. Unfortunately, the Eu-

ler equations often give little information about the actual flow behavior, because solution

are highly non-unique. Another difficulty of high Reynolds number flow is the formation

of singular layers along boundaries and separating streamlines, where the validity of Euler

equations breaks down. Finally, there is the question of instabilities and complex dynam-

ics. It turns out that all those difficulties exist also for the high Weißenberg number limit

of non-Newtonian fluid flow. Indeed, the most severe limitations on successful numerical

simulations are linked to the difficulty of resolving high-stress gradients arising in boundary

layers, along separating streamlines and near corner singularities.

Amongst the more recent computational work, we emphasize that by Aboubacar and

Webster [2] and Aboubacar et al. [1] who have done a comprehensive study of Oldroyd-

B and PTT fluids flowing through sharp and rounded-corner planar contractions, having

highlighted the influence of the elongational properties on the vortex patterns. They employ

a hybrid finite volume/finite element time dependent scheme and present results for the

Page 36: Stability Analysis and Numerical Simulation of Non-Newtonian

24 Chapter 2. Mathematical formulation

Oldroyd-B fluid until We = 2. Meng et al. [48] solved the Oldroyd-B system for creeping

flow (Re = 0) using a spectral element method, and reached the value We1.2. Application

of the spectral element method to viscoelastic fluids are found also in [25].

Alves et al. [6] used an implicit finite volume method based on a time marching pressure-

correction algorithm formulated with a collocated variable arrangement. Due to the fact

that the upwind differencing scheme for the convective terms in the constitutive equations

(in which a cell face stress is given by the corresponding cell center value in the upstream

direction) leads to too much numerical diffusion, they proposed in [5] a new convection

scheme, which damps strongly the formation of any numerical oscillations in regions of high

gradients. The computations have been carried out on very fine meshes, with over one

million degrees of freedom. No upper limit on We was found for the exponential form of the

PTT constitutive model, while an approximate limit of We ≈ 3 was found for the Oldroyd-B

model.

An a posteriori estimate is presented in [52], which makes possible to obtain appropriate

meshes to the finite element approximation of the stationary creeping flow of an Oldroyd

fluid into the four-to-one contraction. The results given here until We = 0.8, indicate that

to obtain a better convergence it is necessary to refine the mesh not only at the re-entrant

corner but also along the downstream part of the domain

For other viscoelastic models, e.g. Giesekus, PTT, the numerical difficulties appeared to

be much less pronounced than in the case of Oldroyd-B fluid (see [64, 33, 6]). Shear-thinning

behavior tends to reduce stress levels in region of high velocity gradients.

2.4 Notes on the high Weißenberg number problem

As it was shown in the previous section, the constitutive equation has a tremendous impact

on the results of numerical simulations and on the stability of the method. Its strong

hyperbolic character requires specific algorithms to solve it. A model including the property

of finite extensibility of polymer molecules like PTT (see e.g. [6, 64]), leads to more stable

algorithms. Generally speaking, computationally simpler models like UCM or Oldroyd-B

models are found more difficult to stabilize. Techniques found efficient for such models,

prove to be also valid for more refined models. Numerical stability being a key aspect in

viscoelastic flow simulations, much effort has been devoted to simple models although they

are known not to reproduce experimentally-observed phenomena in a satisfactory way.

Although the stability of the numerical algorithm has an crucial impact on viscoelastic fluid

flow computations, the decisive factor is the stability of the equation system which describes

the flow. If the continuous equation set becomes unstable then no accurate numerical scheme

should overshoot this fact. As this work is trying to elucidate in the following chapters, the

flow kind is an important factor in obtaining existence and stability of solutions for the

coupled equation system (2.1).

Page 37: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 3

Existence results and finite

element formulation

At the beginning of this chapter, some results concerning the existence of solutions of the

Oldroyd problem with boundary conditions are presented. Next, the weak formulation of

the Oldroyd system is introduced. For the velocity and pressure fields the stable Taylor-

Hood element is chosen. The stationary case will be considered first by presenting the two

main difficulties arising in solving the system by a finite element method: the hyperbolic

character of the stress constitutive equation and the solvability of the system in the case

We = 0. To overcome that, linear discontinuous finite elements for the stress field are

chosen, which satisfy an inf-sup condition in conjunction with the velocity finite element

space. Consequently, the discontinuous Galerkin method will be used for solving the stress

equation.

3.1 Existence results

In [59] Renardy has summarized the known results about the existence of solutions for

the Oldroyd system with suitable boundary conditions. There are basically four types of

existence results:

1. Results on existence locally in time for initial value problems. The solution is a small

perturbation of the initial data.

2. Results on global time existence and asymptotic decay if the initial conditions are

small perturbations of the rest state.

3. Results on existence of steady flows which are small perturbation of the rest state.

Page 38: Stability Analysis and Numerical Simulation of Non-Newtonian

26 Chapter 3. Existence results and finite element formulation

4. Results on existence of steady flows which are small perturbation of the Newtonian

flow.

These results are obtained upon assumptions made for the boundary Γ, the values τΓ, uΓ,

τ0 and u0, the right-hand side f , and data of the parameters Re, We, α and a. Using a

fixed-point method, Renardy [58] has obtained existence of stationary solutions for any value

of α, the other parameters being small.

For particular flows Guillope and Saut [34, 35] have proved a global existence result for the

solution of the Oldroyd problem, in the unsteady case, for small values of α. Videman [71]

has studied existence, regularity and uniqueness of weak, strong and classical solutions in

bounded and unbounded domains for steady and unsteady flows of second- and third-grade

fluids and Oldroyd-B fluids.

Assuming the domain Ω to be a convex polygon, Picasso et Rappaz [54] have proven for

the stationary Oldroyd-B model without the convective terms, the existence of a solution

for small relaxation times. They used continuous piecewise linear finite elements together

with a Galerkin least square method and derived a priori and a posteriori error estimates.

Let us consider the Oldroyd system in the following form

Problem 3.1. Solve in Ω× (0, T ) the system

We

[

∂τ

∂t+ (u · ∇)τ + βa(τ,∇u)

]

+ τ = 2αDu,

Re

[

∂u

∂t+ (u · ∇)u

]

− (1− α)∆u+∇p = div τ + f,

div u = 0,

with the boundary and initial conditions

u = 0 on Γ× (0, T ),

u|t=0 = u0 , τ |t=0 = τ0 in Ω.

In order to make some theoretical numerical analysis of the Oldroyd system, for prob-

lem 3.1 the assumption that uΓ = 0 is considered, which implies that Γ− = ∅ and therefore

no boundary condition (2.3) for τ is needed.

Fernandez-Cara et al. [24] have shown locally in time existence and uniqueness of a solu-

tion to the Oldroyd problem for arbitrary regular data and for small data defined in a large

time interval. This is motivated by the fact that, in general, there is no appropriate energy

estimate for non-Newtonian viscoelastic fluids. It is reasonable to belive that something

must be hidden in the memory of the flow. However, Lions and Masmoudi [44] have shown

that it is possible to find globally in time solutions for arbitrary data, in the case a = 0. In

their argument, they use that for a = 0 energy estimates are available. Accordingly, it is not

reasonable to expect, at least for the moment, results of the same kind for other Oldroyd

models.

Page 39: Stability Analysis and Numerical Simulation of Non-Newtonian

3.1. Existence results 27

In the following, the main results from [24] are given. Assuming that Ω ⊂ RN (N ≥ 2) is a

bounded open set, with smooth boundary Γ = ∂Ω, then for 1 < r, s < +∞ let us introduce

the spaces:

Hr = v ∈ Lr(Ω)N ; ∇ · v = 0 , v · n = 0 on Γ,Vr = Hr ∩W 1,r

0 (Ω)N .

Here, n is a unit vector, normal to Γ and oriented towards the exterior of Ω. Endowed with

the norm of Lr(Ω)N (resp. W 1,r(Ω)N ), Hr (resp. V r) is a reflexive Banach space.

The Helmholtz projector

Pr : Lr(Ω)N → Hr,

is a bounded linear operator characterized by the equality Prv = v0, where v0 is given by

the so called Helmholtz decomposition

v = v0 +∇q, with v0 ∈ Hr, and q ∈ W 1,r(Ω)N .

The Stokes operator

Ar : D(Ar) → Hr,

is defined on D(Ar) = W 2,r(Ω)N ∩ Vr by

Arv = Pr(−∆ v) ∀v ∈ D(Ar).

D(Ar) is a Banach space when equipped with the norm

‖v‖D(Ar) = ‖v‖Hr+ ‖Ar(v)‖Hr

.

Of course, in D(Ar) this norm is equivalent to ‖Ar(v)‖Hrand also to the usual norm in

the Sobolev space W 2,r(Ω)N (because of the smoothness of Γ). The operator −Ar is the

generator of a bounded analytic semigroup of class C0, e−tAr ; t ≥ 0 in Hr. Therefore, one

can introduce the fractional powers Aγr (0 < γ < 1), whose domain satisfies D(Aγr ) ⊃ D(Ar).

The space

Dsr =

v ∈ Hr ;

∫ ∞

0

∥Are−tArv

s

Hrdt <∞

,

associated to the analytic semigroup

e−tAr ; t ≥ 0

, is a Banach space for the norm

‖v‖Dsr

= ‖v‖Hr+

(∫ ∞

0

∥Are−tArv

s

Hrdt

)1/s

.

This space coincides with a real interpolation space between D(Ar) and Hr; one has

D(Ar) ⊂ Dsr ⊂ Hr,

where the embeddings are continuous (and dense) and

‖v‖Dsr≤ C‖v‖1/sHr

‖v‖1−1/sD(Ar) ∀v ∈ D(Ar).

Page 40: Stability Analysis and Numerical Simulation of Non-Newtonian

28 Chapter 3. Existence results and finite element formulation

Dsr is the natural space to choose initial data u0 if looking for a solution in Ls(0, T ;D(Ar)).

Indeed, it can be defined as the space of initial data v ∈ Hr such that the solution to the

Stokes problem

Re∂u

∂t+ (1− α)Aru = 0 a.e. in (0,∞),

u|t=0 = v,

belongs to Ls(0,∞;D(Ar)).

In the proofs of the main theorems presented in [24] some lemmas are used:

Lemma 3.2. Let Ω ⊂ RN with N ≥ 2 be a bounded open set with ∂Ω ∈ C2,µ (0 < µ < 1)

and assume 1 < r, s <∞, T > 0. If

u0 ∈ Dsr , f ∈ Ls(0, T ;Lr)

then there exists a unique function u satisfying

u ∈ Ls(0, T ;D(Ar)) ,∂u

∂t∈ Ls(0, T ;Hr)

and

Re∂u

∂t+ (1− α)Aru = f a.e. in (0, T ),

u|t=0 = u0.

Furthermore,

‖u‖sLs(D(Ar)) +

∂u

∂t

s

Ls(Hr)

≤(

c11− α

)s

(‖u0‖sDsr

+ ‖f‖sLs(Hr)),

where c1 = c1(Re, r, s,Ω).

Lemma 3.3. Let Ω ⊂ RN with N ≥ 2 be a bounded open set with ∂Ω ∈ C1 and assume

N < r < +∞ , 1 < s < +∞ , T > 0. Given

u ∈ Ls(0, T ;D(Ar)) , τ0 ∈W 1,r,

then there exists a unique function1 τ such that

τ ∈ C([0, T ];W 1,r) ,∂τ

∂t∈ Ls(0, T ;Lr),

and

We

[

∂τ

∂t+ (u · ∇)τ + βa(τ,∇u)

]

+ τ = 2αDu a.e. in Ω,

τ |t=0 = τ0,

1since u ∈ Ls(0, T ;D(Ar)), it means that u|Γ = 0 and the inflow boundary is empty, therefore no

boundary condition for τ is needed (see condition (2.3))

Page 41: Stability Analysis and Numerical Simulation of Non-Newtonian

3.2. The stationary Oldroyd system 29

Furthermore,

‖τ‖∞W 1,r ≤(

‖τ‖W 1,r +4α

c2 We

)

exp(c2‖u‖L1(D(Ar))) ≡ Λ,

and∥

∂τ

∂t

s

Lr

≤ c2 Λ

(

‖u‖Ls(Vr) +T 1/s

c2 We,

)

where c1 = c1(a, r).

Theorem 3.4. Let Ω ⊂ R3 be a bounded, connected and open set with Γ ∈ C2,µ (0 < µ < 1)

and assume 3 < r <∞, 1 < s <∞ and T > 0. If

u0 ∈ Dsr , τ0 ∈W 1,r , f ∈ Ls(0, T ;Lr),

then there exist T∗ ∈ (0, T ] and exactly one strong local solution u, p, τ to problem 3.1 in

[0, T∗] (p is unique up to a function depending only on t), with

u ∈ Ls(0, T∗;D(Ar)) ,∂u

∂t∈ Ls(0, T∗;Hr),

τ ∈ C([0, T∗];W1,r) ,

∂τ

∂t∈ Ls(0, T∗;Lr).

Theorem 3.5. Let Ω ⊂ R3 be a bounded, connected and open set with Γ ∈ C2,µ (0 < µ < 1)

and assume 3 < r < ∞, 1 < s < ∞. Then for each T > 0, there exists α0(T ) ∈ (0, 1) such

that, when 0 < α < α0(T ) and the data

u0 ∈ Dsr , τ0 ∈W 1,r , f ∈ Ls(0, T ;Lr),

are sufficiently small in their respective spaces, problem 3.1 possesses exactly one strong

solution u, p, τ in [0, T ] (p is unique up to a function depending only on t), with

u ∈ Ls(0, T ;D(Ar)) ,∂u

∂t∈ Ls(0, T ;Hr),

τ ∈ C([0, T ];W 1,r) ,∂τ

∂t∈ Ls(0, T ;Lr).

In [24] existence, uniqueness, regularity, well-posedness results are given for the evolution

problem 3.1 also for sufficiently small f , that is, in the presence of small perturbations of

the equilibrium state. They comment, that existence of a solution for large data is an open

question.

3.2 The stationary Oldroyd system

For stationary creeping flows, where the inertial term is neglected, the Oldroyd system

problem 3.1 reduces to

Page 42: Stability Analysis and Numerical Simulation of Non-Newtonian

30 Chapter 3. Existence results and finite element formulation

Problem 3.6. Solve in Ω the following system

We [(u · ∇)τ + βa(τ,∇u)] + τ − 2αDu = 0,

−2(1− α)divDu− div τ +∇p = f,

div u = 0,

with the boundary condition u = 0 on Γ.

The solution of the Oldroyd system in the stationary case, problem 3.6, by a finite element

method presents two main difficulties. One is the hyperbolic character of the constitutive

equation considered as a system in τ for u and p fixed. The characteristic lines are the

streamlines, and the components of the stress tensor τ may be considered as quantities

conveyed on these characteristics. The hyperbolic character implies that some upwinding is

needed. The choice of an upwinding technique depends on the choice of finite element space

used to approximate τ as explained below.

The second difficulty encountered in solving problem 3.6 is related to the Stokes problem.

If we want to solve problem 3.6 for We > 0, it is wise to be able to solve it for We = 0

(although We → 0 is a singular perturbation problem). In this case the Oldroyd system

(better called Stokes-Oldroyd system) can be written as

Problem 3.7. Solve in Ω the system

τ − 2αDu = 0,

−2(1− α)divDu− div τ +∇p = f,

div u = 0,

with the boundary condition u = 0 on Γ.

When considering the spaces 2

(3.1)

Θ = L2(Ω)N2

s

V = H1(Ω)N , Vg = v ∈ V ; v = g on Γ,Q = L2(Ω) , Q0 = q ∈ Q ;

Ωq = 0,

and the inner products in L2(Ω)

(3.2)

(τ, σ) =

Ω

(τ : σ)(x) dx, ∀ τ, σ ∈ Θ,

(u, v) =

Ω

(u · v)(x) dx, ∀u, v ∈ V,

(p, q) =

Ω

(pq)(x) dx, ∀ p, q ∈ Q,

2here, the subscript s means symmetrical tensor space

Page 43: Stability Analysis and Numerical Simulation of Non-Newtonian

3.2. The stationary Oldroyd system 31

with the corresponding norms | · |, the Stokes-Oldroyd problem can be formulated in the

following weak form:

Problem 3.8. Find (τ, u, p) ∈ Θ× V0 ×Q0, such that:

(τ, σ) − 2α(Du, σ) = 0,

−(div τ, v) + 2(1− α)(Du,Dv) − (p, div v) = (f, v),

(div u, q) = 0,

for any (σ, v, q) ∈ Θ× V0 ×Q.

Introducing the kernel of the div-operator

V = v ∈ V ; (q, div v) = 0 ∀q ∈ Q, V0 = V ∩ V0,

problem 3.8 is written in the equivalent form:

Problem 3.9. Find (τ, u) ∈ Θ× V0, such that

(τ, σ) − 2α(Du, σ) = 0,

(τ,Dv) + (1− α)(2Du,Dv) = (f, v),

for any (σ, v) ∈ Θ× V0.

Now, let us define the bilinear form A on Θ× V by

A((τ, u), (σ, v)) = (τ, σ) − 2α(Du, σ) + 2α(τ,Dv) + 4α(1− α)(Du,Dv).

Then

(3.3) A((τ, u), (τ, u)) = ‖τ‖2 + 4α(1− α)‖Du‖2

‖Du‖ being a norm on V equivalent to ‖u‖H1 . A is coercive on Θ × V if α ∈ (0, 1) and

problem 3.9 can be approximated by the finite element method without an inf-sup condition

relating τ and u (see [15]). If α = 1 problem 3.9 appears as a mixed problem and an inf-sup

condition is needed.

Consider now a triangulation Th on Ω and the finite element spaces

Θh ∈ Θ , Vh ∈ V , Qh ∈ Q,

and define

Vh = v ∈ Vh ; (div v, q) = 0 ∀q ∈ Qh, Vgh = Vh ∩ Vg .Suppose that, as for the classical Stokes problem with the two unknowns u and p, an inf-sup

condition is satisfied

(3.4) infq∈Qh

supv∈Vh

(q, div v)

‖q‖‖Dv‖ ≥ γ1 > 0.

Using the spaces introduced before, the finite element approximation of problem 3.9 can be

formulated as follows:

Page 44: Stability Analysis and Numerical Simulation of Non-Newtonian

32 Chapter 3. Existence results and finite element formulation

Problem 3.10. Find (τ, u) ∈ Θh × V0h, such that

(τ, σ)− 2α(Du, σ) = 0,

−(τ,Dv) + (1− α)(2Du,Dv) = (f, v),

for any (σ, v) ∈ Θh × V0h.

From (3.3) it follows that if α < 1 problem 3.10 has a unique solution, and that error

estimates are given by Lax-Milgram’s techniques. For α = 1 one supposes that the finite

element spaces Θh and Vh are related by the following inf-sup condition

(3.5) infv∈Vh

supσ∈Θh

(Dv, σ)

‖Dv‖‖σ‖ ≥ γ2 > 0

In [29] it is shown that condition (3.5) may be satisfied by imposing either D(Vh) ⊂ Θh

or the condition that the number of degrees of freedom for σh in each K ∈ Th is not less

than the total number of degrees of freedom for vh in each K ∈ Th. The first case leads to

discontinuous finite elements for the stresses and upwinding by discontinuous Galerkin finite

elements method introduced by Lesaint and Raviart [43]. In the second case one can use

continuous finite elements for the stresses and a streamline upwinding technique of the stress

equation. In the following the first supposition is taken into account. Before formulating

the finite elements method for the Oldroyd system (2.4), let us give a short description of

the discontinuous Galerkin method.

3.3 The discontinuous Galerkin method

The discontinuous Galerkin method originates from Lesaint and Raviart [43]. They have

applied this method to the neutron transport equation. Fortin and Fortin [28] introduced

the discontinuous Galerkin method for the analysis of viscoelastic flows. An advantage of a

discontinuous interpolation is that it can easily satisfy the inf-sup compatibility condition

in contrast to the continuous interpolation as set by Fortin [29]. Another advantage of the

discontinuous interpolation of the extra stresses is that in combination with the GMRES

method efficient preconditioning can be achieved at the element level. A description of the

discontinuous Galerkin method applied to hyperbolic problems is given by Johnson in [36]

and by Brezzi et al. in [17]. This method admits high order accuracy and good stability

properties also if the exact solution is not smooth.

Let us now describe the discontinuous Galerkin method applied to the following scalar

stationary boundary value problem

(3.6)

β · ∇u+ u = f in Ω,

u = g on Γ−.

Page 45: Stability Analysis and Numerical Simulation of Non-Newtonian

3.3. The discontinuous Galerkin method 33

Here Ω is a bounded convex polygonal domain in R2, β is a constant vector with |β| = 1

and Γ− is the inflow boundary defined by

Γ− = x ∈ Γ , β · n(x) < 0 ,

where n(x) is the outward unit normal to Γ at the point x ∈ Γ. For problem (3.6) the

characteristics are straight lines parallel to β, and so the boundary values are prescribed only

on the inflow boundary part Γ−. The solution of the problem (3.6) may be discontinuous

with a jump across a characteristic if, for example, the boundary data g is discontinuous. In

the case that the exact solution is not smooth, standard Galerkin method gives poor results.

The discontinuous Galerkin method is based on using the following finite elements space

defined on the triangulation Th of the domain Ω

Wh = v ∈ L2(Ω) : v|K ∈ Pr(K) ∀K ∈ Th.

That is the space of piecewise polynomials of degree r ≥ 0 with no continuity requirements

across inter-element boundaries. To define this method first some notations are introduced.

The boundary of a triangle K ∈ Th is splitted into an inflow part ∂K− and an outflow part

∂K+ defined by

∂K− = x ∈ ∂K , β · n(x) < 0 ,∂K+ = x ∈ ∂K , β · n(x) ≥ 0 ,

where n(x) is the outward unit normal to ∂K at x ∈ K. Supposing S to be a common side

of two triangles K and K ′, for a function v ∈ Wh which may have a jump discontinuity

across S, one can define the left and right-hand limits v− and v+ in x ∈ S by

v−(x) = limε→0−

v(x+ εβ),

v+(x) = limε→0+

v(x+ εβ).

The jump [v] across S is defined by

[v] = v+ − v−.

The discontinuous Galerkin method for problem (3.6) can now be formulated as seeking a

function uh ∈ Wh according to the following rule:

Problem 3.11. For K ∈ Th, given uh− on ∂K− find uh = uh|K ∈ Pr(K) such that

(3.7) (β · ∇uh + uh, v)K −∫

∂K−

[uh] v+ β · n ds = (f, v)K , ∀v ∈ Pr(K),

where

(w, v)K =

K

w v dx , uh− = g on Γ−.

Page 46: Stability Analysis and Numerical Simulation of Non-Newtonian

34 Chapter 3. Existence results and finite element formulation

This problem admits a unique solution, because (3.7) is nothing but the standard Galerkin

method with weakly imposed boundary conditions in the case of just one element. Thus, if

uh− is given on ∂K−, then uh|K is uniquely determined by (3.7).

Now, relation (3.7) can be written in a more compact form suitable for analysis, as

BK(uh, v) = (f, v)K ∀v ∈ Pr(K),

where

BK(w, v) = (β · ∇w + w, v)K −∫

∂K−

[w] v+n · β ds.

The discontinuous Galerkin method can now be formulated:

Problem 3.12. Find uh ∈Wh such that

(3.8) B(uh, v) = (f, v) ∀v ∈ Pr(K),

where

B(w, v) =∑

K∈Th

BK(w, v),

and uh = g on Γ−.

Clearly the exact solution u satisfies the equation B(u, v) = (f, v), ∀v ∈ Wh (note that

[u]n · β = 0), and thus one has the error equation

B(u− uh, v) = 0 ∀v ∈ Wh.

Let now give the following lemma whose proof can be found in [36]:

Lemma 3.13. For any piecewise smoth function v one has

B(v, v) = |v|2β −1

2

Γ−

v2−|n · β| ds,

with the norm | · |β defined by

|v|2β = ||v||2L2(Ω) +1

2

K

∂K−

[v]2|β · n| ds+1

2

Γ+

v2− β · n ds.

For lemma 3.13 one obtains in the usual way existence and uniqueness of a solution to

the discontinuous Galerkin scheme (3.8), and it is also possible to derive an error estimate

which proves O(hr) convergence in the | · |β-norm. However, this estimate is not the best

possible. One can prove that if δ = Ch for some suitable constant C, then for v ∈ Wh it

holds

B(v, v + δβ · ∇v) ≥ C(||v||2β −∫

γ−

v2− |n · β| ds),

Page 47: Stability Analysis and Numerical Simulation of Non-Newtonian

3.4. Finite element formulation 35

where

||v||2β = |v|2β + h∑

K

||β · ∇v||2K , ||w||K = (w,w)K .

Using this improved stability the following error estimate for the discontinuous Galerkin

method:

Theorem 3.14. If u satisfies (3.6) and uh satisfies (3.8) with r = 0, then there exists a

constant C such that

(3.9) ||u− uh||β ≤ Chr+1/2||u||Hr+1(Ω).

In the case r = 0, for v ∈ Wh one has ||v||β = |v|β , since here β · ∇v = 0 on each K ∈ Th.

Thus, for r = 0 one obtains convergence of order O(h1/2)

Theorem 3.15. If u satisfies (3.6) and uh satisfies (3.8) for r ≥ 0, then there is a constant

C depending on max|Dαu(x)| , |α| = 1 , x ∈ Ω such that

|u− uh|β ≤ Ch1/2||u||H1(Ω).

3.4 Finite element formulation

The flow domain Ω ⊂ RN is supposed to be polygonal and is equipped with an uniformly

regular family of triangulations Th made of triangles K

Ω = ∪K∈Th

K,

and there exists ν0, ν1 such that

(3.10) ν0h < hk < ν1ρk,

where hk is the diameter of K, ρk is the diameter of the greatest ball included in K and

h = maxK∈Thhk.

Let Pk(K) denote the space of polynomials of degree less or equal to k on K ∈ Th. For

the approximation of (u, p) the Taylor-Hood finite element described by the spaces

(3.11)Vh = v ∈ H1(Ω)N ∩ C0(Ω) ; v|K ∈ P2(K)N ∀K ∈ Th,Qh = q ∈ L2(Ω) ∩ C0(Ω) ; q|K ∈ P1(K) ∀K ∈ Th,

will be used. The kernel of the div-operator can be introduced by

(3.12) Vh = v ∈ Vh ; (div v, q) = 0 ∀q ∈ Qh.

It is known that the pair (Vh, Qh) satisfies the inf-sup condition (3.4) (see [31]).

Page 48: Stability Analysis and Numerical Simulation of Non-Newtonian

36 Chapter 3. Existence results and finite element formulation

The stress tensor τ is approximated by linear discontinuous finite elements

Θh = σ ∈ L2(Ω)N2

s ; σ|K ∈ P1(K)N2 ∀K ∈ Th.

In order to describe the approximation of the stress equation by the discontinuous Galerkin

finite element method, following the section before, let us introduce some notations. The

inflow part of the boundary ∂K of element K is defined as

(3.13) ∂K−(u) = x ∈ ∂K; u(x) · nK(x) < 0 ,

where nK(x) is the outward unit normal to K in x ∈ ∂K. Supposing S to be a common

side of two triangles K and K ′, for a function σ ∈ Θh which may have a jump discontinuity

across S, one can define the left and right-hand limits σ− and σ+ in x ∈ S by

σ−(x) = limε→0−

σ(x + εu(x)),

σ+(x) = limε→0+

σ(x + εu(x)),

and σ+ − σ− is the jump across the common side S.

Let us also define the inner products

(τ, σ)h =∑

K∈Th(τ, σ)K ,

〈τ±, σ±〉h,u =∑

K∈Th〈τ±, σ±〉∂K−(u),

〈τ±, σ±〉∂K−(u) =

∂K−(u)

(τ±(u) : σ±(u))|n · u| ds,

and the norm

〈〈τ±〉〉∂K−(u) = 〈τ±, τ±〉1/2∂K−(u).

Then an operator bh on Vh × Vh × Vh is defined by

bh(u, v, w) =1

2((u · ∇)v, w)h −

1

2((u · ∇)w, v)h, bh(u, u, u) = 0,

and an operator Bh on Vh ×Θh ×Θh is defined by

Bh(u, τ, σ) = ((u · ∇)τ, σ)h +1

2(div u τ, σ)h + 〈τ+ − τ−, σ+〉h,u.

The operator Bh has the following properties

Bh(u, τ, σ) = −Bh(u, σ, τ),

Bh(u, τ, τ) =1

2〈〈τ+ − τ−〉〉2h,u,

which implies some coercivity of Bh. For writing the Oldroyd system in a more compact

form let us define the operator gh by

gh(u, τ, σ) = Bh(u, τ, σ) + (βa(τ,∇u), σ)h.

Page 49: Stability Analysis and Numerical Simulation of Non-Newtonian

3.4. Finite element formulation 37

Now, the approximated finite element problem for the Oldroyd system (2.4) can be written

in the following way:

Problem 3.16. Given (τ0, u0) ∈ Θh × Vh, find (τ, u, p) ∈ Θh × Vh ×Q0h such that u = uΓ

on Γ, τ = τΓ on Γ−, u(0) = u0, τ(0) = τ0 and

(3.14)

Wed

dt(τ, σ)h + We gh(u, τ, σ) + (τ, σ)h − 2α(Du, σ)h = (fs, σ)h,

Red

dt(u, v)h + Re bh(u, u, v) + 2(1− α)(Du,Dv)h + (τ,Dv)h

+(p, div v)h = (f, v)h,

(div u, q)h = 0,

for all (σ, v, q) ∈ Θh × V0h ×Qh.To generalize, an additional right-hand side fs was introduced in the stress equation. The

corresponding stationary problem is:

Problem 3.17. Find (τ, u, p) ∈ Θh × Vh ×Q0h such that u = uΓ on Γ, τ = τΓ on Γ− and

(3.15)

We gh(u, τ, σ) + (τ, σ)h − 2α(Du, σ)h = (fs, σ)h,

Re bh(u, u, v) + 2(1− α)(Du,Dv)h + (τ,Dv)h + (p, div v)h = (f, v)h,

(div u, q)h = 0.

for all (σ, v, q) ∈ Θh × V0h ×Qh.Let us now turn back to the stationary creeping flow problem and present some results

concerning the existence of approximate solutions and error bounds given by Baranger and

Sandri in [14], and Najib and Sandri in [51].

Problem 3.18. Find (τ, u, p) ∈ Θh × V0h ×Q0h, such that

We gh(u, τ, σ) + (τ, σ)h = 2α(Du, σ)h,

2(1− α)(Du,Dv)h + (τ,Dv)h − (p, div v)h = (f, v)h,

(div u, q)h = 0,

for any (σ, v, q) ∈ Θh × V0h ×Q0h,

For the stationary creeping problem 3.18 in [51] the following decoupled fix-point iteration

algorithm is proposed:

Algorithm 3.19.

Step 0: At n = 0 start from an initial guess (τ 0, u0, p0) ∈ Θh × V0h ×QhStep 1: un ∈ V0h being known, solve the constitutive equation for the stress τn+1 ∈ Θh

(τn+1, σ)h + We gh(un, τn+1, σ) = 2α(Dun, σ)h, ∀ σ ∈ Θh.

Page 50: Stability Analysis and Numerical Simulation of Non-Newtonian

38 Chapter 3. Existence results and finite element formulation

Step 2: Then, determine the solution (un+1, pn+1) ∈ V0h×Q0h of the Stokes’s like problem:

2(1− α+ α)(Dun+1, Dv)h − (pn+1, div v)h = 2α(Dun, Dv)h − (τn+1, Dv)h + (f, v)h,

(div u, q)h = 0, ∀ (v, q) ∈ V0h ×Qh.

Step 3: Set n← n+ 1 and go to Step 1.

Here, the parameter α is choosen for stability matter such that

(3.16) 1− α+ α > 0.

In the two-dimensional case, under the supposition that the continuous problem 3.6 has a

sufficiently smooth and sufficiently small solution (τ, u, p), in [51] it is shown that problem

problem 3.18 admits a solution (τh, uh, ph), satisfying an error bound in O(h3/2) relative to

(τ, u, p). Furthermore, if the initial approximation of algorithm 3.19 is chosen sufficiently

close to the continuous solution, then the discrete solution (τnh , unh, p

nh) converges linearly to

(τh, uh, ph).

In addition, using a fixed point argument in an appropriate functional setting, Sequeira et

al. [66] have proven existence and uniqueness and obtained error estimates of the approximate

solution for the steady two-dimensional flow of a second-grade fluid. In [7] existence and

uniqueness of the solution is shown for a generalised Oldroyd fluid, under the assumption

of small and suitably regular data. Here, the shear-thinning Oldroyd model, with viscosity

depending on the second invariant of the rate of deformation tensor is chosen to describe

the behavior of blood (see [72]).

Page 51: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 4

Time approximation using the

fractional θ-scheme

The fractional step θ-scheme was first applied to the Navier-Stokes equations by Bristeau

et al. [18]. It was adopted to solve the Navier-Stokes by Bansch [13, 47] and to the Oldroyd

system but only in the case of slow flows by Saramito [63]. The mean idea of this work is to

couple these two tasks for the time discretization of the full nonlinear Oldroyd system.

First, let us give a general description of the basic θ-scheme. Let H be a real valued

Hilbert space and consider a continuous operator A on H and the following initial value

model problem:

Problem 4.1. Find U ∈ L∞(0, T ;H) such that:

mdU

dt+ A(U) = 0,

U(0) = U0,

with given U0 ∈ H.

The idea behind the scheme is to split each time interval [tn, tn+1] of length ∆t = tn+1−tn,into three subintervals [tn, tn+θ∆t], [tn+θ∆t, tn+(1−θ)∆t] and [tn+(1−θ)∆t, tn+1], with

θ ∈ (0, 12 ), as shown in Figure 4.1. Using a decomposition of A of the form A = A1 + A2

one associates with the unknown U the time sequence (Un)n≥0 ≈ U(tn), Un ∈ H , defined

tn tn+θ tn+1−θ tn+1

Figure 4.1: Split of the interval [tn, tn+1] into 3 subintervals.

Page 52: Stability Analysis and Numerical Simulation of Non-Newtonian

40 Chapter 4. Time approximation using the fractional θ-scheme

by the following relations:

(4.1)

mUn+θ − Un

θ∆t+ A1(U

n+θ) = −A2(Un),

mUn+1−θ − Un+θ

(1− 2θ)∆t+ A2(U

n+1−θ) = −A1(Un+θ),

mUn+1 − Un+1−θ

θ∆t+ A1(U

n+1) = −A2(Un+1−θ).

As shown in [18], this scheme applied to the initial value problem 4.1 is unconditionally

stable and on second order accurate if θ = 1−√

2/2.

4.1 Application of the fractional step θ-scheme to the

Oldroyd system

For a solution of the Oldroyd system (2.4)

U(t) = (τ(t), u(t), p(t)),

the time approximation sequence is built

(Un)n≥0 ≈ U(tn), Un = (τn, un, pn).

The application of the procedure defined by (4.1), to the weak form of the flow equations of

an Oldroyd fluid (3.14), enables us to introduce the operators Ai , i = 1, 2 in the following

way:

Criterion 4.2. Let be X = Θh × Vh ×Qh and Ai : X −→ X ′ , ai : X ×X −→ R such that

ai(U,V) = (Ai(U),V)X

were U = (τ, u, p) , V = (σ, v, q) and

a1(U,V) =

ω(τ, σ)− 2α(Du, σ)

(τ,Dv) + 2k(1− α)(Du,Dv) − (p, div v)

(div u, q)

,

a2(U,V) =

We g(u, τ, σ) + (1− ω)(τ, σ)

Re b(u, u, v) + 2(1− k)(1− α)(Du,Dv)

0

,

m = diag (We We We Re Re 0).

Page 53: Stability Analysis and Numerical Simulation of Non-Newtonian

4.1. Application of the fractional step θ-scheme to the Oldroyd system 41

Here, θ = 1−√

2/2 was chosen in order to have second order accuracy, as it will be shown

in (5.61), k and ω are arbitrary parameters in (0, 1) for that moment, their optimal values

are given in (5.62). Criterion 4.2 is a combination of the two splitting methods used in

[18, 32, 11] for the Navier-Stokes equations and in [63] for the Oldroyd system in the case of

slow flows. It is to remark that the term (τ, σ) in the constitutive law and the term (Du,Dv)

in the momentum equation are both split by introducing the splitting parameters ω and k

respectively. This double splitting method leads to the following problem:

Problem 4.3. Given u0 and τ0, then for n > 0 find un+θ, un+1−θ, un+1 in Vh, τn+θ,

τn+1−θ, τn+1 in Θh and pn+θ, pn+1 in Qh such that1

(4.2)

We

θ∆t(τn+θ − τn, σ) + ω(τn+θ, σ)− 2α(Dun+θ, σ) = −We g(un, τn, σ)

−(1− ω)(τn, σ) + (fn+θs , σ),

Re

θ∆t(un+θ − un, v) + (τn+θ , Dv) + 2k(1− α)(Dun+θ , Dv)− (pn+θ, div v)

= −2(1− k)(1− α)(Dun, Dv)−Re b(un, un, v) + (fn+θ, v),

(div un+θ, q) = 0,

(4.3)

We

(1− 2θ)∆t(τn+1−θ − τn+θ, σ) + We g(un+1−θ, τn+1−θ, σ) + (1− ω)(τn+1−θ, σ)

= −ω(τn+θ, σ) + 2α(Dun+θ, σ) + (fn+1−θs , σ),

Re

(1− 2θ)∆t(un+1−θ − un+θ, v) + 2(1− k)(1− α)(Dun+1−θ, Dv)

+Re b(un+1−θ, un+1−θ, v) = −(τn+θ, Dv)− 2k(1− α)(Dun+θ, Dv)

+(pn+θ, div v) + (fn+1−θ, v),

(4.4)

We

θ∆t(τn+1 − τn+1−θ, σ) + ω(τn+1, σ)− 2α(Dun+1, σ) = −(1− ω)(τn+1−θ , σ)

−We g(un+1−θ, τn+1−θ, σ) + (fn+1, σ),

Re

θ∆t(un+1 − un+1−θ, v) + 2k(1− α)(Dun+1, Dv) + (τn+1, Dv)− (pn+1, div v)

= −2(1− k)(1− α)(Dun+1−θ, Dv)−Re b(un+1−θ, un+1−θ, v) + (fn+1, v),

(div un+1, q) = 0.

for all (σ, v, q) ∈ Θh × V0h ×Qh.1for simplicity of notation the index h from the inner products will be drop

Page 54: Stability Analysis and Numerical Simulation of Non-Newtonian

42 Chapter 4. Time approximation using the fractional θ-scheme

4.2 Algorithm

The splitting method described above leads to a new algorithm for solving unsteady flows

of viscoelastic fluids. From equations (4.2)-(4.4) the following algorithm, which allows the

decoupled computation of the stress and velocity-pressure fields, can be deduced2

Algorithm 4.4.

Step 0: At n = 0 start from the initial conditions (τ 0, u0) = (τ0, u0)

Step 1: (τn, un) being known, determine the solution (un+θ, pn+θ) ∈ Vh×Q0h of the Stokes

like problem

(4.5)

λ (un+θ, v) + η (Dun+θ, Dv)− (pn+θ, div v) = λ (un, v)− η1(Dun, Dv)−Re b(un, un, v)− η2(τn, Dv) + η3We g(un, τn, Dv) + (f, v)− η3(fs, Dv),

(div un+θ, q) = 0,

un+θ|Γ = uΓ((n+ θ)∆t), ∀ (v, q) ∈ V0h ×Qh.

Then, determine the solution τn+θ ∈ Θh of the linear problem

(4.6) c1 (τn+θ, σ) = c2 (τn, σ)−We g(un, τn, σ) + c3 (Dun+θ, σ) + (fs, σ), ∀σ ∈ Θh.

Step 2: (τn+θ, un+θ, pn+θ) being known, determine (τn+1−θ , un+1−θ) by computing first the

solution un+1−θ ∈ Vh of the Burgers like problem:

(4.7)

λ1 (un+1−θ, v) + η1 (Dun+1−θ, Dv) + Re b((un+1−θ, un+1−θ, v) = λ2 (un+θ, v)

−λ (un, v) + η1 (Dun, Dv) + Re b(un, un, v),

un+1−θ|Γ

= uΓ((n+ 1− θ)∆t), ∀ v ∈ V0h,

and then, consecutively, the solution τn+1−θ ∈ Θh of the stress transport problem

(4.8)

c4(τn+1−θ, σ) + We g(un+1−θ, τn+1−θ, σ) = c5(τ

n+θ , σ) + c3 (Dun+θ, σ) + (fs, σ),

τn+1−θ|Γ

= τΓ((n+ 1− θ)∆t), ∀ σ ∈ Θh.

Step 3: Repeat Step 1 but replace n+ θ by n+ 1 , and n by n+ 1− θ .

Step 4: Set n← n+ 1 and go to Step 1.

Equation (4.5) was obtained by replacing τn+θ from (4.6) (with3 σ = Dv), into the mo-

mentum equation (4.2)2. By the subtraction of (4.3)2 from (4.2)2 one deduces the Burgers

equation (4.7). However the stress equations (4.6) and (4.8) can be obtained directly from

(4.2)1 and (4.3)1 respectively. To simplify matters the right-hand sides f and fs are consid-

ered in algorithm 4.4 to be time independent. The different coefficients λ, λ1, λ2, c1, c2, c3,

2without excluding the general case, one consider below stationary right-hand sides f and fs

3the definition of the finite elements spaces permit that

Page 55: Stability Analysis and Numerical Simulation of Non-Newtonian

4.2. Algorithm 43

c4, c5, η1, η2 and η3 are dependent on the parameters Re, We, α, k, θ, ω and ∆t according

to the following relations

λ =Re

θ∆t, λ1 =

Re

(1− 2θ)∆t, λ2 = λ+ λ1,

c1 =We

θ∆t+ ω, c2 =

We

θ∆t− (1− ω), c3 = 2α,

c4 =We

(1− 2θ)∆t+ (1− ω), c5 =

We

(1− 2θ)∆t− ω, η = 2k(1− α) +

c3c1,

η1 = 2(1− k)(1− α), η2 =c2c1, η3 =

1

c1.

Thus, by this operator splitting algorithm, one reduces the Oldroyd system to three con-

siderable simpler subproblems. In the first and third step one has to solve a linear, selfadjoint

Stokes problem (4.5), the nonlinearity (u · ∇)u is treated explicitly. For the stress tensor,

one has to solve at this step a linear problem like (4.6). In the second step one has to solve

for the velocity field a nonlinear problem (4.7) of Burgers type. Here the divergence free

condition is dropped and the pressure gradient is taken from the previous time step. For

the stress tensor one has to solve a transport problem (4.8), which is solved by means of the

discontinuous Galerkin method.

Using the algorithm 4.4 three major numerical difficulties of the Oldroyd system are de-

coupled: the treatment of the solenoidal condition, the nonlinearity in the momentum equa-

tion, given by the velocity transport term, and the stress transport term in the constitutive

equation.

4.2.1 Well-posedness of the subproblems

The subproblem of Stokes type (4.5) remains well-posed because of η > 0 for all α ∈ [0, 1],

We > 0 and ∆t > 0. Especially for high Weißenberg numbers one has

limWe→∞

η = 2λk(1− α) > 0 ∀ α ∈ [0, 1).

The subproblem of Burgers type (4.7) is well-posed because all coefficients of the left-hand

side terms are positive.

According to [62] a sufficient condition for existence of weak solutions of the stress trans-

port subproblem (4.8), would be

c4 − 2|a|‖Du‖∞ > 0,

where ‖ · ‖∞ is the norm of L∞(Ω)N2

. This condition results from imposing that the

symmetric part of the left-hand side operator of (4.8) is positive definite in the sense of

Friedrichs. Thus, for a 6= 0, a sufficient condition for the time step size would be

(4.9) ∆t ≤ limWe→∞

We

(1− 2θ)(2|a|We ‖Du‖∞ − 1 + ω)=

1

2|a| (1− 2θ)‖Du‖∞,

Page 56: Stability Analysis and Numerical Simulation of Non-Newtonian

44 Chapter 4. Time approximation using the fractional θ-scheme

whereas for a = 0, no restriction for the time step size arises. However, if the stability

condition (5.75) is not satisfied4, then in some situations restriction (4.9) would not be

helpful for achieving convergence of the algorithm 4.4, based on the θ-scheme, as well as of

any other time discretization algorithm.

4.2.2 Fixed-point iteration scheme for the stationary Oldroyd sys-

tem

We give here a fixed-point iteration scheme for solving the Oldroyd system in the stationary

case corresponding to problem 3.17. Starting from an initial guess (τ 0, u0), let us consider

the following fixed-point iteration algorithm:

Algorithm 4.5.

Step 0: At n = 0 start from the initial guess (τ 0, u0)

Step 1: τn, un being known, determine the solution (un+1, pn+1) ∈ Vh ×Q0h of the Stokes

like problem:

(4.10)

2(1−α+χα+ε)(Dun+1, Dv)h − (pn+1, div v)h = 2ε(Dun, Dv)h −Re b(un, un, v)h

+χWe g(∇un, τn, Dv)h + (χ− 1)(τn, Dv)h + (f, v)h − χWe (fs, Dv)h,

(div u, q)h = 0,

un+1|Γ

= uΓ, ∀ (v, q) ∈ V0h ×Qh.

Step 2: Then, determine the solution τn+1 ∈ Θh of the stress transport problem

(4.11)

(τn+1, σ)h + We g(∇un+1, τn+1, σ)h = 2α(Dun+1, σ)h + (fs, σ)h,

τn+1|Γ

= τΓ, ∀ σ ∈ Θh.

Step 3: Set n← n+ 1 and go to Step 1.

Here, the index n denotes the iteration step and has no time meaning as in the whole

text. Regarding equation (4.10)1 one observes that the extreme case χ = 0 corresponds

to the situation when the equation of motion (3.15)2 is not combined with the constitutive

law (3.15)1, whereas the case χ = 0 corresponds to the situation when the constitutive law

is subtracted from the equation of motion. The constant ε is additionally introduced for

reasons of stability5, such that the coefficient of (Dun+1, Dv)h in (4.10)1 is larger than the

coefficient of (Dun+1, σ)h in (4.11). Thus ε has to fulfill

1− α+ χα+ ε > α.

4see the stability analysis in section 5.2 and the remark 5.85similar as condition (3.16) for algorithm 3.19

Page 57: Stability Analysis and Numerical Simulation of Non-Newtonian

4.2. Algorithm 45

In particular, for ε = 0 and6 χ = η3, equation (4.10)1 is equivalent to (4.2)1 for k = 1 and

∆t→∞. Also, equation (4.11) is equivalent to (4.3)2 for ω = 0 and ∆t→∞. It is notable

that if algorithm algorithm 4.4 converges to a stationary solution, then this solution is the

same as the solution obtained by the fixed-point algorithm algorithm 4.5 with χ = η3.

Restricting algorithm 4.5 to creeping flows and considering χ = 0 and ε = α, then this

algorithm is similar to algorithm 3.19 when changing the first and second steps.

6here, η3 is that defined for algorithm 4.4 corresponding to the time step width ∆t used used there

Page 58: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 59: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 5

Stability analysis

In the first part of this chapter a spectral analysis of the continuous linearized Oldroyd system

and of the time discretization scheme is presented. In the second part the influence of the

βa-term on the stability of the constitutive stress equation and of the whole Oldroyd system

is examined. The last part deals with the stability of the instationary linearized Oldroyd

system, fully discretized by the finite element method and the fractional step θ-scheme.

5.1 Spectral analysis of the linearized Oldroyd system

The first proposal of this section is to investigate the eigenvalues of the linearized continuous

Oldroyd system in the spectral space. After that, the eigenvalues of the fractional step θ-

scheme applied to this Oldroyd system will be studied.

5.1.1 Spectral analysis of the linearized continuous Oldroyd system

The linearization of the Oldroyd system (2.4), by neglecting the convective terms and the

β-term, will get the following problem:

Problem 5.1. Solve in Ω× R+ the system

We∂τ

∂t+ τ − 2αDu = fs,

Re∂u

∂t− 2(1− α) divDu− div τ +∇p = f,

div u = 0,

which is supposed to the boundary and initial conditions

u = 0 on Γ× R+,

u|t=0 = u0 , τ |t=0 = τ0 in Ω,

Page 60: Stability Analysis and Numerical Simulation of Non-Newtonian

48 Chapter 5. Stability analysis

For the linearized Oldroyd problem 5.1, the Fourier transform of the unknown fields was

formally considered in the two-dimensional case:

(5.1)

τ(x, t) =

R2

exp (−iξ · x)τ(ξ, t) dξ,

u(x, t) =

R2

exp (−iξ · x)u(ξ, t) dξ,

p(x, t) =

R2

exp (−iξ · x)p(ξ, t) dξ,

which leads to the following system

(5.2)

We∂τmj∂t

+ τmj + iα(ξmuj + ξj um) = (fs)mj ,

Re∂uj∂t

+ iξmτmj + (1− α)|ξ|2uj − iξj p = fj ,

ξmum = 0.

Here f and fs are the corresponding Fourier transform of the right-hand sides f and fs,

respectively. Multiplying equation (5.2)2 by ξj and using the equality (5.2)3, the pressure

can be written in the form

(5.3) p =iξ · f + ξ · (ξ · τ )

|ξ|2 .

Thus, one obtains

(5.4)

We∂τmj∂t

+ τmj + iα(ξmuj + ξj um) = (fs)mj ,

Re∂uj∂t

+ iξmτmj + (1− α)|ξ|2uj − iξjξmξs|ξ|2 τsm = gj ,

ξmum = 0,

where

g = f + iξ · (ξ · f)

|ξ|2 ,

and the indeces m, j ∈ 1, 2.Consider now the unknown vector

(5.5) U = (τxx, τxy, τyy, ux, uy)

and let for the following spectral analysis of the linear Oldroyd system (5.4) the right-hand

sides f and fs to be zero. Now, system (5.4) can be put in the form

(5.6)∂U

∂t+ AU = 0,

Page 61: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 49

with

(5.7) A =

[

1WeI3 0

0 1Re I2

][

I3 B

C l I2

]

=

[

1WeI3

1WeB

1ReC

lRe I2

]

.

The submatrices B and C in (5.7) are defined as follows

(5.8) B = iα

2ξ1 0

ξ2 ξ10 2ξ2

, C =i

|ξ|2

[

ξ1ξ22 −ξ2z −ξ1ξ22

−ξ21ξ2 ξ1z ξ21ξ2

]

,

where |ξ| is the length of the wave vector, l = (1− α)|ξ|2 and z = ξ21 − ξ22 .

The characteristic polynomial of the matrix A is

(

1

We− λe

)2 (l

Re− λe

) |ξ|4ξ1ξ2

[(

1

We− λe

)(

l

Re− λe

)

+ α|ξ|2]

= 0.

Hence, their eigenvalues are

(5.9) λe1,2 =1

We, λe3 =

l

Re, λe4,5 =

1

2

(

l

Re+

1

We±√

∆e

)

with

∆e =

(

l

Re− 1

We

)2

− 4α|ξ|2ReWe

=(1− α)2|ξ|4

Re2 − 2(1 + α)|ξ|2ReWe

+1

We2 .

The eigenvalues λj , j ∈ 1, 2, 3 are clearly real and positive, so for stability of the system

(5.6) it remains to show that the other two eigenvalues λj , j ∈ 4, 5 have real parts, which

are also positive. To do this, two cases can be distinguished. First, if ∆e < 0 the task is

fulfilled because

R(λe4,5) =l

Re+

1

We> 0.

Secondly, if ∆e ≥ 0 then λ4,5 ∈ R and

λe4,5 > 0 ⇔ l

Re+

1

We> −

(1− α)2|ξ|4Re2 − 2(1 + α)|ξ|2

ReWe+

1

We2 ⇔ l > −α|ξ|2.

So, it was shown that the real parts of all the five eigenvalues are positive, and thereby the

continuous linear Oldroyd system is unconditionally stable.

Further, a description of the solution of system (5.6) is given. The solution U at time t

can be expressed by

U(t) = exp (−At)U(0),

and expanding the exponential function in a Taylor series one gets

U(t) =

(

I5 −At+1

2A

2t2 + ...

)

U(0).

Page 62: Stability Analysis and Numerical Simulation of Non-Newtonian

50 Chapter 5. Stability analysis

For the calculation of the powers of matrix A the matrices Λ and M will be introduced in

the following way

Λ =1

|ξ|2

[

ξ21 ξ1ξ2

ξ1ξ2 ξ22

]

, M =

2ξ21ξ22 −2ξ1ξ2z −2ξ21ξ

22

−ξ1ξ2z z2 ξ1ξ2z

−2ξ21ξ22 2ξ1ξ2z 2ξ21ξ

22

.

One can see that the matrices Λ and M are singular with

det Λ = 0 , rankM = 1.

It is also easy to show that

CB = −α|ξ|2(I2 − Λ) , BC = − α

|ξ|2M,

M2 = |ξ|4M, CM = |ξ|4C, MB = −iα|ξ|2

2ξ1ξ22 −2ξ21ξ2

−ξ2z ξ1z

−2ξ1ξ22 2ξ21ξ2

.

With this considerations the square of matrix A2 is

A2 =

1

We2 I3 +1

ReWeBC

(

1

We2 +l

ReWe

)

B

(

1

We2 +l

ReWe

)

C

(

l

Re

)2

I2 +1

ReWeCB

=

1

We2 I3 −α

|ξ|2ReWeM

(

1

We2 +l

ReWe

)

B

(

1

We2 +l

ReWe

)

C

(

(

l

Re

)2

− α|ξ|2ReWe

)

I2 +α|ξ|2ReWe

Λ

.

Due to the fact that ξ · u = 0, consistent with (5.4)3, it results in

Λu = 0 , MBu = |ξ|4Bu,

and so

A2U =

1

We2 I3 −α

|ξ|2ReWeM

(

1

We2 +l

ReWe

)

B

(

1

ReWe+

l

Re2

)

C

(

(

l

Re

)2

− α|ξ|2ReWe

)

I2

U.

Applying this argument recursively, the n-th power of matrix A applied to the unknown

vector U can be written in the form

AnU =

[

enI3 + anM bnB

cnC dnI2

]

U,

Page 63: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 51

where the sequences an≥1, bn≥1, cn≥1, dn≥1 and en≥1 are defined by

a1 = 0, b1 =1

We, c1 =

1

Re, d1 =

l

Re, e1 =

1

We,

an+1 =anWe− α

We| ξ|2 cn,

bn+1 =bn + dn

We,

cn+1 =en + l cn + |ξ|4an

Re,

dn+1 =ldn − α|ξ|2bn

Re,

en+1 =enWe

.

Now, the time dependent solution U at two points of time, say tn and tn+1 with step size

∆t = tn+1 − tn, are related by

Un+1 = Ωe U

n.

Here Ωe is the asymptotic damping factor defined as

Un+1 =

(

I5 −A∆t+1

2A

2∆t2 + ...

)

Un.

Our main aim in the next subsection is to demonstrate the second order time accuracy of

the θ-scheme time discretization applied to the linearized Oldroyd system. For this task it

is necessary to see how the damping factor behaves relative to the powers of ∆t. Breaking

down at terms of second order, the asymptotic damping factor becomes

(5.10) Ωe ≈[

K3I3 +KMM KBB

KCC K2I2

]

,

where the coefficients Ki , i ∈ 2, 3,M,B,C are defined by

(5.11)

K3 = 1− e1∆t+1

2e2∆t

2 = 1− 1

We∆t+

1

2We2 ∆t2,

KM = −a1∆t+1

2a2∆t

2 = − α

2|ξ|2ReWe∆t2,

KB = −b1∆t+1

2b2∆t

2 = − 1

We∆t+

1

2

(

1

We2 +l

ReWe

)

∆t2,

KC = −c1∆t+1

2c2∆t

2 = − 1

Re∆t+

1

2

(

1

ReWe+

l

Re2

)

∆t2,

K2 = 1− d1∆t+1

2d2∆t

2 = 1− l

Re∆t+

1

2

(

(

l

Re

)2

− α|ξ|2ReWe

)

∆t2.

Page 64: Stability Analysis and Numerical Simulation of Non-Newtonian

52 Chapter 5. Stability analysis

5.1.2 Spectral analysis of the θ-scheme for the linearized Oldroyd

system

The goal of this subsection is to analyze the stability properties and accuracy of the fractional

step θ-scheme corresponding to the linearized version of the Oldroyd system. So, applying

the θ-scheme to problem 5.1 with zero right-hand sides f and fs, it provides the following

problem:

Problem 5.2. Given u0 and τ0, then for n > 0 find un+θ, un+1−θ, un+1, τn+θ, τn+1−θ,

τn+1, pn+θ and pn+1, defined in Ω and with adequate boundary conditions, such that

(5.12)

We

θ∆t(τn+θ − τn) + ωτn+θ − 2αDun+θ = −(1− ω)τn,

Re

θ∆t(un+θ−un)− 2k(1− α)divDun+θ − div τn+θ +∇pn+θ

= 2(1− k)(1− α)divDun,

div un+θ = 0,

(5.13)

We

(1− 2θ)∆t(τn+1−θ − τn+θ) + (1− ω)τn+1−θ = −ωτn+θ + 2αDun+θ,

Re

(1− 2θ)∆t(un+1−θ − un+θ)− 2(1− k)(1− α)divDun+1−θ

= 2k(1− α)divDun+θ + div τn+θ −∇pn+θ,

(5.14)

We

θ∆t(τn+1 − τn+1−θ) + ωτn+1 − 2αDun+1 = −(1− ω)(τn+1−θ, σ),

Re

θ∆t(un+1 − un+1−θ)− 2k(1− α)divDun+1 − div τn+1 +∇pn+1

= 2(1− k)(1− α)divDun+1−θ,

div un+1 = 0.

For all n ≥ 0 the Fourier transform of the unknown fields will be considered formally

τn(x) =

R2

exp (−iξ · x)τn(ξ) dξ,

un(x) =

R2

exp (−iξ · x)un(ξ) dξ,

pn(x) =

R2

exp (−iξ · x)pn(ξ) dξ.

Introducing these Fourier transforms into the equation (5.12)-(5.14) and after eliminating

Page 65: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 53

the pressure in a similar way as in (5.3), one obtains

(5.15)

We

θ∆t(τn+θmj − τnmj) + ωτn+θ

mj + iα(ξmun+θj + ξj u

n+θm ) = −(1− ω)τnmj ,

Re

θ∆t(un+θj − unj ) + iξmτ

n+θmj − i

ξjξmξs|ξ|2 τn+θ

sm + k(1− α)|ξ|2un+θj

= −(1− k)(1− α)|ξ|2unj ,ξmu

n+θm = 0,

(5.16)

We

(1−2θ)∆t(τn+1−θmj − τn+θ

mj ) + (1−ω)τn+1−θmj = −iα(ξmu

n+θj + ξj u

n+θm )− ωτn+θ

mj ,

Re

(1− 2θ)∆t(un+1−θj − u+θ

j ) + (1− k)(1− α)|ξ|2un+1−θj

= −iξmτn+θmj + i

ξjξmξs|ξ|2 τn+θ

sm − k(1− α)|ξ|2un+θj ,

(5.17)

We

θ∆t(τn+1mj − τn+1−θ

mj ) + ωτn+1mj + iα(ξmu

n+1j + ξj u

n+1m ) = −(1− ω)τn+1−θ

mj ,

Re

θ∆t(un+1j − un+1−θ

j ) + iξmτn+1mj − i

ξjξmξs|ξ|2 τn+1

sm + k(1− α)|ξ|2un+1j

= −(1− k)(1− α)|ξ|2un+1−θj ,

ξmun+1m = 0.

To simplify the notations further

θ′ = 1− 2θ

will be used. The systems (5.15)-(5.17) corresponding to the time discretization scheme, can

be re-written in the following compact form:

(5.18)

A1Un+θ = A2U

n,

A3Un+1−θ = A4U

n+θ,

A5Un+1 = A6U

n+1−θ.

Here, the matrices Ai, i ∈ 1, 2, 3, 4, 5, 6 are defined by the relations

A1 =

[

aI3 B

C dI2

]

, A2 =

[

a1I3 0

0 d1I2

]

, A3 =

[

a′I3 0

0 d′I2

]

,

A4 =

[

a′1I3 −B−C d′1I2

]

, A5 = A1, A6 =

[

a1I3 0

0 d1(I2 − Λ)

]

,

Page 66: Stability Analysis and Numerical Simulation of Non-Newtonian

54 Chapter 5. Stability analysis

with the coefficients a, a1, a, a′1 and d, d1, d, d

′1 given by means of

(5.19)

a =We

θ∆t+ ω, a1 =

We

θ∆t− (1− ω),

a′ =We

θ′∆t+ (1− ω), a′1 =

We

θ′∆t− ω,

d =Re

θ∆t+ kl, d1 =

Re

θ∆t− (1− k)l,

d′ =Re

θ′∆t+ (1− k)l, d′1 =

Re

θ′∆t− kl.

Denoting

g =a+ a′1

We=d+ d′1

Re=

1

∆t

(

1

θ+

1

θ′

)

, G =

[

We I3 0

0 Re I2

]

,

it is easy to show that

A4 = g G−A1.

The Schur complement of matrix a I3 in the block matrix A1 is

S = d I2 −1

aCB,

with the determinant

detS = d2 + dα

a|ξ|2.

Denoting now

m =d

detS, p =

α

a detS|ξ|2,

the inverse of the Schur complement becomes

S−1 = mI2 + pΛ,

and, because ΛC = 0, it fulfills

(5.20) S−1C = mC, BS−1C = −m α

|ξ|2M.

Using the Schur complement decomposition the inverse of matrix A1 can be written as

A−11 =

[

I31

aB

0 I2

][ 1

aI3 0

0 S−1

][

I3 01

aC I2

]

=

1

a

(

I3 +1

aBS−1C

)

−1

aBS−1

−1

aS−1C S−1

.

Due to (5.20) and denoting

D = I3 −m

a

α

|ξ|2M,

Page 67: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 55

the inverse of the matrix A1 can be written in the compressed form

A−11 =

1

aD −1

aBS−1

−maC S−1

.

From the continuity equations (5.98)3 and (5.96)3 corresponding to the n -th and (n+ θ) -th

time step the following relations hold

ξ · un = 0 ⇒ Λun = 0 ⇒ MBun = |ξ|4Bun,ξ · un+θ = 0 ⇒ Λun+θ = 0 ⇒ MBun+θ = |ξ|4Bun+θ,

and the properties for the inverse of the Schur complement

S−1un = mun, S−1un+θ = mun+θ,

BS−1un = mBun, BS−1un+θ = mBun+θ.

In accordance with the previous relations one has

Un+θ = A−1

1 A2Un =

a1

aD −md1

aB

−ma1

aC md1I2

Un,

and

Un+1 = A−11 A6A

−13 A4U

n+θ = A−11

(

A2 −[

0 0

0 d1Λ

])

1

a′I3 0

01

d′I2

(gG−A1) Un+θ

=

(

A−11 A2 −A−1

1

[

0 0

0 d1Λ

])

g

We

a′I3 0

0Re

d′I2

1

a′I3 0

01

d′I2

A1

Un+θ

=

gA−1

1

We a1

a′I3 0

0Re d1

d′I2

−A−1

1

a1

a′I3 0

0d1

d′I2

A1 +A−11

0 0

0d1

d′Λ

A1

Un+θ

= g

Wea1

aa′D −Re

d1

ad′BS−1

−Wema1

aa′C Re

d1

d′S−1

Un+θ −A−11

a1

a′I3 0

0d1

d′(I2 − Λ)

A1U

n+θ

= g

Wea1

aa′D −Re

md1

ad′B

−Wema1

aa′C Re

md1

d′I2

Un+θ −A−11

a1

a′I3 0

0d1

d′(I2 − Λ)

A1U

n+θ.

Page 68: Stability Analysis and Numerical Simulation of Non-Newtonian

56 Chapter 5. Stability analysis

The dependence between the unknown fields at two consecutive time steps is given by

Un+1 = Ω Un,

where the damping matrix Ω is

Ω = g

Wea1

aa′D −Re

md1

ad′B

−Wema1

aa′C Re

md1

d′I2

a1

aD −md1

aB

−ma1

aC md1I2

1

aD −1

aBS−1

−maC S−1

a21

a′I3 0

0d21

d′(I2 − Λ)

.

Taking into account that

D2 = I3 −m

a

α

|ξ|2(

2− m

aα|ξ|2

)

M,

CD =(

1− m

aα|ξ|2

)

C,

DBun =(

1− m

aα|ξ|2

)

Bun,

the damping matrix can be written in the compressed form

(5.21) Ω =

[

k3I3 + kMM kBB

kCC k2I2

]

.

Here, the coefficients kj , j ∈ 2, 3,M,B,C are defined as follows

(5.22)

k3 =a21a

′1

a2a′, y = mg

(

Wea1

a2a′α|ξ|2 −Re

d1

d′

)

,

k2 = −md1

(

y +d1

d′

)

, kM =ma1

a2

(

y − 2a1a

′1

aa′− a1

a′

)

α

|ξ|2 ,

kC =ma1

a

(

y − a1a′1

aa′

)

, kB =md1

a

(

y − a1a′1

aa′+d1

d′− a1

a′

)

.

The characteristic polynomial of the damping matrix Ω is

det(Ω− λI5) = (k3 − λ)2(k2 − λ)[

(k3 − λ)(k2 − λ) + kM (k2 − λ)|ξ|4 + αkBkC |ξ|2]

,

and has the roots

(5.23)

λ1,2 = k3,

λ3 = k2,

λ4,5 =P1 ±

P 21 − 4P0

2,

Page 69: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 57

whereP1 = k3 + k2 + kM |ξ|4,P0 = k3k2 + k2kM |ξ|4 + αkBkC |ξ|2.

Stability

The numerical stability of any time discretization requires that the absolute value of the

eigenvalues of the damping factor are less than 1, i.e.

|λj | ≤ 1, j ∈ 1, ..., 5.

The condition |λ1,2| ≤ 1 requires that

(5.24) |k3| ≤ 1,

and the condition |λ3| ≤ 1 requires that

(5.25) |k2| ≤ 1.

If P 21 − 4P0 < 0 (this is possible only if P0 > 0), then

(5.26) |λ4,5| =√

P0,

and for stability one needs to show that

(5.27) P0 ≤ 1.

Else if P 21−4P0 ≥ 0, the stability condition |λ4,5| ≤ 1 requires that |P1| ≤ 2 and |P1|−P0 ≤ 1.

But, because it will be demonstrated that (5.27) is true, it is sufficiently to show that

(5.28) |P1| < 1 + P0.

As it will be shown in the next subsections (see (5.61)-(5.62)), the optimal choice of the

splitting parameters for the θ-scheme are

(5.29) θ = 1−√

2

2, k = ω =

1− 2θ

1− θ ,

and for these values we try to demonstrate the relations (5.25)-(5.28). For the following

steps, let us denote by r, w and γ the quantities

(5.30) r =l∆t

Re, w =

∆t

We, γ =

1

a

α

1− α.

The coefficient k3 can be written as a function of w

k3(w) =(1− (1− ω)θw)

2(1− ωθ′w)

(1 + ωθw)2(1 + (1− ω)θ′w)

Page 70: Stability Analysis and Numerical Simulation of Non-Newtonian

58 Chapter 5. Stability analysis

10−5

100

105

−0.5

0

0.5

1

w

k 3

Figure 5.1: Dependence of k3 on w.

after replacing in (5.22) the quantities (5.19). Then condition (5.24) is reduced to the two

inequalities

ω(1− ω)θ2θ′w3 + (2ω − 1)θ2w2 + w ≥ 0,

ω(1− ω)(2ω − 1)θ2θ′w3 +(

(ω2 + (1− ω)2)θ2 + 4ω(1− ω)θθ′)

w2

+(2ω − 1)(2θ − θ′)w + 2 ≥ 0,

which are true because from the definitions (5.29) of θ, θ′ and ω one obtains positive coef-

ficients of the powers of w. Also, in Fig. 5.1 the dependence of k3 on the rate of the time

step size and the Weißenberg number is represented, which shows that k3 ∈ (−0.8, 1] and

consequently the condition (5.24) is fulfilled.

Let us introduce the following notations

(5.31)

φ = Wea1

aa′∆t, χ =

a1

a, ψ =

2a1a′1

aa′+a1

a′,

φ = φθ + θ′

θθ′, ϕ =

a21

aa′, χ ψ = 2k3 + ϕ.

Taking into account the relations (5.19), then all the quantities defined in (5.31) are func-

tions1 of w = ∆tWe . Now, the quantities defined in (5.25)-(5.28) will be written as fractions

(5.32) k2 =Z(k2)

N, P0 =

Z(P0)

N, P1 =

Z(P1)

N.

1the dependence of φ and ϕ on w is represented in Fig. A.1

Page 71: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 59

Here, Z(·) are the numerators of the fractions from (5.32) and N is the denominator of all

those fractions, which takes on the expression

N = [1 + (k + γ)θr]2[1 + (1− k)θ′r].

Because γ ≥ 0 and 0 < k < 1 it follows that

(5.33) N ≥ 1.

Further, all the numerators Z(·) and the denominator N in (5.32) are polynomials of order

two in γ and of order three in r, of the form

(5.34) (c32γ2 + c31γ + c30)r

3 + (c22γ2 + c21γ + c20)r

2 + (c12γ2 + c11γ + c10)r + (c02γ

2 + c01γ + c00)

for r ≥ 0, γ ≥ 0 and with the coefficients cji , i ∈ 0, 1, 2 and j ∈ 0, 1, 2, 3 depending on

the functions φ, φ, ψ, χ, ϕ defined in (5.31) and on θ, θ′, k, which are related by (5.29).

A simple but tedious computation yields

(5.35) N :

c32 = (1− k)θ2θ′, c31 = 2k(1− k)θ2θ′, c30 = k2(1− k)θ2θ′,c22 = θ2, c21 = 2(kθ + (1− k)θ′)θ, c20 = k(kθ + 2(1− k)θ′)θ,c11 = 2θ, c10 = 2kθ + (1− k)θ′, c00 = 1, c12 = c20 = c10 = 0.

(5.36) Z(k2) :

c31 = (1− k)2(φ− 1)θ2θ′, c30 = −k(1− k)2θ2θ,′

c21 = (1− k)(φ(θ − θ′) + 2θ′)θ, c20 = (1− k)(2kθ′ + (1− k)θ)θ,c11 = −(φθ + θ′), c10 = −(kθ′ + 2(1− k)θ), c00 = 1,

c32 = c22 = c12 = c02 = c01 = 0.

(5.37) Z(P0) :

c31 = ϕ(1− k)2θ2θ′, c30 = −kk3(1− k)2θ2θ′, c21 = −2ϕ(1− k)θθ′,c20 = k3(1− k)(2kθ′ + (1− k)θ)θ, c10 = −k3(kθ

′ + 2(1− k)θ),c11 = ϕθ′, c00 = k3, c32 = c22 = c12 = c02 = c01 = 0.

(5.38) Z(P1) :

c31 = (1− k)[(1− k)(φ− 1)− kϕ]θ2θ′,

c30 = (1− k)k(k3k − (1− k))θ2θ′,

c21 = (1− k)[χ(θ + θ′) + φ(θ − θ′) + 2θ′ − 2ϕθ′]θ,

c20 = (1− k)[2kθ′ + (1− k)θ]θ + k3k[kθ + 2(1− k)θ′]θ,c11 = −[φθ + χ(θ + θ′) + θ′ + ϕθ], c32 = c22 = c12 = c02 = c01 = 0,

c10 = k3(2kθ + (1− k)θ′)− [kθ′ + 2(1− k)θ], c00 = 1 + k3.

Page 72: Stability Analysis and Numerical Simulation of Non-Newtonian

60 Chapter 5. Stability analysis

Due to (5.33) for (5.25)-(5.28), one has to show that all the quantities

Z(1 + k2) = N + Z(k2),

Z(1− k2) = N − Z(k2),

Z(1 + P0) = N + Z(P0),

Z(1− P0) = N − Z(P0),

Z(1 + P0 − P1) = N + Z(P0)− Z(P1),

Z(1 + P0 + P1) = N + Z(P0) + Z(P1),

are positive.

The numerator of 1 + k2 can be written in the form (5.34) with the coefficients

(5.39)

c32 = (1− k)θ2θ′, c31 = (1− k)(2k + (1− k)(φ− 1))θ2θ′,

c30 = k(1− k)(2k − 1)θ2θ′, c22 = θ2,

c21 = [2kθ + 4(1− k)θ′ + (1− k)φ(θ − θ′)]θ,c20 = [k2 + (1− k)2]θ2 + 4k(1− k)θ′θ, c11 = 2θ − θ′ − φθ,c10 = (2k − 1)(2θ − θ′), c12 = c02 = c01 = 0, c00 = 2.

One can calculate easily and see in Fig. A.2 that all coefficients given in (5.39) are positive

apart from the coefficient c11. Because c2j , j = 1, 2, 3 are positive it implies that also

c22γ2 + c21γ + c20 is positive and the function

(5.40)f(r, γ) = p2r

2 + p1r + p0

= (c22γ2 + c21γ + c20)r

2 + (c11γ + c10)r + 2,

has a minimum relative to r. To demonstrate that this minimum is positive, it suffices to

show that

(5.41) 4p2p0 − (p1)2 = (8c22 − (c11)

2)γ2 + (8c21 − 2c11c10)γ + 8c20 − (c10)

2 ≥ 0.

The zero order coefficient is 8c20− (c10)2 ≈ 0.2, and the coefficients of the second and the first

order are represented as functions of w in Fig. A.6 and are positive in the whole range of

their variation.

The numerator of 1− k2 can be written in the form (5.34) with the coefficients:

(5.42)

c32 = (1− k)θ2θ′, c31 = (1− k)(2k − (1− k)(φ − 1))θ2θ′, c30 = k(1− k)θ2θ′,

c22 = θ2, c21 = [2kθ − (1− k)φ(θ − θ′)]θ, c20 = (2k − 1)θ2,

c11 = 1 + φθ, c10 = 1, c12 = c02 = c01 = c00 = 0.

By means of Fig. A.2 it results that all coefficients appearing in (5.42) are positive. So,

condition (5.25) is fulfilled.

Page 73: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 61

The numerator of 1 + P0 can be written in the form (5.34) with the coefficients:

(5.43)

c32 = (1− k)θ2θ′, c31 = (1− k)[2k + (1− k)ϕ]θ2θ′, c30 = k(1− k)[k − k3(1− k)]θ2θ′,c22 = θ2, c21 = 2[kθ + (1− k)(1− ϕ)θ′]θ,

c20 = k[kθ + 2(1− k)θ′] + k3(1− k)[2kθ′ + (1− k)θ]θ, c11 = 2θ + ϕθ′,

c10 = 2kθ + (1− k)θ′ − k3[kθ′ + 2(1− k)θ], c12 = c02 = c01 = 0, c00 = 1 + k3.

Calculations and the left part of Fig. A.3 show that all coefficients given in (5.43) are positive.

The numerator of 1− P0 can be written in the form (5.34) with the coefficients:

(5.44)

c32 = (1− k)θ2θ′, c31 = (1− k)[2k − (1− k)ϕ]θ2θ′, c30 = k(1− k)[k + k3(1− k)]θ2θ′,c22 = θ2, c21 = 2[kθ + (1− k)(1 + ϕ)θ′]θ, c11 = 2θ − ϕθ′,c20 = k[kθ + 2(1− k)θ′]− k3(1− k)[2kθ′ + (1− k)θ]θ,c10 = 2kθ + (1− k)θ′ + k3[kθ

′ + 2(1− k)θ], c12 = c02 = c01 = 0, c00 = 1− k3.

One can calculate easily and see in the right part of Fig. A.4 that all coefficients given in

(5.44) are positive. So, the condition (5.27) is fulfilled.

The numerator of 1 + P0 + P1 can be written in the form (5.34) with the coefficients

(5.45)

c32 = (1− k)θ2θ′, c31 = (1− k)[2k + (1− 2k)ϕ+ (1− k)(φ− 1)]θ2θ′,

c30 = (1− k)(1 + k3)k(2k − 1)θ2θ′, c22 = θ2

c21 =[

2kθ + 4(1− k)θ′ − 4(1− k)ϕθ′ + (1− k)(χ(θ + θ′) + φ(θ − θ′))]

θ,

c20 = (1 + k3)[(k2 + (1− k)2)θ + 4k(1− k)θ′]θ, c10 = (1 + k3)(2k − 1)(2θ − θ′),

c11 = 2θ − θ′ + ϕ(θ′ − θ)− (φθ + χ(θ + θ′)), c12 = c02 = c01 = 0, c00 = 2(1 + k3).

Also, all coefficients given in (5.45) are positive (see Fig. A.5), apart from the coefficient c11for w ≤ w∗, where w∗ < 6.0. Now, for w ≤ w∗, it is a similar demonstration as before in

(5.40). Because c2j , j = 1, 2, 3 are positive it implies that also c22γ2 + c21γ+ c20 is positive and

the function

(5.46)f(r, γ) = p2r

2 + p1r + p0

= (c22γ2 + c21γ + c20)r

2 + (c11γ + c10)r + c00,

has a minimum fmin = f(rmin, γ) towards r. On the one hand if rmin < 0 than f(r, γ) ≥f(0, γ) = c00 > 0. On the other hand, if rmin ≥ 0 than f(r, γ) ≥ fmin. To demonstrate that

fmin is positive, it suffices to show that

(5.47) 4p2p0 − (p1)2 = (4c22c

00 − (c11)

2)γ2 + (4c21c00 − 2c11c

10)γ + 4c20c

00 − (c10)

2 ≥ 0.

The coefficients of zero, first and second order of the polynomial (5.46) are represented as

functions of w in Fig. A.7 and are positive in the interval w ∈ [0,w∗].

Page 74: Stability Analysis and Numerical Simulation of Non-Newtonian

62 Chapter 5. Stability analysis

The numerator of 1 + P0 − P1 can be written in the form (5.34) with the coefficients

c31 = (1− k)[2k + ϕ− (1− k)(φ− 1)]θ2θ′, c30 = (1− k)(1− k3)kθ2θ′, c22 = θ2,

c32 = (1− k)θ2θ′, c21 = [2kθ − (1− k)(χ(θ + θ′) + φ(θ − θ′)]θ,c20 = (1− k3)(2k − 1)θ, c11 = 1 + ϕ(θ + θ′) + (φθ + χ(θ + θ′)),

c10 = 1− k3, c12 = c02 = c01 = c00 = 0.

Obviously, c32, c22 are positive and because |k3| ≤ 1, according to (5.28), also c30, c

20 and c10

are positive. From the right column of Fig. A.5, one can see that the coefficients c31, c21, c

11

are positive for all values of w.

Strong stability

For demonstrating that the θ-scheme applied to the Oldroyd system is strongly stable, it

is necessary to show that the absolute value of the eigenvalues of the damping factor are

asymptotically smaller than 1. First, one proves that

(5.48) lim∆t→∞

|λi(∆t, |ξ|)| < 1, i ∈ 1, 2, 3, 4, 5.

Firstly one can observe that under the assumptions k > 12 and ω > 1

2 , in the limit ∆t→∞the eigenvalues (5.23) become

(5.49) lim∆t→∞

|λ1,2| = lim∆t→∞

|k3| =1− ωω

< 1,

(5.50) lim∆t→∞

|λ3| = lim∆t→∞

|k2| =ω(1− k)(1− α)

ωk(1− α) + α< 1,

(5.51) lim∆t→∞

P1 = − [ω(1− k) + (1− ω)k]α

ωk(1− α) + α,

(5.52) lim∆t→∞

P0 =(1− ω)(1− k)(1− α)

ωk(1− α) + α< 1.

If P 21 − 4P0 < 0 then according to (5.26) one gets

(5.53) lim∆t→∞

|λ4,5|2 = lim∆t→∞

P0 < 1.

Else if P 21 − 4P0 ≥ 0 by means of (5.28) it is sufficiently to show that

(5.54) lim∆t→∞

|P1| < 1 + lim∆t→∞

P0.

Page 75: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 63

Via the relations (5.52) and (5.51) it is easy to prove that (5.54) is true, and so

lim∆t→∞

|λ4,5|2 < 1.

A much stronger condition than (5.48) is the requirement that all eigenvalues λi,

i ∈ 1, 2, 3, 4, 5 fulfill

(5.55) ∀ε > 0 ∃∆t0, ξ0 : |λi(∆t, |ξ|)| ≤ q + ε, 0 < q < 1, ∀∆t ≥ ∆t0 ∀|ξ| ≥ |ξ0|.

To demonstrate the second condition for strong stability, one observes that the quantity k3 is

independent of |ξ|, and that the numerators and denominators of k2, P0, P1 are polynomials

of third degree in r, of the form (5.34), whose coefficients are functions of w, and thereby

of ∆t. When |ξ| is large (|ξ| > |ξ0|) then r is also large, and only the coefficients of the 3th

order terms in r control the corresponding polynomials.

After simplifying the fractions with (1 − k)θ2θ′, the controlling terms of the numerators

and denominator of k2, P0 and P1 are

(5.56)

N = γ2 + 2γk + k2 ≥ k2 (γ ≥ 0),

Z(k2) = (1− k)[γ(φ− 1)− k],Z(P0) = (1− k)(γϕ− k3k),

Z(P1) = γ[(1− k)(φ− 1)− kϕ] + k3k2 − k(1− k).

By virtue of Fig. 5.1 and Fig. A.1 one can observe that for large ∆t (or large w) the function

k3 is asymptotical decreasing to the value −(1 − ω)/ω and the functions ϕ and φ increase

asymptotically to (1− ω)/ω and 0 respectively. Hence,

(5.57) ∀ε > 0 ∃∆t0 :

k3 +1− ωω

< ε,

ϕ− 1− ωω

< ε, |φ| < ε ∀∆t > ∆t0.

The asymptotical behavior of k3 demonstrates that the requirement (5.55) is fulfield for λ1,2.

For the eigenvalue λ3 the validity of (5.55) is equivalent to the fact that there exists

0 < q < 1 so that qN ± Z(k2) > 0. By virtue of (5.56) one has, that

qN − Z(k2) = γ2q + γ[2qk − (1− k)(φ − 1)] + k[qk + (1− k)],

which is positive if 0 < q < 1. On the other hand

qN + Z(k2) = γ2q + γ[2qk + (1− k)(φ − 1)] + k[qk − (1− k)],

which, taking into account the asymptotical behavior of φ given by (5.57), is also positive

for

max

1− kk

,(1− k)(1 + ε)

2k

< q < 1.

Page 76: Stability Analysis and Numerical Simulation of Non-Newtonian

64 Chapter 5. Stability analysis

It remains to prove of the requirement (5.55) for the other two eigenvalues λ4,5. If Z(P1)2−

4Z(P0) < 0, then one needs to show that

|λ4,5| =Z(P0)

N≤ q.

On the one hand the quantity

qN − Z(P0) = γ2q + γ[2kq − (1− k)ϕ] + k[kq + (1− k)k3]

is positive if 12 (1− k)2/k2 < q < 1 and on the other hand the quantity

qN + Z(P0) = γ2q + γ[2kq + (1− k)ϕ] + k[kq − (1− k)k3],

is also positive if 0 < q < 1 due to the fact that ϕ > 0 and k3 < 0 for large ∆t.

Else if Z(P1)2 − 4Z(P0) ≥ 0 then |λ4,5| ≤ q is equivalent to

Z(P1)2 − 4Z(P0) ≤ 2 Nq − |Z(P1)| ⇔

|Z(P1)| ≤ 2Nq,

|Z(P1)|q < Nq2 + Z(P0).

Due to the fact that Z(P0) < qN , if the last inequality is satisfied, then also the first is true.

Thus one has to show only that |Z(P1)|q < Nq2 + Z(P0) is true. This is equivalent to

(5.58)Nq2 + Z(P0)− Z(P1)q = γ2q2 + γ[2kq2 + ((1− k) + kq)ϕ+ (1− k)q(1− φ)]

+k[kq2 + (1− k)q − k3((1− k) + kq)],

and

(5.59)Nq2 + Z(P0) + Z(P1)q = γ2q2 + γ[2kq2 + ((1− k)− kq)ϕ+ (1− k)q(φ− 1)]

+k[kq2 + (1− k)q − k3((1− k)− kq)].

From (5.57) one deduces that for w > ∆t0We it holds that ϕ > 0, 1−φ > 1−ε and−k3 >

1−ωω −ε.

So, in the first expression (5.58) all the coefficients of γ are positive for any q ∈ (0, 1). Further,

the coefficients of first and zero order in γ from the second expression (5.59) can be evaluated

for (1− k)/k < q < 1, in the following maner

2kq2 + ((1−k)−kq)ϕ+ (1−k)q(φ− 1) > q[2(1− k) + (1− k)(φ− 1)] + ((1− k)− kq)ϕ> q(1− k)(1 + φ)− ε > 0,

kq2 + (1− k)q − k3((1− k)− kq) > kq2 + (1− k)q + ( 1−ωω − ε)((1− k)− kq) > 0.

Consequently, when q ∈ ((1 − k)/k, 1), then the strong stability requirement (5.55) is

fulfilled for all the five eigenvalues.

Page 77: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 65

Accuracy

To prove the accuracy of the θ-scheme applied to the linearized Oldroyd system (5.96)-(5.98),

one must compare the damping factor Ω given in (5.21) with the damping factor Ωe given in

(5.10) corresponding to the continuous linearized Oldroyd system (5.1). More precisely one

must compare the coefficients ki , i ∈ 2, 3,M,B,C from (5.22) corresponding to the damping

factor Ω with the coefficients Ki , i ∈ 2, 3,M,B,C corresponding to the the damping factor

Ωe given in (5.11). For this task one needs to expand the coefficients (5.22) around ∆t = 0.

First let us consider the following expansions of type (A.1) around ∆t = 0 for the inverse of

quantities defined in (5.19)

1

a= θ

∆t

We− ω

(

θ∆t

We

)2

+ ω2

(

θ∆t

We

)3

+ O(∆t4),

1

a′= θ′

∆t

We− (1− ω)

(

θ′∆t

We

)2

+ (1− ω)2(

θ′∆t

We

)3

+ O(∆t4),

1

d= θ

∆t

Re− kl

(

θ∆t

Re

)2

+ (kl)2(

θ∆t

Re

)3

+ O(

∆t4)

,

1

d′= θ′

∆t

Re− (1− k)l

(

θ′∆t

Re

)2

+ ((1− k)l)2(

θ′∆t

Re

)3

+ O(∆t4).

With the help of these Taylor expansions and those of (A.2)-(A.3), and taking into account

that θ′ = 1−2θ, the coefficients kj , j ∈ 2, 3, B, C,M from (5.22) can be approximated until

the terms of second order in ∆t

k3 ≈ 1− ∆t

We+ [(1− θ)2 − (2θ2 − 4θ + 1)ω]

(

∆t

We

)2

,

kM ≈ −θ(2− θ) α

|ξ|2∆t2

ReWe,

kB ≈ −∆t

We+ θ(2− θ)l ∆t2

ReWe+ [(1− θ)2 − ω(2θ2 − 4θ + 1)]

(

∆t

We

)2

,

kC ≈ −∆t

Re+ θ(2− θ)l ∆t2

ReWe+ l[(1− θ)2 − k(2θ2 − 4θ + 1)]

(

∆t

Re

)2

,

k2 ≈ 1− l∆tRe− α l

1− αθ(2− θ)∆t2

ReWe+ l2[(1− θ)2 − k(2θ2 − 4θ + 1)]

(

∆t

Re

)2

.

One can observe that the first and zeroth order terms in ∆t from the coefficients ki , i ∈2, 3,M,B,C are the same as those of the coefficients Ki. By comparing the second order

terms, it is easy to deduce that if

(5.60) 2θ2 − 4θ + 1 = 0

Page 78: Stability Analysis and Numerical Simulation of Non-Newtonian

66 Chapter 5. Stability analysis

then also these terms are equal. The parameter θ was initially chosen in the interval (0, 1),

therefore only the solution

(5.61) θ = 1−√

2

2

is acceptable.

The conclusion is that the fractional step θ-scheme applied to the linear Oldroyd system

is second order accurate if the parameter θ satisfies condition (5.60), otherwise the time

splitting scheme is only first order accurate.

Choice of the splitting parameters k and ω

In general, the splitting parameters k and ω have to be chosen in order to have the possibility

to invert the same matrix at each partial time step of the integration procedure (5.18).

Therefore it is required that

A1 θ∆t = A3 θ′∆t,

i.e.

a θ = a′ θ′ ⇔ ω θ = (1− ω)θ′,

d θ = d′ θ′ ⇔ k θ = (1− k)θ′.Thus, the optimal choice of the splitting parameters k and ω is

(5.62) ω = k =θ′

1− θ .

Plots of the eigenvalues

In Fig. 5.2 the dependence of the eigenvalues λ3, λ4, λ5 given in (5.23) on the time step size

by Weißenberg number are plotted for different values of α ∈ 0, 0.2, 0.5, 0.99. These plots

correspond to two values, 0.01 and 100, of the wave vector |ξ| multiplied by the rate We/Re

and to the splitting parameters k and ω defined in (5.62). The eigenvalues λ1,2 were already

represented in Fig. 5.24.

In these figures one can observe that the absolute value of all eigenvalues of the damping

matrix (5.21) are less than 1 in the whole variation of the time step size along the positive

axis, just as demonstrated before in this subsection. Moreover, the absolute value of all the

eigenvalues increases asymptotically to one for time step sizes tending to zero, as expected

for a good time discretization scheme.

Page 79: Stability Analysis and Numerical Simulation of Non-Newtonian

5.1. Spectral analysis of the linearized Oldroyd system 67

10−5

100

105

1010

−0.5

0

0.5

3

α = 0α = 0.2α = 0.5α = 0.99

10−5

100

105

1010

−0.5

0

0.5

3

α = 0α = 0.2α = 0.5α = 0.99

10−5

100

105

10100

0.2

0.4

0.6

0.8

1|λ

4|

α = 0α = 0.2α = 0.5α = 0.99

10−5

100

105

10100

0.2

0.4

0.6

0.8

1|λ

4|

α = 0α = 0.2α = 0.5α = 0.99

10−5

100

105

10100

0.2

0.4

0.6

0.8

1|λ

5|

α = 0α = 0.2α = 0.5α = 0.99

10−5

100

105

10100

0.2

0.4

0.6

0.8

1|λ

5|

α = 0α = 0.2α = 0.5α = 0.99

Figure 5.2: Representation of the eigenvalues λ3, λ4, λ5 as functions of ∆tWe at different

values of α for (left) |ξ|WeRe = 0.01; (right) |ξ|We

Re = 100 .

Page 80: Stability Analysis and Numerical Simulation of Non-Newtonian

68 Chapter 5. Stability analysis

5.2 Contribution of the β-term

5.2.1 Influence of the β-term on the stability of the constitutive

equation

The next step in the evaluation of the stability properties of the Oldroyd system in the con-

tinuous case is to study the influence of the β-term. For this task, first the pure constitutive

law formulated as a stress problem will be examined.

Problem 5.3. Given a velocity field u, solve in Ω×R+ the following equation for the stress

tensor τ

(5.63)∂τ

∂t+ (u · ∇)τ + βa(τ,∇u) +

1

Weτ =

WeD,

with the boundary condition τ|Γ−

= τΓ and the initial condition τ|t=0= τ0.

Before that, let us remember some simple stability properties of linear partial-differential

equations.

Linear stability analysis

Let us consider the initial-value problem for a scalar field ϕ(x, t)

(5.64)

∂ϕ

∂t+ λϕ = 0, t > 0, 0 ≤ x ≤ x

ϕ(0, x) = ϕ0(x),

whose solution is

(5.65) ϕ(x, t) = ϕ0(x) e−λ t.

For λ ≥ 0 the solution is bounded in time, but if λ < 0 the solution is blowing up.

Generally, the stable regime is given by the condition that the real part of λ is positive.

(5.66) R (λ) ≥ 0,

and for more details the reader is referred to [55].

Further let us consider the initial-boundary-value problem for a scalar field ϕ(x, t)

(5.67)

∂ϕ

∂t+ c

∂ϕ

∂x+ λϕ = 0, t > 0, 0 < x ≤ x

ϕ(x, 0) = ϕ0(x),

ϕ(0, t) = ϕ1(t).

We consider the case c ≥ 0, so that the waves are traveling from small x to large x. The

Page 81: Stability Analysis and Numerical Simulation of Non-Newtonian

5.2. Contribution of the β-term 69

xx

t

x = c t

R1

R2

Figure 5.3: The characteristics x = ct+ x0 in the plane x − t.

characteristics are given by

x = ct+ x0.

The characteristic through the origin, splits the x − t plane into two regions, see Fig. 5.3.

In the region R1, x > ct, the solution is determined by the initial condition at t = 0 and the

characteristics originate from the x-axis. In the region R2, x < ct, the solution is given by

the boundary condition at x = 0. Here the characteristics originate from the t-axis. Thus,

the solution is determined by where the characteristics come from and is given by

(5.68) ϕ(x, t) =

ϕ0(x− ct)e−λt, x > ct,

ϕ1

(

t− xc

)

e−λxc , t > x

c .

In the region R1 for the current problem, the solution depends on time by means of the

function ϕ0 and by exp−λt which is upper bounded either by 1 if λ ≥ 0 or by exp−λx/cif λ < 0. In the region R2 the solution depends on time only through the function ϕ1. Hence,

for problem (5.67) the solution is bounded if the functions ϕ0 and ϕ1 are bounded.

In the limit case c→ 0 the region R1 will cover the whole computational domain and the

solution (5.68) will tend towards solution (5.65) and instability can arise if condition (5.66)

is not satisfied.

Generally, if c is not constant it can occur that the characteristics are closed curves in

the region R1. In this situation, if condition (5.66) is not satisfied, the solution will grows

unlimited in time along those characteristics.

Page 82: Stability Analysis and Numerical Simulation of Non-Newtonian

70 Chapter 5. Stability analysis

Stability of the constitutive equation without the stress convective term

By neglecting the stress transport term (u ·∇)τ in equation (5.63) the initial-boundary-value

problem 5.3 becomes an initial-value problem.

Problem 5.4. Given a velocity field u, solve in Ω×R+ the following equation for the stress

tensor τ

(5.69)∂τ

∂t+ βa(τ,∇u) +

1

Weτ =

WeD,

with the initial condition τ|t=0= τ0.

It is common in this work, in the two-dimensional case, that the symmetrical stress ten-

sor is written as a vector field with three components [τxx τxy τyy]T . So, equation (5.69)

transforms to

(5.70)∂

∂t

τxxτxyτyy

+ A

τxxτxyτyy

= R.

Here, A is a 3× 3 matrix corresponding to the linear term in τ and the β-term, and R is a

three component vector corresponding to the right-hand side

(5.71) R =2α

We

Dxx

Dxy

Dyy

.

Taking into account that the velocity field u satisfies the continuity equation (1.4), the matrix

A can be set in the form:

(5.72) A =

1

We− 2aDxx 2(−aDxy +Wyx) 0

−(aDxy +Wyx)1

We−aDxy +Wyx

0 −2(aDxy +Wyx)1

We+ 2aDxx

.

The three eigenvalues of matrix A depend on the velocity field and the Weißenberg number

in the following way

(5.73)

λs =1

We,

λ± =1

We± 2√

a2(D2xx +D2

xy)−W 2yx,

Page 83: Stability Analysis and Numerical Simulation of Non-Newtonian

5.2. Contribution of the β-term 71

and the corresponding eigenvectors are

vs =

[−aDxx +Wyx

aDxx, 1,

aDxx +Wyx

aDxx

]

, if a 6= 0; vs = [1, 0, 1] , if a = 0,

v± =

−aDxx +Wyx

aDxx ±√

a2(D2xx +D2

xy)−W 2yx

, 1,aDxx +Wyx

aDxx ∓√

a2(D2xx +D2

xy)−W 2yx

.

The stability requirement (5.66) means that the real parts of all three eigenvalues λs, λ+

and λ− have to be positive. In the case that

a2(D2xx +D2

xy)−W 2yx ≥ 0,

one obtains an upper stability limit of the Weißenberg number

(5.74) Wecr =(

2√

a2(D2xx +D2

xy)−W 2yx

)−1

.

If a = 0, the radicand in (5.73) is always negative, so no stability boundary for the Weißen-

berg number exists. In the case of Oldroyd A or B model (|a| = 1) the condition

(5.75) R (λ−) = R(

1

We− 2√

−det(∇u))

≥ 0,

must be fulfilled to have stability. This condition shows, that whenever the determinant of

the velocity gradient tensor becomes negative in any point of the domain, it appears the

possibility that the stability requirement (5.66) will not be fulfilled in that subdomain and,

as it will be exemplified in Chapter 7, the computations break down.

Stability of the constitutive equation

We turn now to the initial-boundary-value problem 5.3. Here, the stress convective term

(u · ∇)τ influences the solution of problem 5.3.

Let us consider the Fourier transform τ of the unknown field τ

(5.76) τ(x, t) =

R2

exp (−iξ · x)τ (ξ, t) dξ.

By writing the symmetrical stress tensor τ as a vector field then equation (5.63), transformed

in the spectral space, becomes

(5.77)∂

∂t

τxxτxyτyy

+ A

τxxτxyτyy

= R.

Here, matrix A differs from matrix A given in (5.72) by

(5.78) A = A− i(uxξ1 + uyξ2)I3

Page 84: Stability Analysis and Numerical Simulation of Non-Newtonian

72 Chapter 5. Stability analysis

and R is the right-hand side vector (5.71). The imaginary contribution on the diagonal of

matrix A does not influence the real parts of the eigenvalues. That means that the real parts

of the eigenvalues of matrix A are the same as the real parts of the eigenvalues of matrix A.

For problem 5.3 the characteristics equation is

d x

ux=

d y

uy= d t,

therefore, the characteristics are the streamlines. Following the same argumentation as for

the example problem (5.67), instability can arise in problem 5.3 in two situations. The one

situation is if the velocity field is zero in any point of the computational domain where the

stability condition (5.66) is violated. Better said, if there is a stagnation point in the part of

the domain where the real part of at least one eigenvalue of the matrix A, defined in (5.72),

is negative. The other situation is if streamlines are closed curves in the subdomain where

(5.66) is violated. Both situations are illustrated in chapter 7.

5.2.2 Influence of the nonlinearity β on the stability of the Oldroyd

system

In this subsection, the Oldroyd system without the convective terms is considered:

Problem 5.5. Solve in Ω× R+ the following system of equations

∂τ

∂t+ βa(τ,∇u) +

1

Weτ − 2α

WeD = fs,

∂u

∂t− 2(1− α)

RedivD − 1

Rediv τ +

1

Re∇p = f,

div u = 0,

with boundary and initial conditions

u|Γ = u|Γ , τ|Γ−

= τ|Γ−

,

u|t=0= u0, τ|t=0

= τ0.

Let (τ0, u0, p0) be a stationary solution of problem 5.5, and (τ , u, p) be a small pertur-

bation of the stationary solution such that

(5.79)

τ(x, t) = τ0(x) + τ (x, t),

u(x, t) = u0(x) + u(x, t),

p(x, t) = p0(x) + p(x, t).

Introducing the dispartment (5.79) in problem 5.5, after neglecting the second order pertur-

bation terms, the following linearized problem for the perturbed fields is obtained:

Page 85: Stability Analysis and Numerical Simulation of Non-Newtonian

5.2. Contribution of the β-term 73

Problem 5.6. Solve in Ω× R+ for τ , u and p the following system of equations

(5.80)

∂τ

∂t+ βa(τ ,∇u0) + βa(τ

0,∇u) +1

Weτ − 2α

WeDu = 0,

∂u

∂t− 2(1− α)

RedivDu− 1

Rediv τ +

1

Re∇p = 0,

div u = 0,

with boundary condition u|Γ = 0 and initial conditions u|t=0= 0 and τ|t=0

= 0.

We consider now formally the Fourier transforms (τ , u, p) of the unknown fields (τ , u, p)

(5.81)

τ(x, t) =

R2

exp (−iξ · x)τ (ξ, t) dξ,

u(x, t) =

R2

exp (−iξ · x)u(ξ, t) dξ,

p(x, t) =

R2

exp (−iξ · x)p(ξ, t) dξ.

Similar as in the previous subsections, the system (5.80) can be written as an initial value

system of type (5.6) with

(5.82) U = (τxx, τxy, τyy, ux, uy),

and the matrix A of the form

(5.83) A5 =

A1

WeB + B

1

ReC

1− αRe

‖ξ‖2I2

.

The matrices B and C are those defined in (5.8), the matrix A is similar to (5.72), but with

D and W corresponding to Du0 and Wu0, respectively and B is defined as

B = i

−2aτ0xxξ1 − (1 + a)τ0

xyξ2 (1− a)τ0xyξ1

1

2[(1− a)τ0

xx − (1 + a)τ0yy]ξ2

1

2[(1− a)τ0

yy − (1 + a)τ0xx]ξ1

(1− a)τ0xyξ2 −2aτ0

yyξ2 − (1 + a)τ0xyξ1

.

The case a = 1

In the case of a = 1, for the Oldroyd-B model, one eigenvalue of the matrix A5 is

λξ =1− αRe

‖ξ‖2,

Page 86: Stability Analysis and Numerical Simulation of Non-Newtonian

74 Chapter 5. Stability analysis

and the other four eigenvalues can be obtained from the solutions X of the polynomial

equation

(

1

We− λ)(

1− αRe

‖ξ‖2 − λ)

[

4 det (∇u0) +

(

1

We− λ)2]

‖ξ‖2

+

(

1

We− λ)2 α

We‖ξ‖2 −

[

ξ12τ0xx + 2ξ1ξ2τ

0xy + ξ22τ

0yy

]

‖ξ‖2Re

+

(

1

We− λ)

[

2Dxxξ1ξ2 −Dxyz +Wyx‖ξ‖2] [

−ξ1ξ2τ0xx + zτ0

xy + ξ1ξ2τ0yy

] 2

Re

− 4α

ReWe[Dxxz + 2Dxyξ1ξ2]

2 +4

Re[Dxxz + 2Dxyξ1ξ2]

[Dxy‖ξ‖2 −Wyxz]τ0xy

+[Dxxξ1 + (Dxy +Wyx)ξ2]ξ1τ0xx + [−Dxxξ2 + (Dxy −Wyx)ξ1]ξ2τ

0yy

= 0.

By fixing the direction of the wave vector to ξ = (ξ1, 0), the previous equation becomes

(5.84)

(

1

We− λ)(

1− αRe‖ξ‖2 − λ

)

[

4 det (∇u0) +

(

1

We− λ)2]

+

(

1

We− λ)2( α

We− τ0

xx

)

+ 2

(

1

We− λ)

(Wyx −Dxy)τ0xy −

WeD2xx

+4Dxx

[

Dxxτ0xx + (Dxy −Wyx)τ

0xy

]

‖ξ‖2Re

= 0,

if the wave vector is ξ = (0, ξ2), the previous equation becomes

(5.85)

(

1

We− λ)(

1− αRe‖ξ‖2 − λ

)

[

4 det (∇u0) +

(

1

We− λ)2]

+

(

1

We− λ)2( α

We− τ0

yy

)

− 2

(

1

We− λ)

(Wyx +Dxy)τ0xy −

WeD2xx

+4Dxx

[

Dxxτ0yy − (Dxy +Wyx)τ

0xy

]

‖ξ‖2Re

= 0,

whereas if the wave vector is ξ = (ξ1,±ξ1), one has

(5.86)

(

1

We− λ)(

1− αRe

‖ξ‖2 − λ)

[

4 det (∇u0) +

(

1

We− λ)2]

+

(

1

We− λ)2(

α

We− τ0

xx ± 2τ0xy + τ0

yy

2

)

‖ξ‖2Re

+

(

1

We− λ)

(Dxx ±Wyx)(

τ0yy − τ0

xx

) ‖ξ‖2Re− 4D2

xy

α

We

‖ξ‖2Re

±2Dxy

[

2Dxyτ0xy + (Dxx ±Dxy ±Wyx)τ

0xx ± (Dxy ∓Dxx −Wyx)τ

0yy

] ‖ξ‖2Re

= 0.

Page 87: Stability Analysis and Numerical Simulation of Non-Newtonian

5.2. Contribution of the β-term 75

It is interesting to observe that if the wave vector tends to zero, one obtains the reduced

equation

(5.87) λ

(

1

We− λ)

[

4 det (∇u0) +

(

1

We− λ)2]

= 0,

which possess the solutions

λ0 = 0, λs =1

We, λ± =

1

We± 2√

−det(∇u0).

However, λs and λ± are the eigenvalues of the pure stress constitutive equation (5.73). That

means, the eigenvalues of the Oldroyd system without the convective terms (5.80) cannot

be better than the eigenvalues of the stress equation without the convective term (5.69).

Accordingly, the stability requirement for the Oldroyd system (5.66) implies nothing less

than the stability limit of the Weißenberg number for the stress constitutive equation Wecr.

At this point, it is to remark that the restriction imposed in [54] on the relaxation time

for the test example used there, is nothing else than the stability restriction (5.75) founded

in this work.

The case a = 0

The eigenvalue analysis of matrix A5, (5.83), in the case a = 0, provides two eigenvalues

given by

λξ =1− αRe

‖ξ‖2, λs =1

We,

and the other three eigenvalues are solutions of the following equation

1

2

(

1

We− λ)2(

1− αRe

‖ξ‖2 − λ)

+ 2W 2yx

(

1− αRe

‖ξ‖2 − λ)

+

(

1

We− λ)(

α

We

‖ξ‖22− τxyξ1ξ2+(τyy−τxx)

z

4

)

1

Re+Wyx [τxyz + (τyy−τxx)ξ1ξ2]

1

Re= 0.

By fixing the direction of the wave vector to ξ = (ξ1, 0) or ξ = (0, ξ2), the previous equation

becomes

(5.88)

1

2

(

1

We− λ)2 (

1− αRe

‖ξ‖2 − λ)

+ 2W 2yx

(

1− αRe

‖ξ‖2 − λ)

+

(

1

We− λ)(

α

We± τyy − τxx

2

) ‖ξ‖22Re

±Wyxτxy‖ξ‖2Re

= 0,

whereas if the wave vector is ξ = (ξ1,±ξ1), one has

(5.89)

1

2

(

1

We− λ)2 (

1− αRe

‖ξ‖2 − λ)

+ 2W 2yx

(

1− αRe

‖ξ‖2 − λ)

+

(

1

We− λ)

( α

We∓ τxy

) ‖ξ‖22Re

±Wyx(τyy − τxx)‖ξ‖22Re

= 0.

Page 88: Stability Analysis and Numerical Simulation of Non-Newtonian

76 Chapter 5. Stability analysis

Influence of the convective terms on the stability of the Oldroyd system

Introducing the dispartments (5.79) in the full Oldroyd system (2.4), after neglecting the

second order perturbation terms, the following linearized problem for the perturbed fields is

obtained:

Problem 5.7. Solve in Ω× R+ for τ , u and p the following system of equations

∂τ

∂t+ (u0 · ∇)τ + (u · ∇)τ0 + βa(τ ,∇u0) + βa(τ

0,∇u) +1

Weτ − 2α

WeDu = 0,

∂u

∂t+ (u0 · ∇)u+ (u · ∇)u0 − 2(1− α)

RedivDu− 1

Rediv τ +

1

Re∇p = 0,

div u = 0,

with boundary condition u|Γ = 0 and initial conditions u|t=0= 0 and τ|t=0

= 0.

Introducing the Fourier transforms (5.81) problem 5.7 can be written as an initial value

problem of type (5.6) with the unknown vector U defined by (5.82) and the matrix A of the

form

A5 = A5 − i(u0xξ1 + u0

yξ2) I5 +D5,

where A5 is defined by (5.83) and

D5 =

[

0 B

0 D

]

, B =

∂τ0xx

∂x

∂τ0xx

∂y

∂τ0xy

∂x

∂τ0xy

∂y

∂τ0yy

∂x

∂τ0yy

∂y

, D =

∂u0x

∂x

∂u0x

∂y

∂u0y

∂x

∂u0y

∂y

.

The convective terms (u0 ·∇)τ and (u0 ·∇)u contributes only to the diagonal of the matrix

A5 through pure imaginary terms of the form i(u0xξ1 +u0

yξ2). These imaginary terms do not

influence the real parts of the eigenvalues. Compared to the influence of the convective term

on the pure stress equation with given velocity field, in the case of the Oldroyd system, the

convective terms contribute also with the expression (u·∇)τ 0 and (u·∇)u0, which include the

addition of matrix D5 to A5. The real elements of matrix D5 could have a certain influence

on the real parts of the eigenvalues. Thus the real parts of the eigenvalues of matrix A5

could differ from the real parts of the eigenvalues of matrix A5. This result was expected,

since in the full Oldroyd system both the stress and the velocity are transported along the

streamlines.

Stability of the time semi-discretized constitutive equation

Our purpose now, is to analyze the influence of the β-term on the stability of the time

discretization scheme. Therefore, let us consider the stress equation without the stress

Page 89: Stability Analysis and Numerical Simulation of Non-Newtonian

5.2. Contribution of the β-term 77

convective term (5.69) with given stationary velocity field and α = 0. Due to the complex

calculations necessary for the θ-scheme time discretization, at this point the analysis is

restricted to the explicit Euler scheme. Applying the explicit Euler time discretization to

equation (5.70)

τxxτxyτyy

n+1

= (I3 −∆tA)

τxxτxyτyy

n

The eigenvalues of the damping matrix I3 −∆tA are

(5.90)

λs = 1− ∆t

We,

λ± = 1−∆t

(

1

We± 2√

a2(D2xx +D2

xy)−W 2yx

)

,

To have a stable scheme, the absolute value of the eigenvalues (5.90) are to be less than 1.

Thus the time step must fulfill the condition

∆t < 2 We.

If the radicand is positive, then for a 6= 0 the upper stability limit Wecr (5.74), will be

obtained also for the time approximation scheme.

The θ-scheme time discretization contains two explicit steps, so it is to anticipate that

for the stability of the θ-scheme one needs, if a = ±1, nothing less than the fulfillment of

condition (5.75) in any point of the computational domain.

Naturally, if the stress convective term is taken into account, this stability restriction is

essential only in the two special situations, in which the streamlines are closed curves or

there exists a stagnation point in the region where condition (5.75) is violated.

Remark 5.8.

Let us consider the semi-discretized stress equation (4.8) from the second step of algo-

rithm 4.4 in the form

B

τxxτxyτyy

n+1−θ

= c5

τxxτxyτyy

n+θ

+ R.

The matrix B is given by

B =

c4 − 2aDxx 2(−aDxy +Wyx) 0

−(aDxy +Wyx) c4 −aDxy +Wyx

0 −2(aDxy +Wyx) c4 + 2aDxx

,

Page 90: Stability Analysis and Numerical Simulation of Non-Newtonian

78 Chapter 5. Stability analysis

R is some right-hand side vector and the coefficients ci, i ∈ 4, 5 are

c4 =1

(1− 2θ)∆t+

1− ωWe

, c5 =1

(1− 2θ)∆t− ω

We.

The eigenvalues of the matrix B are

λs = c4,

λ± = c4 ± 2√

a2(D2xx +D2

xy)−W 2yx,

and by imposing that all eigenvalues are to be positive, it yields

∆t ≤ We

(1− 2θ)(

2|a|We · R(√

a2(D2xx +D2

xy)−W 2yx

)

− 1 + ω) .

Hence, condition (4.9), declared in [62] as sufficient for solution existence, could be verified.

But for any time discretization scheme one can see that the absolute values of the damping

matrix eigenvalues are less than 1, and so the choice of ∆t to be sufficientely small is not a

guarantee of stability of the numerical scheme.

Page 91: Stability Analysis and Numerical Simulation of Non-Newtonian

5.3. A priori stability estimation of the linear Oldroyd problem 79

5.3 A priori stability estimation of the linear Oldroyd

problem

5.3.1 A priori stability estimation of the linear Oldroyd problem in

weak form

The weak formulation of the linear Oldroyd problem 5.1 can be written with the spaces and

inner products defined in (3.1)-(3.2) as:

Problem 5.9. Find (τ, u, p) ∈ Θ× V0 ×Q0 such that

(5.91)

Wed

dt(τ, σ) + (τ, σ)− 2α(Du, σ) = (fs, σ),

Red

dt(u, v) + 2(1− α)(Du,Dv) + (τ,Dv)− (p, div v) = (f, v),

(div u, q) = 0,

for all (σ, v, q) ∈ Θ× V0 ×Q.

The results concerning the stability of the weak linear Oldroyd problem 5.9 are given by

the following theorem.

Theorem 5.10. For α ∈ [0, 1) there exist positive coefficients Ci, i ∈ 0, 1, 2, 3, 4 such that

for problem 5.9 the following stability condition holds

(5.92)We

2

d

dt‖τ‖2 +

Re

2C0

d

dt‖u‖2 + C1‖τ‖2 + C2‖Du‖2 ≤ C3‖fs‖2 + C4‖f‖2.

Step 1: Taking σ = τ in (5.91)1, v = u in (5.91)2, q = p in (5.91)3 and adding the three

equations, one obtains

(5.93)We

2

d

dt‖τ‖2+

Re

2

d

dt‖u‖2+‖τ‖2+2(1−α)‖Du‖2 = (2α−1)(τ,Du)+(fs, τ)+(f, u).

Taking into account that for bounded Ω and u ∈ V0 the Poincare inequality holds

‖u‖ ≤ D0‖Du‖,

with D0 = D0(Ω), and using Young’s inequalities for the right-hand side terms

(fs, τ) ≤1

4ε1‖fs‖2 + ε1‖τ‖2,

(f, u) ≤ 1

4D20ε2‖f‖2 + ε2‖Du‖2,

(2α− 1)(τ,Du) ≤ |2α− 1|2ε

‖τ‖2 +|2α− 1|ε

2‖Du‖2,

equation (5.93) transforms into the inequality (5.92) where the coefficients Ci, i ∈ 0, 1, 2, 3, 4are given by

C0 = 1, C1 = 1− ε1 −|2α− 1|

2ε, C2 = 2(1− α)− ε2 −

|2α− 1|ε2

, C3 =1

4ε1, C4 =

1

4ε2.

Page 92: Stability Analysis and Numerical Simulation of Non-Newtonian

80 Chapter 5. Stability analysis

For α ≤ 0.5 the coefficients C1 and C2 are positive when taking ε = 1, 0 < ε1 ≤ 0.5 and

0 < ε2 ≤ 0.5. In the case that α > 0.5 one has to choose

0 < ε1 < 1− 2α− 1

2ε, 0 < ε2 < 2(1− α)− 2α− 1

2ε.

But such ε1,2 exists only if 4α2 + 4α − 7 < 0, which implies α < 0.9142. This result is

unsatisfactory for α near 1, therefore a second approach was chosen in the following.

Step 2: Let α be in (0, 1). Taking σ = τ in (5.91)1, v = u in (5.91)2 and multiplying this

equation by 2α, q = p in (5.91)3 and adding the three equations, one obtains

(5.94)We

2

d

dt‖τ‖2 +

Reα

2

d

dt‖u‖2 + ‖τ‖2 + 2α(1− α)‖Du‖2 = (fs, τ) + α(f, u).

Using Young’s inequalities for the right-hand side terms, combined with the Poincare in-

equality, for ε1 = 1/2 and ε2 = 1− α, equation (5.94) transforms into the inequality (5.92)

where the coefficients Ci, i ∈ 0, 1, 2, 3, 4 are given by

C0 = α, C1 = 1, C2 = α(1− α), C3 =1

2, C4 =

α

4D20(1− α)

.

For α = 1, as one can observe by means of relation (5.94), a priori stability holds only if

no external forces f act on the fluid.

5.3.2 A priori stability estimation of the fractional θ-scheme cou-

pled with the finite element approximation

For the Navier-Stokes equations, Kloucek and Rys in [41] show conditional stability and

convergence of the fractional step θ-scheme coupled with the finite element approximation.

Muller-Urbaniak [49] stated, also for the Navier-Stokes equations, the unconditional stability

and a suboptimal error estimate for semi-discretization in time and the linear case. The

stability analysis presented in this section is based on the stability analysis of the θ-scheme

applied to the Navier-Stokes equations in [41] and is used to obtain a priori estimates for

the linearized Oldroyd system.

However, the stability results from [41] are given for the full Navier-Stokes system, due

to the strong coupling of the momentum equation and the constitutive law in the Oldroyd

system, here not only the nonlinear terms in the constitutive law, but also the nonlinear

velocity term in the momentum equation were neglected. The finite element formulation of

the linear Oldroyd problem 5.1 can be written accordingly to problem 3.16 as:

Problem 5.11. Find (τ, u, p) ∈ Θh × V0h ×Q0h such that 2

(5.95)

Wed

dt(τ, σ) + (τ, σ) − 2α(Du, σ) = (fs, σ),

Red

dt(u, v) + 2(1− α)(Du,Dv) + (τ,Dv) − (p, div v) = (f, v),

(div u, q) = 0,

2the index h from the inner products will be drop for simplicity of the notations

Page 93: Stability Analysis and Numerical Simulation of Non-Newtonian

5.3. A priori stability estimation of the linear Oldroyd problem 81

for all (σ, v, q) ∈ Θh × V0h ×Qh.

The θ-scheme algorithm 4.4 corresponding to the linearized Oldroyd system problem 5.11

provides the following system of equations:

Problem 5.12. Given u0 and τ0, then for n > 0 find un+θ, un+1−θ, un+1 in V0h, τn+θ,

τn+1−θ, τn+1 in Θh and pn+θ, pn+1 in Qh such that

(5.96)

We

θ∆t(τn+θ− τn, σ) + ω(τn+θ, σ)− 2α(Dun+θ, σ) = −(1− ω)(τn, σ) + (fns , σ),

Re

θ∆t(un+θ − un, v) + k(1− α)(2Dun+θ, Dv) + (τn+θ , Dv)− (pn+θ, div v)

= −(1− k)(1− α)(2Dun, Dv) + (fn, v),

(div un+θ, q) = 0,

(5.97)

We

θ′∆t(τn+1−θ − τn+θ , σ) + (1− ω)(τn+1−θ, σ) = −ω(τn+θ, σ)

+2α(Dun+θ, σ) + (fns , σ),

Re

θ′∆t(un+1−θ − un+θ, v) + (1− k)(1− α)(2Dun+1−θ, Dv)

= −k(1− α)(2Dun+θ, Dv)− (τn+θ , Dv) + (pn+θ, div v) + (fn, v),

(5.98)

We

θ∆t(τn+1 − τn+1−θ, σ) + ω(τn+1, σ)− 2α(Dun+1, σ) = −(1− ω)(τn+1−θ, σ)

+(fn+1s , σ),

Re

θ∆t(un+1 − un+1−θ, v) + k(1− α)(2Dun+1, Dv) + (τn+1, Dv)− (pn+1, div v)

= −(1− k)(1− α)(2Dun+1−θ, Dv) + (fn+1, v),

(div un+1, q) = 0,

for all (σ, v, q) ∈ Θh × V0h ×Qh.

The results concerning the stability of the scheme comprised in problem 5.12 are given by

the following theorem.

Theorem 5.13. Under the assumption that θ ∈ (0, 0.5], k ∈ (0, 1), α ∈ [0, 1), δ ∈ (0, 0.5],

D2 > 0 and that the time step ∆t and mesh size h are connected with the data by the stability

condition

(5.99)∆t

h2≤ min

Re

2(1− α)(1− k)D2 θ,

Re(3− k)δ4(1− α)[(1− k)2 +D2]θθ′2

,

Page 94: Stability Analysis and Numerical Simulation of Non-Newtonian

82 Chapter 5. Stability analysis

then for arbitrary N > 0 there exist positive coefficients Ci, i ∈ u, τ, f, fs, 0, 1, ..., 10, such

that the following a priori stability estimate holds

(5.100)

Cu‖uN+1‖2 + Cτ‖τN+1‖2 + C1∆t

N+1∑

n=0

‖Dun‖2 + C2∆t

N∑

n=0

‖Dun+θ‖2

+C3∆t

N∑

n=0

‖Dun+1−θ‖2 + C4

N∑

n=0

(

‖un+θ − un‖2 + ‖un+1 − un+1−θ‖2)

+C5

N∑

n=0

‖un+1−θ − un+θ‖2 + C6∆t

N∑

n=0

‖Dun+θ −Dun‖2

+2θ∆tN+1∑

n=1

‖τn‖2 + C7∆tN∑

n=0

‖τn+1−θ‖2 + C8∆tN∑

n=0

‖τn+θ‖2

+Cτ

N∑

n=0

‖τn+θ − τn‖2 + C9

N∑

n=0

‖τn+1 − τn+1−θ‖2 + C10

N∑

n=0

‖τn+1−θ − τn+θ‖2

≤ Cu ‖u0‖2 + Cτ‖τ0‖2 + Cf∆t

N+1∑

n=0

‖fn‖2 + Cfs∆t

N+1∑

n=0

‖fns ‖2.

The proof of this theorem is structured in six main steps

Step 1: By taking v = un+θ in (5.96)2 and q = pn+θ in (5.96)3 one gets

Re

θ∆t‖un+θ‖2 + 2k(1− α)‖Dun+θ‖2

=Re

θ∆t(un, un+θ)− 2(1− k)(1− α)(Dun, Dun+θ)− (τn+θ , Dun+θ) + (fn, un+θ).

Using Schwarz’s equality, one obtains the following formulation after multiplication with

2θ∆t

Re‖un+θ‖2 + 2(1 + k)(1− α)θ∆t‖Dun+θ‖2

= Re‖un‖2 −Re‖un+θ − un‖2 − 2(1− k)(1− α)θ∆t‖Dun‖2

+2(1− k)(1− α)θ∆t‖Dun+θ −Dun‖2 − 2θ∆t(τn+θ, Dun+θ) + 2θ∆t(fn, un+θ).

Due to the fact that the finite element spaces Vh and Vh, defined in (3.11)-(3.12), are finite

dimensional they can be equipped with equivalent norms induced by L2(Ω) and H10 (Ω).

Moreover, since condition (3.10) holds, from this equivalence results the inequality

(5.101) D1‖un+θ − un‖2 ≤ ‖Dun+θ −Dun‖2 ≤ D2h−1‖un+θ − un‖2,

where the constants D1 and D2 are positive and D2 depends on the degree of the polynomial

Page 95: Stability Analysis and Numerical Simulation of Non-Newtonian

5.3. A priori stability estimation of the linear Oldroyd problem 83

approximation. From the previous equality follows

(5.102)

Re‖un+θ‖2 −Re‖un‖2 + 2θ∆t(τn+θ, Dun+θ) + 2(1 + k)(1− α)θ∆t‖Dun+θ‖2

+2(1− k)(1− α)θ∆t‖Dun‖2 +

[

Re− 2(1− k)(1− α)D2 θ∆t

h2

]

‖un+θ − un‖2

≤ 2θ∆t(fn, un+θ).

Step 2: By taking v = un+1−θ in (5.97)2 one gets

Re

θ′∆t‖un+1−θ‖2 + 2(1− k)(1− α)‖Dun+1−θ‖2 =

Re

θ′∆t(un+θ, un+1−θ)

−2k(1− α)(Dun+θ, Dun+1−θ)− (τn+θ, Dun+1−θ) + (pn+θ, div un+1−θ) + (fn, un+1−θ).

Now, considering v = un+1−θ − un+θ in equation (5.96)2 one obtains

(pn+θ, div un+1−θ) = (pn+θ, div(un+1−θ − un+θ)) =Re

θ∆t(un+θ − un, un+1−θ − un+θ)

+(τn+θ, Dun+1−θ −Dun+θ) + 2(1− k)(1− α)(Dun, Dun+1−θ −Dun+θ)

+2k(1− α)(Dun+θ, Dun+1−θ −Dun+θ)− (fn, un+1−θ − un+θ).

By adding the two previous identities we get, after multiplication by 2θ′∆t, the following

relation

Re‖un+1−θ‖2 + Re‖un+1−θ − un+θ‖2 + 4(1− k)(1− α)θ′∆t‖Dun+1−θ‖2

+4k(1− α)θ′∆t‖Dun+θ‖2 + 2θ′∆t(τn+θ, Dun+θ)

= Re‖un+θ‖2 + 4(1− k)(1− α)θ′∆t(Dun, Dun+1−θ −Dun+θ)

+2Reθ′

θ(un+θ − un, un+1−θ − un+θ) + 2θ′∆t(fn, un+θ).

Denoting in the following b =(

θ′

θ

)2

and using Young’s inequalities,

2θ′

θ(un+θ − un, un+1−θ − un+θ) ≤ b

δ‖un+θ − un‖2 + δ‖un+1−θ − un+θ‖2

4(1− k)(1− α)θ′∆t(Dun, Dun+1−θ −Dun+θ)

≤ 1

δRe

(

2(1− k)(1− α)θ′∆t

h

)2

‖Dun‖2 + δRe‖un+1−θ − un+θ‖2

with 0 < δ ≤ 12 . Hence, the relation

(5.103)

Re ‖un+1−θ‖2 + 4k(1− α)θ′∆t‖Dun+θ‖2 + 2θ′∆t(τn+θ , Dun+θ)

+Re (1− 2δ)‖un+1−θ − un+θ‖2 + 4(1− k)(1− α)θ′∆t‖Dun+1−θ‖2

≤ Re ‖un+θ‖2 +Reb

δ‖un+θ − un‖2 +

1

δRe

(

2(1− k)(1− α)θ′∆t

h

)2

‖Dun‖2

+2(1− 2θ)∆t(fn, un+θ).

Page 96: Stability Analysis and Numerical Simulation of Non-Newtonian

84 Chapter 5. Stability analysis

is obtained. Because the term ‖un+θ−un‖2 is on the right-hand-side, it has to be estimated.

The best possibility is to get it from (5.96)2 with v = un+θ − un.The equations for the pressure (5.96)3 with q = pn+θ, and (5.98)2 with q = pn+θ and n

in place of n+ 1 gives

(pn+θ, div(un+θ − un)) = 0.

From these settings one gets after multiplying by θ∆t

Re‖un+θ − un‖2 + 2k(1− α)θ∆t‖Dun+θ −Dun‖2 + θ∆t(τn+θ , Dun+θ −Dun)= −2(1− α)θ∆t(Dun, Dun+θ −Dun) + θ∆t(fn, un+θ − un).

Using at this point (5.101) together with the Young’s inequality, one obtains

−2(1− α)θ∆t(Dun, Dun+θ −Dun) ≤ D2

2Re

(

2(1− α)θ∆t

h

)2

‖Dun‖2 +Re

2‖un+θ − un‖2

which implies the inequality

Re

2‖un+θ − un‖2 + 2k(1− α)θ∆t‖Dun+θ −Dun‖2 + θ∆t(τn+θ , Dun+θ −Dun)

≤ D2

2Re

(

2(1− α)θ∆t

h

)2

‖Dun‖2 + θ∆t(fn, un+θ − un).

After multiplying the last relation by 2b/δ and adding it to the inequality (5.103), one gets

the following inequality

(5.104)

Re ‖un+1−θ‖2 + 4(1− α)θ′∆t[

(1− k)‖Dun+1−θ‖2 + k‖Dun+θ‖2]

+Re(1− 2δ)‖un+1−θ − un+θ‖2 +4kb(1− α)θ∆t

δ‖Dun+θ −Dun‖2

+2θ′∆t(τn+θ, Dun+θ) +2bθ∆t

δ(τn+θ, Dun+θ −Dun)

≤ Re ‖un+θ‖2 +(1− k)2 +D2

δRe

(

2(1− α)θ′∆t

h

)2

‖Dun‖2

+2θ′∆t(fn, un+θ) +2bθ∆t

δ(fn, un+θ − un).

Step 3:By taking v = un+1 in (5.98)2 and q = pn+1 in (5.98)3 one gets the equality

Re

θ∆t‖un+1‖2 + 2k(1− α)‖Dun+1‖2 + (τn+1, Dun+1)

=Re

θ∆t(un+1−θ, un+1)− 2(1− k)(1− α)(Dun+1−θ, Dun+1) + (fn+1, un+1).

Page 97: Stability Analysis and Numerical Simulation of Non-Newtonian

5.3. A priori stability estimation of the linear Oldroyd problem 85

Similar to the procedure from Step 1, one obtains here

(5.105)

Re‖un+1‖2 −Re‖un+1−θ‖2 + 2θ∆t(τn+1, Dun+1)

+2(1 + k)(1− α)θ∆t‖Dun+1‖2 + 2(1− k)(1− α)θ∆t‖Dun+1−θ‖2

+

[

Re− 2(1− k)(1− α)D2 θ∆t

h2

]

‖un+1 − un+1−θ‖2 ≤ 2θ∆t(fn+1, un+1).

The results of these three steps, namely (5.102), (5.104) and (5.105), provide the desired

inequality

(5.106)

Re(

‖un+1‖2 − ‖un‖2)

+ 2(1− α)[(1 + k)θ + 2k(1− 2θ)]∆t‖Dun+θ‖2

+

[

2(1− k)(1− α)θ∆t− (1− k)2 +D2

δRe

(

2(1− α)θ′∆t

h

)2]

‖Dun‖2

+2(1− k)(1− α)(2− 3θ)∆t‖Dun+1−θ‖2 + 2(1 + k)(1− α)θ∆t‖Dun+1‖2

+

[

Re− 2(1− k)(1− α)D2 θ∆t

h2

]

(

‖un+θ − un‖2 + ‖un+1 − un+1−θ‖2)

+Re(1− 2δ)‖un+1−θ − un+θ‖2 +4kb(1− α)θ∆t

δ‖Dun+θ −Dun‖2

+2(1− θ)∆t(τn+θ , Dun+θ) +2bθ∆t

δ(τn+θ, Dun+θ−Dun) + 2θ∆t(τn+1, Dun+1)

≤ 2(1− θ)∆t(fn, un+θ) +2bθ∆t

δ(fn, un+θ − un) + 2θ∆t(fn+1, un+1).

In the next steps the stress equations will be considered.

Step 4: When taking σ = τn+θ in (5.96)1 one gets(

We

θ∆t+ ω

)

‖τn+θ‖2 − 2α(Dun+θ, τn+θ) =

(

We

θ∆t− (1− ω)

)

(τn+θ , τn) + (fns , τn+θ).

After multiplying by θ∆t and using Schwarz’s equality one has

(5.107)

[We + (1 + ω)θ∆t] ‖τn+θ‖2 − 4αθ∆t(Dun+θ, τn+θ)

= [We− (1− ω)θ∆t] ‖τn‖2 − [We− (1− ω)θ∆t] ‖τn+θ − τn‖2 + θ∆t(fns , τn+θ).

Step 5: By taking σ = τn+1−θ in (5.97)1 one gets(

We

θ′∆t+ 1− ω

)

‖τn+1−θ‖2 =

(

We

θ′∆t− ω

)

(τn+1−θ , τn+θ) + 2α(Dun+θ, τn+1−θ)

+(fns , τn+1−θ).

After multiplying by θ′∆t and using Schwarz’s equality one has

(5.108)[We + (2− ω)(1− 2θ)∆t] ‖τn+1−θ‖2 = 4αθ′∆t(Dun+θ, τn+1−θ)

+ [We− ωθ′∆t][

‖τn+θ‖2 − ‖τn+1−θ − τn+θ‖2]

+ θ′∆t(fns , τn+1−θ).

Page 98: Stability Analysis and Numerical Simulation of Non-Newtonian

86 Chapter 5. Stability analysis

Step 6: By taking σ = τn+1 in (5.98)1 and similar as step 4 one gets

(5.109)[We + (1 + ω)θ∆t] ‖τn+1‖2 − 4αθ∆t(Dun+1, τn+1) = θ∆t(fn+1

s , τn+1)

+ [We− (1− ω)θ∆t] ‖τn+1−θ‖2 − [We− (1− ω)θ∆t] ‖τn+1 − τn+1−θ‖2.Adding now the equalities (5.107)-(5.109) corresponding to the Steps 4,5 and 6 one obtains

the following identity for the stresses

[We + (1 + ω)θ∆t] ‖τn+1‖2 + [(2− ω)θ′ + (1− ω)θ]∆t ‖τn+1−θ‖2

+[ω(1− θ) + (1 + ω)θ]∆t ‖τn+θ‖2 + [We− ωθ′∆t]‖τn+1−θ − τn+θ‖2

+[We− (1− ω)θ∆t](

‖τn+1 − τn+1−θ‖2 − ‖τn+θ − τn‖2 − ‖τn‖2)

= 4αθ∆t[

(Dun+1, τn+1) + (Dun+θ, τn+θ)]

+ 4αθ′∆t (Dun+θ, τn+1−θ)

+θ∆t(fns , τn+θ) + θ′∆t(fns , τ

n+1−θ) + θ∆t(fn+1s , τn+1).

By adding the last stress identity to the velocity inequality (5.106) multiplied by 2α and

using the following Young’s inequalities

2(τn+1−θ − τn+θ, Dun+θ) ≤ (1− α)‖Dun+θ‖2 +1

1− α∆t ‖τn+1−θ − τn+θ‖2,

2(τn+θ, Dun+θ −Dun) ≤ (1− α)‖Dun+θ −Dun‖2 +1

1− α‖τn+θ‖2,

we obtain

2αRe(

‖un+1‖2 − ‖un‖2)

+ [We− (1− ω)θ∆t](

‖τn+1‖2 − ‖τn‖2)

+4α(1− α)

[

1− k − 2(1− α)b(1− k)2 +D2

δRe

θ∆t

h2

]

θ∆t‖Dun‖2

+2α(1− α) 2[(1 + k)θ + 2kθ′]− θ′∆t ‖Dun+θ‖2

+4α(1− α)(1− k)(2− 3θ)∆t ‖Dun+1−θ‖2 + 2α(2 + k)(1− α)θ∆t ‖Dun+1‖2

+2α

[

Re− 2(1− k)(1− α)D2 θ∆t

h2

]

(

‖un+θ − un‖2 + ‖un+1 − un+1−θ‖2)

+2αRe(1− 2δ)‖un+1−θ − un+θ‖2 + 2α(1− α)b

δ(4k − 1)θ∆t‖Dun+θ −Dun‖2

+[(2− ω)θ′ + (1− ω)θ]∆t ‖τn+1−θ‖2 +

(

ω + θ − α

1− α2bθ

δ

)

∆t ‖τn+θ‖2

+2θ∆t‖τn+1‖2 +

[

We−(

ω +2α

1− α

)

θ′∆t

]

‖τn+1−θ − τn+θ‖2

+[We− (1− ω)θ∆t](

‖τn+1 − τn+1−θ‖2 + ‖τn+θ − τn‖2)

≤ 4α(1− θ)∆t(fn, un+θ) + 2α2bθ∆t

δ(fn, un+θ − un) + 4αθ∆t(fn+1, un+1)

+θ∆t(fns , τn+θ) + θ′∆t(fns , τ

n+1−θ) + θ∆t(fn+1s , τn+1).

Page 99: Stability Analysis and Numerical Simulation of Non-Newtonian

5.3. A priori stability estimation of the linear Oldroyd problem 87

For the right-hand side terms Young’s inequalities are used:

2(fn, un+θ) ≤ (1− α)θ

1− θ‖Dun+θ‖2 +

1

(1− α)

1− θθ‖fn‖2,

2(fn, un+θ − un) ≤ (1− α)‖Dun+θ −Dun‖2 +1

1− α‖fn‖2,

2(fn+1, un+1) ≤ (1− α)‖Dun+1‖2 +1

1− α‖fn+1‖2,

(fns , τn+θ) ≤ ε‖τn+θ‖2 +

1

4ε‖fns ‖2,

(

θ′

θfns + fn+1

s , τn+1−θ

)

≤ ε‖τn+θ‖2 +b

4ε‖fns ‖2 +

1

4ε‖fn+1s ‖2,

(fn+1s , τn+1 − τn+1−θ) ≤ ε‖τn+1 − τn+1−θ‖2 +

1

4ε‖fn+1s ‖2,

which give the following relation

2αRe(

‖un+1‖2 − ‖un‖2)

+ [We− (1− ω)θ∆t](

‖τn+1‖2 − ‖τn‖2)

+4α(1− α)

[

1− k − 2(1− α)b(1− k)2 +D2

δRe

θ∆t

h2

]

θ∆t‖Dun‖2

+2α(1− α) [(1 + 2k)θ + (4k − 1)θ′] ∆t ‖Dun+θ‖2

+4α(1− α)(1− k)(2− 3θ)∆t ‖Dun+1−θ‖2 + 2α(1 + k)(1− α)θ∆t ‖Dun+1‖2

+2α

[

Re− 2(1− k)(1− α)D2 θ∆t

h2

]

(

‖un+θ − un‖2 + ‖un+1 − un+1−θ‖2)

+2αRe(1− 2δ)‖un+1−θ − un+θ‖2 + 4α(1− α)b

δ(2k − 1)θ∆t‖Dun+θ −Dun‖2

+[(2− ω)θ′ + (1− ω − ε)θ]∆t ‖τn+1−θ‖2 +

(

ω + θ(1− ε)− α

1− α2bθ

δ

)

∆t ‖τn+θ‖2

+2θ∆‖τn+1‖2 +

[

We−(

ω +2α

1− α

)

θ′∆t

]

‖τn+1−θ − τn+θ‖2

+[We− (1− ω + ε)θ∆t]‖τn+1 − τn+1−θ‖2 + [We− (1− ω)θ∆t]‖τn+θ − τn‖2

≤ 2α

1− α∆t

[(

(1− θ)2θ

+2b

δθ

)

‖fn‖2 + θ‖fn+1‖2]

+θ∆t

[

(1 + b)‖fns ‖2 + 2‖fn+1s ‖2

]

.

Summing up the above inequalities for n = 0, 1, 2, ..., N , the desired stability estimation

Page 100: Stability Analysis and Numerical Simulation of Non-Newtonian

88 Chapter 5. Stability analysis

(5.100) is arrived, where the coefficients are defined by

Cu = 2αRe,

Cτ = We− (1− ω)θ∆t,

C1 = 4α(1− α)θ

[

1− k − 2(1− α)b(1− k)2 +D2

δRe

θ∆t

h2

]

θ + 2α(1− α)(1 + k)θ,

C2 = 2α(1− α) [(1 + 2k)θ + (4k − 1)θ′] ,

C3 = 4α(1− α)(1− k)(2− 3θ),

C4 = 2α

[

Re− 2(1− k)(1− α)D2 θ∆t

h2

]

,

C5 = 2αRe(1− 2δ),

C6 = 4α(1− α)b

δ(2k − 1)θ,

C7 = (2− ω)θ′ + (1− ω − ε)θ,

C8 = ω + θ(1− ε)− α

1− α2bθ

δ,

C9 = We− (1− ω + ε)θ∆t,

C10 = We−(

ω + 2α

1− α

)

θ′∆t,

Cf =2α

1− α

(

(1− θ)2θ

+2b

δ+ θ

)

,

Cfs=

θ

4ε(3 + b).

The coefficients Cu , C2, , C3 , C6 , C7 , Cf and Cfsare positive for every value θ ∈ (0, 0.5],

k ∈ (0, 1) and α ∈ [0, 1). If δ ≤ 0.5 then also coefficient C5 is positive. For ∆t sufficiently

small, fulfilling

∆t ≤ min

We

(1− ω)θ,

We

(1− ω + ε)θ

,

also the coefficients Cτ and C9 are positive. The coefficients C8 and C10 contain the term

−α/(1− α), so for α not very close to 1, fulfilling

α

1− α ≤δ

2bθ[ω + θ(1− ε)],

and ∆t sufficiently small, they are positive. Most problematically are the coefficients C1

and C4, which contain the ratio ∆t/h2, but in virtue of the restriction (5.99) these two

coefficients are also positive.

Page 101: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 6

Implementation aspects

The numerical implementation of the Oldroyd system uses the program package Albert.

This is a library, which provides all elementary functions for an efficient finite element

solver. The starting point for my program was an Albert based Navier-Stokes solver. The

numerical results presented in this work were generated by an own extended version of the

previously mentioned package. For an overview of the range of Albert’s capabilities, see

e.g. [65].

The structured two-dimensional meshes for the four-to-one contraction problem were han-

dled based on my own code combined with the mesh generator Triangle. Local and global

refinements of the used meshes were generated by the bisection algorithm described in [11].

6.1 Solution of the subproblems

In the following subsections, some details concerning the solution procedure of the subprob-

lems from algorithm 4.4 are given. The treatment of the Stokes and Burgers subproblems

is similar to the procedure used in [12, 13] for Navier-Stokes equation system.

6.1.1 Solution of the Stokes problem

The Stokes subproblem of type (4.5) which has to be solved in the first and third sub-steps

of the algorithm 4.4, can be written in the form

Problem 6.1. Find (u, p) ∈ Vgh ×Q0h such that

(6.1)

(u, v) + γ (Du,Dv)− (p, div v) = l(v),

(div u, q) = 0,

for all (v, q) ∈ V0h ×Qh. Here l ∈ V ′0h denotes the functional given by the explicit terms of

(4.5), p = p/λ and γ = η/λ.

Page 102: Stability Analysis and Numerical Simulation of Non-Newtonian

90 Chapter 6. Implementation aspects

The corresponding Schur complement operator T : Q0h → Q0h will be defined as follows

Problem 6.2. For given p ∈ Q0h solve for χp ∈ V0h the equation

(χp, v) + γ (Dχp, Dv) = (p, div v),

for all v ∈ V0h and define T p ∈ Q0h by

(T p, q) = (divχp, q),

for all q ∈ Qh.

It is easy to see that the operator T is symmetric and positive definite. Moreover

Problem 6.3. (u, p) ∈ Vgh ×Q0h is a solution of problem 6.1 if and only if

(6.2) (T p, q) = −(div u, q),

for all q ∈ Qh and

u = u+ χp,

where u ∈ Vgh is defined by

(u, v) + γ (Du,Dv) = l(v),

for all v ∈ V0h.

This follows from the simple identity

(u+ χp, v) + γ (D(u+ χp), Dv) = (u, v) + γ (Du,Dv) + (χp, v) + γ (Dχp, Dv)

= l(v) + (p, div v).

Thus, solving (6.1) reduces to solving (6.2), an operator equation in Q0h with a symmetric,

positive definite operator T . Furthermore, since the inf-sup condition (3.4) is fulfilled, it

follows that for fixed γ the operator T is an isomorphism with ‖T‖L2→L2 , ‖T−1‖L2→L2

bounded independently of h. Thus a conjugate gradient method applied to T seems to be

an optimal solver. However, for γ → 0, i.e. ∆t → 0 or Re → ∞, the Schur complement

degenerates and the condition number

cond(T ) := ‖T‖L2→L2 · ‖T−1‖L2→L2

blows up. One expects a growth of

cond(T ) = O

(

1

h2 + γ

)

,

in the case of a quasi-uniform triangulation with the mesh size h. Therefore a preconditioner

S : Q0h → Q0h as proposed in [18] was used

Sr := γ r + ϕr,

Page 103: Stability Analysis and Numerical Simulation of Non-Newtonian

6.1. Solution of the subproblems 91

with ϕr ∈ Q0h fulfilling

(6.3) (∇ϕr,∇q) = (r, q) ∀q ∈ Q0h.

The evaluation of S is inexpensive, since it involves only the inversion of a scalar Poisson

problem. Furthermore the solution of this problem is sought in the pressure space Q0h which

is much smaller than the velocity space Vh.

Details about the numerical treatment of the Stokes problem by means of the precon-

ditioned conjugate gradient algorithm can be found in Glowinski and Pironneau [32] and

Bristeau et al. [18].

The main aim in this work is to examine the influence of the Weißenberg number on

the Oldroyd problem, and therefore the computations were restricted to relatively small

Reynolds numbers, i.e. Re ≤ 100. To reach a tolerance error of 10−9 − 10−10 in the Stokes

step, one needs 20-30 iteration steps.

6.1.2 Solution of the Burgers problem

To solve the nonlinear subproblem (4.7) one use a preconditioned GMRES method [61, 12].

For N ∈ N, M ∈ RN×N regular matrix, b ∈ R

N let us consider the following linear problem

Problem 6.4. Find x ∈ RN such that

M x = b.

For a given k ∈ N and an initial guess x0 ∈ RN the GMRES algorithm determines an

approximate solution x ∈ RN by

(6.4) x = arg minz∈K‖S(b−M(x0 + z))‖RN ,

with the Krylov spaceK := spanSr0, SMr0, ...SMk−1r0, the residual r0 := S(b−Mx0),

the Euclidian norm ‖ · ‖RN and a suitable preconditioner S ∈ RN×N .

The minimization in (6.4) is based on finding an orthonormal basis of the spaceK. Usually

the dimension k of the Krylov space K is chosen small and several restarts of GMRES are

performed using the last iterate as initial value for the new start until the desired accuracy

is achieved.

The nonlinear equation (4.7) may be written as

Problem 6.5. Find u ∈ Vgh such that

(6.5) (Au, v) + (N(u)u, v) := γ1 (u, v) + γ2 (Du,Dv) + b(u, u, v) = l(v),

for all v ∈ V0h, where N(v)w = (v · ∇)w, γ1 = λ1/Re, γ2 = η1/Re and l ∈ V ′0h denotes the

functional given by the explicit terms of (4.7).

Page 104: Stability Analysis and Numerical Simulation of Non-Newtonian

92 Chapter 6. Implementation aspects

Denoting by uh ∈ RN the vector of nodal values of the discrete solution in Vgh and

N = dim(Vgh), then the discrete form of equation (6.5) is

(6.6) Ahuh +Nh(uh)uh, = lh.

In order to apply the GMRES algorithm to (6.6) the nonlinearity Nh(·) will be frozen and

updated at every restart. That is, in the p-th restart of GMRES one defines

M := Ah +Nh(up−1h ),

where up−1h is the (p−1)-th iterate and the initial guess is setting as uph,0 := up−1

h . It turns out

that for transient flows a simple diagonal scaling as preconditioner S is sufficient. Moreover

the dimension k of the Krylov space may be chosen quite small and the convergence of the

method is rather fast, that means the number of restarts is small. In our computations k is

setting to 10 and the number of restarts usually lies between 2-10, even when the dimension

N of the velocity space Vh is of the order1 105 − 106.

6.1.3 Solution of the stress convective problem

For the stress tensor one has to solve a linear problem in the first and third steps of algo-

rithm 4.4 and a transport problem (4.8) in the second step.

Problem 6.6. Find τ ∈ Θh such that

(6.7) (Nτ, σ) := (τ, σ) + γ g(u, τ, σ) = l(σ),

with given τΓ on the inflow boundary part, for all σ ∈ Θh. Here γ = We/c4 or γ = 0 for

(4.8) and (4.6) respectively, and l ∈ Θ′h denotes the functional given by the explicit terms of

the corresponding equations.

Equation (6.7) was solved by a preconditioned GMRES method. Typical values for k are

in the range 10-50, the number of restarts usually lies between 2-3, even when the dimension

N of the stress space θh is of the order2 106. Usually the diagonal preconditioner was used

in the solver for the stress equations. However, the 3 × 3 block ILU preconditioner was

implemented.

6.1.4 Approximation of the boundary conditions

If the boundary conditions are defined by

u = g on Γ,

Γ

g · n dΓ = 0,

1for the square domain [0, 1]2 the uniform refinement at h = 2−7 gives 49665 degrees of freedom for each

velocity componente2for the square domain [0, 1]2 the uniform refinement at h = 2−7 gives 73728 degrees of freedom for each

stress componente

Page 105: Stability Analysis and Numerical Simulation of Non-Newtonian

6.2. Assembly of the element matrices 93

it is of fundamental importance to approximate g by gh such that∫

Γ

gh · n dΓ = 0.

The construction of such a gh follows [18, 32]. For simplicity, g is supposed continuous over

Γ. Let be V γh the space of the trace of those functions vh belonging to Vh given by

V γh = µh; µh = vh|Γ , vh ∈ Vh.

Actually, if Vh is defined by (3.11), V γh is also the space of those functions defined over Γ,

taking their values in RN , N = 2, 3, continuous over Γ and piecewise quadratic over the

edges of the triangulation Th contained in Γ.

The problem is to construct an approximation gh of g such that

(6.8) gh ∈ V γh ,∫

Γ

gh · n dΓ = 0.

If Πhg is the unique element of V γh , obtained by piecewise quadratic interpolation of g over

Γ, i.e. obtained from the values taken by g at those nodes of Th belonging to Γ, one usually

nave∫

ΓΠhg · n dΓ 6= 0. To overcome this difficulty, one proceeds as follows:

Step 1: Define an approximation nh of n as the solution of the following variational

problem in V γh

(6.9) nh ∈ V γh ,∫

Γ

nh · µh dΓ =

Γ

n · µh dΓ, ∀µh ∈ V γh .

Problem (6.9) is equivalent to a linear system whose matrix is sparse, positive definite,

well-conditioned and quite easy to compute.

Step 2: Then define gh by

(6.10) gh = Πhg −(∫

Γ

Πhg · n dΓ/∫

Γ

n · nh dΓ)

nh.

It is easy to check that (6.9) and (6.10) implies (6.8).

6.2 Assembly of the element matrices

The Oldroyd system (2.4) leads to the subproblems (4.5)-(4.8), which have to be solved with

the help of the finite element method presented in chapter 4. For the unknown velocity,

pressure and stress fields, one uses continuous quadratic ψi, i ∈ 1, 2, ..., 6, continuous

linear φı, ı ∈ 1, 2, 3 and discontinuous linear ξı finite elements respectively. That means

that on each simplex K belonging to the triangulation T of the domain Ω, one has the

notation

(6.11) u|K =

Nψ−1∑

i=0

[

uix

uiy

]

ψi, p|K =

Nφ−1∑

ı=0

pıφı, τ|K =

Nξ−1∑

ı=0

[

τ ıxx τ ıxy

τ ıxy τ ıyy

]

ξı.

Page 106: Stability Analysis and Numerical Simulation of Non-Newtonian

94 Chapter 6. Implementation aspects

A detailed description of the data structures and subroutines for matrix and vector assembly

used by Albert, are found in [65].

6.3 Implementation of the jump-terms

As a consequence of using discontinuous elements for the stress, the discontinuous Galerkin

method, described in Chapter 4, introduced in the material constitutive equation (3.14)1 a

jump term

(6.12) 〈τ+ − τ−, σ+〉h,u =

∂K−(u)

(τ+ − τ−, σ+)|n · u| ds.

So, for solving the stress problems (4.6) and (4.8) one has to implement the jump of the

stress tensor (6.12) along the inflow part of the boundary ∂K of each element K as defined

in (3.13). The combinations of different equations of the systems (4.2)-(4.4), which leads to

the algorithm (4.5)-(4.8), introduce also jump terms in the motion equations like

(6.13) 〈τ+ − τ−, Dv〉h,u =

∂K−(u)

(τ+ − τ−, Dv)|n · u| ds.

The first step is to determine the inflow part of the boundary ∂K for each simplex K of the

triangulation T. Let be S a side of the triangular element K. Since using quadratic elements

i0

i1

im0 1

s

S

n

Figure 6.1: Degrees of freedom for the side S.

for the velocity, it is easy to determine the product u · n|S as a quadratic function of the

reference segment parameter s ∈ [0, 1] in the way described below. The basis functions ψ

for the Lagrange element of second order corresponding to the barycentric coordinates λ of

the segment [0, 1] are

ψi0 = λ0(2λ0 − 1),

ψi1 = λ1(2λ1 − 1),

ψim = 4λ0λ1,

Page 107: Stability Analysis and Numerical Simulation of Non-Newtonian

6.4. List of own implementations and program modifications 95

whereλ0 = 1− s,λ1 = s.

With these considerations it results

u · n|S(s) =[

s2(2ui0 + 2ui1 − 4uim) + s(−3ui0 − ui1 + 4uim) + ui0]

· n.

Now, knowing the velocity field and the unit normal n of S, it is simple to determine the

parts of the side S which belongs to ∂K−(u).

6.3.1 Computational costs

The computations were carried out on a Compaq Tru64 UNIX V5.1B (Rev. 2650) with 4096

MB memory, 2 cpus AlphaServer ES45 Model 2 alpha and 1000 MHz frequency; and on an

AMD Opteron(tm) Processor 246 with 4045 MB memory.

The time necessary for the computations of the Oldroyd system is problem dependent.

For the test problems described in details in chapter 7 the time consumed per time step was

varying between 3 and 50 seconds. For example for the first EOC test problem at h = 2−5

the computational time at starts is 12 seconds and after reaching the convergence state one

need only 3 seconds per timestep. By the lid driven cavity 11-20 seconds per timestep were

neccessary on mesh with size h = 2−6. The four-to-one planar contraction problem has

consuming 200-300 seconds on mesh M1 and approximatively 1000 seconds on mesh M2 per

timestep.

6.4 List of own implementations and program modifi-

cations

1. Design of the stress tensor and the involving fields.

2. Assembly of the Oldroyd system:

– the transport stress equation (4.8) and the linear stress equation (4.6),

– the appropriate terms in the Stokes like problem (4.5) and in the Burgers like

problem (4.7).

3. Implementation of the discontinuous Galerkin method

4. Implementation of help routines for the stress equations

– 3× 3 block-matrix with 3 block-vector multiplication,

– stress gmres routine for solving the stress equations,

– 3× 3 block-matrix ILU preconditioner.

Page 108: Stability Analysis and Numerical Simulation of Non-Newtonian

96 Chapter 6. Implementation aspects

5. Integration of L2-error estimation and of the error corresponding to the discontinuous

Galerkin method (3.9) for the stress tensor components.

6. Data output function albert toolbox for handling with the Matlab’s PDEtoolbox rou-

tines.

7. Scripts and batch programs with own parameter phrases for running jobs in the back-

ground.

8. Implementation of the approximate boundary conditions in the program package Navier.

Page 109: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 7

Numerical examples

In this chapter the author first proves for three simple test examples the stability boundaries

deduced in chapter 5. For the first example, the velocity field comprises a stagnation point

and it will be shown that as soon as the Weißenberg number excides the stability limit (5.74)

instability arises and the existing steady state solution cannot be numerically reached. In the

second example, the streamlines are, in part, closed curves. For high Weißenberg numbers

instability appears in the Oldroyd-B problem and grows along that streamlines on which

the condition (5.75) is violated. For the third example, no limitations on the Weißenberg

number exist due to the fact that the streamlines leave the computational domain and the

velocity field is nonzero in any point.

Then, we examine two benchmark problems: the lid driven cavity and the four-to-one

planar contraction problem. For the lid driven cavity the streamlines are closed curves and

the four-to-one planar contraction comprise a streamline at the outer boundary along that

the velocity is zero. By means of the stability analysis presented in chapter 5, the author

tries to explain why the computations of the Oldroyd-B fluid blows up at high Weißenberg

numbers.

7.1 Experimental order of convergence

The experimental order of convergence (EOC) is defined by

EOCh :=1

log 2

ErrhErrh/2

,

with the error

Errh = limt→∞

‖v(·, t)− vh(·, t)‖,for an unknown variable v and grid size h (scale resolution). The aim of EOC tests is to prove

for stationary solutions, the correctness of the algorithm implementation and the theoretical

stability boundaries.

Page 110: Stability Analysis and Numerical Simulation of Non-Newtonian

98 Chapter 7. Numerical examples

For EOC tests we need to know an analytical stationary solution of the equation or

equation system under study. Such a demand is quite difficult for the Oldroyd system, so

we built up the right-hand sides of the equations such that the analytical given functions

(uex, τex, pex) satisfies the Oldroyd system, in the following manner:

Problem 7.1. Given the velocity field uex, solve in Ω×R+ the following system of equations

for (τ, u, p)

(7.1)

∂τ

∂t+ (u · ∇)τ + βa(τ,∇u) +

1

Weτ − 2α

WeD = fs(uex, τex),

∂u

∂t+ (u · ∇)u− 2(1− α)

RedivD − 1

Redivτ +

1

Re∇p = f(uex, τex, pex),

div u = 0.

Let be as boundary conditions the values of the exact stationary solution on the boundary

u|Γ = uex|Γ , τ|Γ−

= τex|Γ−

,

and as initial conditions for velocity and stress the exact stationary solutions

u|t=0= uex, τ|t=0

= τex.

The right-hand sides f and fs are given by the left-hand side terms of the corresponding

equation, builded with τex, uex and pex.

The analytical velocity field uex is choose to be divergence free. Numerical tests were made

for different mesh refinements starting from a two triangle mesh of the unit square domain.

When h is the mesh width, then h = 2−n corresponds to a mesh with 22n−1 triangular

elements. For the examples given in this work, the most used mesh width is h = 2−6 and

for consistency of the numerical results also tests on a h = 2−7 refined mesh were done.

7.1.1 EOC tests for example 1

For the first example the computational domain is the unit square, Ω = [0, 1]2. The velocity,

stress components and pressure used here are

(7.2)

uex =(

sin(π2 x) sin(π2 y) , cos(π2 x) cos(π2 y))

,

τxx, xy, yyex = pex = cos(π2 x) sin(π2 y).

In Fig. 7.1 the corresponding velocity field and streamlines are plotted on a mesh with

resolution h = 2−6.

Page 111: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 99

Figure 7.1: For example 1, (left) velocity field and (right) streamlines on mesh with h = 2−6.

Stress equation without convective term for a = 1

First, the stress equation without the convective transport term was analyzed, in the case

of the Oldrouyd-B fluid (a = 1).

Problem 7.2. Given the velocity field uex by (7.2), solve in Ω×R+ the following equation

for the stress tensor τ

(7.3)∂τ

∂t+ βa(τ,∇uex) +

1

Weτ − 2α

WeDuex = fs(uex, τex),

supplemented by the initial condition for stress τ|t=0= τex. The right-hand side fs is given

by the left-hand side terms builded with uex and τex from (7.2).

The parameter α does not appear in the terms which comprise the unknown stress field,

and therefore it will not influence the stability of equation (7.3). In this subsection the

parameter α was fixed to the value 0.89.

In the case of the Oldrouyd-B fluid, the eigenvalue λ−, for the test example (7.2), is

λ− =1

We− π

cos2(π

2x) sin2(

π

2y)− sin2(

π

2x) cos2(

π

2y).

This eigenvalue becomes at first zero in the point with coordinates (x, y) = (0.0, 1.0), when

the Weißenberg number achieved its critical value. One can calculate analytically the critical

value from (5.74), which gives

(7.4) Wecr =1

π= 0.3183.

Page 112: Stability Analysis and Numerical Simulation of Non-Newtonian

100 Chapter 7. Numerical examples

After this critical state, the eigenvalue λ− becomes negative in a more and more larger region.

In Fig. 7.2 isolines of λ− in the negative domain are plotted for We ∈ 0.2, 0.4, 0.6, 1. The

eigenvalues and eigenvectors corresponding to the A-matrix of equation (7.3) for this example

problem in the corner point (0.0, 1.0) are

(7.5)

λ0 =1

We→ v0 = [0 , 1 , 0],

λ+ =1

We+ π → v+ = [0 , 0 , 1],

λ− =1

We− π → v− = [1 , 0 , 0].

So, when λ− becomes negative, then the component τxx of the stress tensor will be first

affected by instabilities. The instability in the other stress components τxy and τyy appear

much later due to their coupling with τxx through the β-term.

Table 7.1: Numerical tests at We = 0.3, a = 1, α = 0.89 for the stress equation the without

convective term

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

4 5.639e-03 2.01 1.081e-03 2.00 8.869e-04 2.00

5 1.399e-03 2.00 2.704e-04 2.00 2.219e-04 2.00

6 3.491e-04 2.00 6.761e-05 2.00 5.549e-05 2.00

7 8.726e-05 1.690e-05 1.387e-05

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

4 2.150e-02 1.60 9.127e-03 1.52 7.803e-03 1.52

5 7.105e-03 1.55 3.190e-03 1.51 2.729e-03 1.51

6 2.425e-03 1.53 1.121e-03 1.50 9.597e-04 1.50

7 8.421e-04 3.952e-04 3.383e-04

Table 7.1 contains EOC tests for problem 7.2 applied to an Oldroyd-B fluid at We = 0.3,

which is subcritical. The L2 and dg norm (which corresponds to the discontinuous Galerkin

finite elements method, (3.9)) for the stress components together with the corresponding

EOC, are tabulated for different levels of mesh refinement. The time step ∆t, for this

example, is taken as 0.01. For subcritical values the tests yields good EOC values. For

supercritical values of the Weißenberg number, as expected, no convergence of the numerical

scheme was achieved, even if the time step width was scaled down.

Figures 7.3 and 7.3 shows the τxx isolines at We = 0.3 and We = 0.31, respectively,

which are close to Wecr but subcritical. Independently of the mesh width, convergence was

Page 113: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 101

50 100 150 200 250 300

50

100

150

200

250

300

0

We = 0.32

50 100 150 200 250 300

50

100

150

200

250

300

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.4

−0.5

−0.6

We = 0.4

50 100 150 200 250 300

50

100

150

200

250

300

0

0

0

−0.4

−0.4

−0.4

−0.7

−0.7

−1

−1

−1.2

−1.4

We = 0.6

50 100 150 200 250 300

50

100

150

200

250

300

0

0

0

0

−0.5

−0.5

−0.5

−1

−1

−1

−1.5

−1.5

−1.8

−2

We = 1.0

Figure 7.2: Eigenvalue λ− for example 1 at different supercritical We.

achieved. One can see that from point (0.0, 1.0) a small nonconformity is emanated which

decreases with increasing the mesh resolution. When We is supercritical this nonconformity

transforms more and more into a perturbation as one can see in Fig. 7.5, which leads to the

blowing up of the computations.

To have a better insight about this perturbation, the values of τxx along the diagonal

y = 1 − x of the square computational domain were plotted. For subcritical We (Fig. 7.6

and Fig. 7.7) there are no instabilities, but for supercritical We the instabilities are increasing

with time in amplitude and broadening (Fig. 7.8 and Fig. 7.9).

Figure 7.12 shows the time evolution of the norm ||τn+1xx − τnxx||L2 for different We at mesh

with size h = 2−6, whereas Fig. 7.10 and Fig. 7.11 shows the time evolution of the L2-error

norms for the stress tensor components at the same mesh width at We = 0.3 and We = 0.32,

respectively. For supercritical Weißenberg numbers, one can observe in Fig. 7.11 the growth

of the τxx stress component in time which induces the blowing up of the computations.

Page 114: Stability Analysis and Numerical Simulation of Non-Newtonian

102 Chapter 7. Numerical examples

Figure 7.3: Isolines of τxx for the stationary solution of the stress equation without the

convective term at a = 1 and We = 0.3 on mesh with size (left) h = 2−6; (right) h = 2−7.

Figure 7.4: Isolines of τxx for the stationary solution of the stress equation without the

convective term at a = 1 and We = 0.31 on mesh with size (left) h = 2−6; (right) h = 2−7.

Figure 7.5: Isolines for the instable solution τxx of the stress equation without the convective

term, at time t = 10 and mesh size h = 2−6 for a = 1 and (left) We = 0.32;

(middle) We = 0.4 ; (right) We = 0.5.

Page 115: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 103

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

position on y = 1−x

τ xx −

stre

ss c

ompo

nent

h = 2−6

h = 2−7

Figure 7.6: Stationary τxx solution of the stress equation without the convective term along

the line y = 1− x on two meshes of size h ∈ 2−6, 2−7, for a = 1 and We = 0.3.

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

position on y = 1−x

τ xx −

stre

ss c

ompo

nent

h = 2−6

h = 2−7

Figure 7.7: Stationary τxx solution of the stress equation without the convective term along

the line y = 1− x on two meshes of size h ∈ 2−6, 2−7, for a = 1 and We = 0.31.

Page 116: Stability Analysis and Numerical Simulation of Non-Newtonian

104 Chapter 7. Numerical examples

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

position on y = 1−x

τ xx −

stre

ss c

ompo

nent

t = 60

t = 75

exact solution

Figure 7.8: Time growth of the τxx instabilities along the line y = 1− x on mesh with size

h = 2−6 for We = 0.32, in the stress equation without the convective term.

0 0.2 0.4 0.6 0.8 1−30

−20

−10

0

10

20

30

position on y = 1−x

τ xx −

stre

ss c

ompo

nent

t = 15exact solution

Figure 7.9: Time growth of the τxx instabilities along the line y = 1− x on mesh with size

h = 2−6 for We = 0.4, in the stress equation without the convective term.

Page 117: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 105

0 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

x 10−4

||τxx

−τxxh ||

L2

||τxy

−τxyh ||

L2

||τyy

−τyyh ||

L2

t −time

erro

r no

rm

Figure 7.10: Time evolution of the L2-error norms of the solutions of the stress equation

without convective term, for mesh size h = 2−6 at a = 1 and We = 0.3.

0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5x 10

−3

||τxx

−τxxh ||

L2

||τxy

−τxyh ||

L2

||τyy

−τyyh ||

L2

t −time

erro

r no

rm

Figure 7.11: Time evolution of the L2-error norms of the solutions of the stress equation

without convective term, for mesh size h = 2−6 at a = 1 and We = 0.32.

Page 118: Stability Analysis and Numerical Simulation of Non-Newtonian

106 Chapter 7. Numerical examples

0 10 20 30 40

10−10

10−8

10−6

10−4

10−2

We = 1.0We = 0.5We = 0.4We = 0.32We = 0.3We = 0.25

t −time

||τxxn+

1 −τ xxn

|| L 2 −no

rm

Figure 7.12: Time evolution of the norm ||τn+1xx − τnxx||L2 by computing the stress equation

without convective term for example 1, at a = 1, h = 2−6, ∆t = 10−2 and different We.

0 5 10 15 20 25 30

10−12

10−10

10−8

10−6

10−4

t −time

||τxxn+

1 −τ xxn

|| −

norm We = 0.3

We = 0.4We = 0.5We = 0.6We = 1.0

Figure 7.13: Time evolution of the error norm ||τn+1xx − τnxx||L2 by computing the stress

equation for a = 1 and different We at h = 2−6 and ∆t = 10−2.

Page 119: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 107

Stress constitutive equation for a = 1

In this subsection, the stress constitutive equation formulated as the following transport

(convective) problem was analyzed:

Problem 7.3. Given the velocity field uex by (7.2), solve in Ω×R+ the following equation

for the stress tensor τ

∂τ

∂t+ (uex · ∇)τ + βa(τ,∇uex) +

1

Weτ − 2α

WeD(uex) = fs(uex, τex),

supplemented by the boundary condition τ|Γ−

= τex|Γ−

and the initial condition for stress

τ|t=0= τex. The right-hand side fs is given by the left-hand side terms builded with uex and

τex defined in (7.2).

Table 7.2: Example 1 for a = 1 and We = 0.4

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

4 1.448e-03 2.13 4.699e-04 1.95 4.431e-04 1.93

5 3.316e-04 2.10 1.218e-04 1.96 1.163e-04 1.94

6 7.754e-05 2.08 3.126e-05 1.97 3.028e-05 1.96

7 1.833e-05 2.07 7.955e-06 1.98 7.791e-06 1.97

8 4.361e-06 - 2.012e-06 - 1.986e-06 -

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

4 5.761e-03 1.58 5.235e-03 1.50 5.166e-03 1.49

5 1.932e-03 1.53 1.853e-03 1.50 1.839e-03 1.49

6 6.684e-04 1.51 6.559e-04 1.50 6.532e-04 1.50

7 2.340e-04 1.51 2.320e-04 1.50 2.315e-04 1.50

8 8.240e-05 - 8.208e-05 - 8.198e-05 -

Table 7.2 contains EOC tests for problem 7.3 applied to an Oldroyd-B fluid at We = 0.4.

The L2 and dg norms for the stress components together with the corresponding EOC, were

tabulated for different levels of mesh refinement and time step ∆t = 0.01. It is noticeable

that although We = 0.4 is supercritical (Wecr = 1/π = 0.3183 according to (7.4)), one still

obtains very good EOC results.

In the numerical tests of the convective problem 7.3, convergence is achieved also for

supercritical We. As shown in Fig. 7.13, the error norm ||τn+1xx − τnxx||L2 decrease in time

even for We = 1, which means that a stationary state will be reached. But before the

error norm is beginning to decrease, one observes that the curves have a hump. This hump

becomes larger by increasing the Weißenberg number.

Page 120: Stability Analysis and Numerical Simulation of Non-Newtonian

108 Chapter 7. Numerical examples

In Figs. 7.14 - 7.17, the time evolution of the L2-error norms ||τxx−τhxx||L2 , ||τxy−τhxy||L2

and ||τyy−τhyy||L2 was plotted at different values of We. For subcritical We = 0.3 all the three

curves decrease monotonous and reached the stationary state. At supercritical We = 0.32

one observes a lightly increment on the curve ||τxx−τhxx||L2 before it passes into the stationary

state. By higher supercritical We the increment on the curve ||τxx − τhxx||L2 is more and

more larger, but for all that the curve is passing into the stationary state.

0 2 4 6 8

3

3.5

4

4.5

5

5.5

6

6.5x 10

−5

||τxx

−τxxh ||

L2

||τxy

−τxyh ||

L2

||τyy

−τyyh ||

L2

t −time

erro

r no

rm

Figure 7.14: Stress L2-error norms for the stress transport problem, on mesh with size

h = 2−6 at We = 0.3.

Now, the reached stationary state of the stress field will be considered. In Fig. 7.18 the

isolines of the stress components were plotted at We = 1. Only the component τxx was

perturbed on the upper element layer, the other two stress components, τxy and τyy, are not

affected. These results are explained further on.

The corner (0.0, 1.0) of the computational domain is a stagnation point for the velocity field

defined in (7.2). For supercritical We this corner is situated in the subdomain with negative

real part of λ−. From relations (7.5) the eigenvector corresponding to the eigenvalue λ− is

the unit vector in the xx direction, i.e. the negative λ− affects only the stress component

τxx.

For the stress transport problem 7.3, as well as for the stress problem without convection,

problem 7.2, the instabilities arise at the stagnation point (0.0, 1.0). But problem 7.3 is an

initial-boundary value problem where the stress is transported along the streamlines which

were mapped in Fig. 7.1. The inflow boundary, for the example 1, is at y = 0.0. So, if

a streamline traverses a region where the stability requirement (5.66) is not fulfilled, the

stress fields for the stress transport problem 7.3 were not perturbed. But the boundary lines

Page 121: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 109

0 5 10 15 20

3

3.5

4

4.5

5

5.5

6x 10

−5

||τxx

−τxxh ||

L2

||τxy

−τxyh ||

L2

||τyy

−τyyh ||

L2

t −time

erro

r no

rm

Figure 7.15: Stress L2-error norms for the stress transport problem, on mesh with size

h = 2−6 at We = 0.32.

x = 0.0 and y = 1.0 belongs to the streamline which contains the stagnation point (0.0, 1.0).

Here the τxx instability arises and will be transported along that streamline, respectively it

will be propagated on the upper element layer where that streamline lies.

In Fig. 7.19 and Fig. 7.20 the τxx isolines were plotted for We ∈ 0.4, 0.5, 0.6 on two

meshes with mesh width h = 2−5 and h = 2−6, respectively. These figures confirm that the

perturbation arises only in the upper element layer. What happens in the upper element

layer? The instability leads to the deviation from the exact solution as shown in Fig. 7.21

and Fig. 7.22.

For the stress transport problem 7.3 related to example 1, one concludes that the computa-

tions converges to a stationary solution not only for subcritical We, but also for supercritical

We. Nevertheless, for supercritical We, the stationary computed solution differs from the ex-

act solution at the stress component τxx. These differences appears only in the upper element

layer where the streamline comprising the stagnation point lies. For small supercritical We

the deviation from the expected solution is hardly observable because the convective term

(uex · ∇)τ is dominant compared with the linear terms in τ , and convergence was achieved

before the negative eigenvalue has time to produce visible perturbations. However, when

increasing the supercritical We the influence of this linear terms, better said the influence

of the negative eigenvalue λ−, become visible, and although the computations converges the

reached steady state is not the exact steady state solution.

Page 122: Stability Analysis and Numerical Simulation of Non-Newtonian

110 Chapter 7. Numerical examples

0 5 10 15 20

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8x 10

−5

||τxx

−τxxh ||

L2

||τxy

−τxyh ||

L2

||τyy

−τyyh ||

L2

t −time

erro

r no

rm

Figure 7.16: Stress L2-error norms for the stress transport problem, on mesh with size

h = 2−6 at We = 0.4.

0 5 10 15 200

0.5

1

1.5

2

2.5

3x 10

−3

||τxx

−τxxh ||

L2

||τxy

−τxyh ||

L2

||τyy

−τyyh ||

L2

t −time

erro

r no

rm

Figure 7.17: Stress L2-error norms for the stress transport problem, on mesh with size

h = 2−6 at We = 0.6.

Page 123: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 111

Figure 7.18: Isolines of the three stress components (left) τxx; (middle) τxy; (right) τyy for

the stress transport equation at a = 1, We = 1 for mesh size h = 2−6.

Figure 7.19: Isolines of the τxx stress components for mesh size h = 2−5 at (left) We = 0.4;

(middle) We = 0.5; (right) We = 0.6.

Figure 7.20: Isolines of the τxx stress components for mesh size h = 2−6 at (left) We = 0.4;

(middle) We = 0.5; (right) We = 0.6.

Page 124: Stability Analysis and Numerical Simulation of Non-Newtonian

112 Chapter 7. Numerical examples

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

x −position along y = 0.985

τ xx −

stre

ss c

ompo

nent

We = 1.0We = 0.6exact solution

Figure 7.21: Deviation of τxx from the exact solution in the upper element layer along the

line y = 0.985 for mesh size h = 2−6 for We = 0.6 and We = 1, in the stress equation with

a = 1.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

x −position along y = 1

τ xx −

stre

ss c

ompo

nent

We = 1.0

We = 0.6

exact solution

Figure 7.22: Deviation of τxx from the exact solution in the upper element layer along the

line y = 1.0 for mesh size h = 2−6 for We = 0.6 and We = 1, in the stress equation with

a = 1.

Page 125: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 113

Oldroyd system without the β-term

In this subsection, EOC tests for the Oldroyd system given by the problem 7.1 without the

β-term, as well as, without and with the convective terms were done.

Numerical tests were carried out for different values of the Weißenberg number, by keeping

the other parameter constant at the values Re = 1.0 and α = 0.89. Just like the analysis

from chapter 5, for the Oldroyd system without the β-term no stability limits exist. The

computations confirmed that convergence to the steady state solution was achieved for any

value of the Weißenberg number.

Tables 7.3 and 7.4 contain the results of the EOC tests for We = 100 for the linear Oldroyd

system and for the Oldroyd system without the β-term, respectively. For these numerical

tests the time step was taken as ∆t = 0.01.

Table 7.3: EOC test example 1 for the linear Oldroyd system at We = 100

Ref. ‖p− ph‖L2 EOC ‖p− ph‖H1 EOC

3 2.410e-03 2.14 1.195e-01 1.07

4 5.437e-04 2.11 5.677e-02 1.05

5 1.251e-04 2.06 2.729e-02 1.03

6 2.993e-05 - 1.332e-02 -

Ref. ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

3 1.324e-04 2.96 5.947e-03 1.99

4 1.694e-05 2.98 1.488e-03 2.00

5 2.140e-06 2.99 3.719e-04 2.00

6 2.689e-07 - 9.295e-05 -

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

3 4.255e-03 1.98 3.168e-03 2.06 3.999e-03 1.99

4 1.076e-03 1.99 7.581e-04 2.03 1.001e-03 1.99

5 2.703e-04 2.00 1.846e-04 2.02 2.503e-04 2.00

6 6.773e-05 - 4.550e-05 - 6.259e-05 -

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

3 2.943e-02 1.48 1.950e-02 1.50 2.643e-02 1.48

4 1.049e-02 1.49 6.854e-03 1.50 9.433e-03 1.49

5 3.723e-03 1.49 2.414e-03 1.50 3.353e-03 1.49

6 1.318e-03 - 8.520e-04 - 1.189e-03 -

Page 126: Stability Analysis and Numerical Simulation of Non-Newtonian

114 Chapter 7. Numerical examples

The L2 and H1 norm of pressure and velocity fields and the L2 and dg norm (3.9) for the

stress components together with the corresponding EOC were registered in the tables for

different levels of mesh refinement.

Table 7.4: EOC test example 1 for the Oldroyd system without the β-term at We = 100

Ref. ‖p− ph‖L2 EOC ‖p− ph‖H1 EOC

3 2.272e-03 2.12 1.147e-01 1.06

4 5.217e-04 2.07 5.485e-02 1.02

5 1.235e-04 2.02 2.694e-02 1.00

6 3.044e-05 - 1.341e-02 -

Ref. ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

3 4.383e-04 2.85 1.690e-02 1.90

4 6.078e-05 2.86 4.524e-03 1.88

5 8.320e-06 2.80 1.226e-03 1.78

6 1.198e-06 - 3.564e-04 -

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

3 2.545e-03 2.09 2.528e-03 2.09 2.552e-03 2.09

4 5.943e-04 2.06 5.900e-04 2.06 5.957e-04 2.07

5 1.415e-04 2.04 1.406e-04 2.04 1.418e-04 2.04

6 3.428e-05 - 3.409e-05 - 3.432e-05 -

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

3 1.470e-02 1.49 1.469e-02 1.49 1.469e-02 1.49

4 5.227e-03 1.49 5.225e-03 1.49 5.226e-03 1.49

5 1.853e-03 1.50 1.852e-03 1.50 1.853e-03 1.50

6 6.560e-04 - 6.560e-04 - 6.560e-04 -

Page 127: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 115

Oldroyd-B system without the stress convective term

In the next calculations, EOC tests for the Oldroyd-B system without the stress convective

term for stress (u · ∇)τ were done.

Problem 7.4. Given the velocity field uex by (7.2), solve in Ω × R+ the following system

of equations for τ, u and p

(7.6)

∂τ

∂t+ βa(τ,∇u) +

1

Weτ − 2α

WeDu = fs(uex, τex),

∂u

∂t+ (u · ∇)u− 2(1− α)

RedivDu− 1

Rediv τ +

1

Re∇p = f(uex, τex, pex),

div u = 0,

supplemented by the boundary condition

u|Γ = uex|Γ ,

and the initial conditions

u|t=0= uex, τ|t=0

= τex.

The right-hand sides f and fs are given by the left-hand sides terms builded with the sta-

tionary τex, uex and pex defined in (7.2).

Apart from the convective term for velocity (u · ∇)u, the equation system (7.6) is treated

from the point of view of subsection 5.2.2. The term (u · ∇)u will have only an imaginary

contribution on the diagonal of matrix A5, (5.83), and will not influence the real parts of

whose eigenvalues. It is known that for the EOC test example (7.2), the critical Weßenberg

number is Wecr = 1/π = 0.3183, (5.74). Also from the analysis in subsection 5.2.2 we

have to expect that the eigenvalues of A5 will not be better than those of matrix A, (5.72),

corresponding to the pure stress equation. The critical state appears at the corner (0.0, 1.0),

so we take a look to the eigenvalues and eigenvectors corresponding to the A5-matrix in this

corner. For the wave vectors ξ = (ξ1, 0) or ξ = (0, ξ2), one eigenvalue is λξ and the other

four fulfills equations (5.84) and (5.85) at the corner

[

(

1

We− λ)2

− π2

]

[

λ2 − λ(

(1− α)‖ξ‖2Re

+1

We

)

+

(

1

We− 1

) ‖ξ‖2Re

]

= 0,

and are given by

λ1,2 =1

2

(

(1− α)‖ξ‖2Re

+1

We

)

±√

(

(1− α)‖ξ‖2Re

+1

We

)2

− 4

(

1

We− 1

) ‖ξ‖2Re

,

λ± =1

We± π.

Page 128: Stability Analysis and Numerical Simulation of Non-Newtonian

116 Chapter 7. Numerical examples

The real part of the eigenvalues λ1 and λ+ is always positive, the real part of λ2 is positive

as long as We < 1, but the real part of the eigenvalue λ− becomes negative already at

Wecr = 1/π. The corresponding eigenvector of eigenvalue λ− is given by

v− = [1 , 0 , 0 , 0 , 0].

So, similar as in the analysis of the pure stress equation without the convective term (7.3),

when λ− becomes negative, then the τxx stress component will be affected by instabilities.

At the corner (0.0, 1.0) the equation (5.86), corresponding to the wave vector ξ = (ξ1, ξ1),

has a solution equal to 1/We and the other three solutions are given through the equation

(7.7)

λ3 − λ2

(

2

We+ (1− α)

‖ξ‖2Re

)

+ λ

[(

2− αWe

− 2

) ‖ξ‖2Re

+1

We2 − π2

]

−(

1

We2 −2

We− (1− α)π2

) ‖ξ‖2Re

= 0.

The dependence of the real part of the smallest solution of equation (5.86) on the wave

vector length square by Re, at the point (x, y) = (0.0, 1.0), was plotted in Fig. 7.23 for

α = 0.89 and different values of the Weißenberg number. One can see that the smallest

solution becomes negative only for values of We larger than Wecr.

10−1

100

−0.1

0

0.1

0.2

0.3

0.4We = 0.3We = 0.31We = 0.32We = 0.325We = 0.33

||ξ||2/Re

λ −

eige

nval

ue

Figure 7.23: Dependence of the real part of the smallest solution of equation (7.7) on the

wave vector length square by Re, for α = 0.89 and different We.

At the same corner, equation (5.86) corresponding to the wave vector ξ = (ξ1,−ξ1), has

likewise a solution equals to 1/We and the other three solutions are given by the equation

Page 129: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 117

(7.8)

λ3 − λ2

(

2

We+ (1− α)

‖ξ‖2Re

)

+ λ

[

2− αWe

‖ξ‖2Re

+1

We2 − π2

]

−(

1

We2 − (1− α)π2

) ‖ξ‖2Re

= 0.

The solutions of equation (7.8) are positive as long as We is smaller than 1/π, independently

of α.

In conclusion, for the Oldroyd-B system are expected instabilities caused by the eigenval-

ues corresponding to the wave vectors ξ = (ξ1, 0) or ξ = (0, ξ2) for values of the Weißenberg

number larger than Wecr = 1/π = 0.318, similarly as for the pure stress equation. The

following numerical tests, were showing that this stability limit is acute.

Numerical tests were carried out for the Reynolds number fixed to Re = 1.0 and mainly

for two values of the parameter α. The value α = 0.89 is the most used value in the 4:1

contraction benchmark problem, and α = 0.41 correspond to the MIT Boger fluid (see [67]).

The numerical experiments for α = 0.89 show that when neglecting the stress transport

term in the Oldroyd system, convergence of the program can be obtained only below the

critical value of the Weißenberg number, likewise as for the pure stress equation without the

stress convective term. The instability of the stress field appears, for supercritical We, in the

region near the corner (x, y) = (0.0, 1.0), where the real part of the eigenvalue λ− becomes

negative, as it can be observed in Fig. 7.24.

Figure 7.24: Isolines of τxx for mesh size h = 2−6 for α = 0.89, corresponding to stationary

state for (left) We = 0.3; (middle) We = 0.31; and (right) We = 0.32 at time t = 90.

In Fig. 7.25, the time evolution of the norms ‖τn+1xx − τnxx‖L2 at α = 0.41 and α = 0.89,

were represented for two subcritical values of the Weißenberg number We ∈ 0.3, 0.31 and

for the supercritical value We = 0.32. At the same parameters, the time evolution of the

error norm ‖τxx−τhxx‖L2 was represented in Fig. 7.26 and Fig. 7.27. For the subcritical values

the program converges to the stationary state but for supercritical values no convergence

was achieved and the computations were blowing up. The stress component τxx was plotted

Page 130: Stability Analysis and Numerical Simulation of Non-Newtonian

118 Chapter 7. Numerical examples

in Fig. 7.28 and Fig. 7.29 for α = 0.89 along the diagonal y = 1−x of the square domain at

We = 0.31 and We = 0.32, respectively. For subcritical values of We the steady state solution

agrees with the exact solution, but for the supercritical value We = 0.32 the instability exists

and growths in time until the computations blows up.

0 20 40 60 80 10010

−11

10−10

10−9

10−8

10−7

10−6

10−5

t −time

||τn+

1xx

−τn xx

|| −

norm

We = 0.3

We = 0.31

We = 0.32

0 20 40 60 80 10010

−11

10−10

10−9

10−8

10−7

10−6

10−5

t −time

||τn+

1xx

−τn xx

|| −

norm

We = 0.3

We = 0.31

We = 0.32

α = 0.41

α = 0.89

Figure 7.25: Time evolution of the norm ‖τn+1xx − τnxx‖L2 in the Oldroyd-B system without

the stress transport term at different We for mesh size h = 2−6 and α ∈ 0.41, 0.89.

Page 131: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 119

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−3

We = 0.3

We = 0.31

We = 0.32

t −time

||τxx

−τh xx

|| L 2 −er

ror

norm

Figure 7.26: Time evolution of the error norm ‖τxx−τhxx‖L2 in the Oldroyd-B system without

the stress transport term at α = 0.41.

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3

We = 0.3

We = 0.31

We = 0.32

t −time

||τxx

−τh xx

|| L 2 −er

ror

norm

Figure 7.27: Time evolution of the error norm ‖τxx−τhxx‖L2 in the Oldroyd-B system without

the stress transport term at α = 0.89.

Page 132: Stability Analysis and Numerical Simulation of Non-Newtonian

120 Chapter 7. Numerical examples

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

position along y = 1−x

τ xx −

stre

ss c

ompo

nent

numerical solution

exact solution

Figure 7.28: Steady solution τxx of the Oldroyd-B system without the stress transport term,

along the line y = 1− x for h = 2−6 at α = 0.89 and We = 0.31.

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

position along y = 1−x

τ xx −

stre

ss c

ompo

nent

t = 50

t = 80

exact solution

Figure 7.29: Numerical values of τxx in the Oldroyd-B system without the stress transport

term, along the line y = 1− x for h = 2−6 at α = 0.89 and We = 0.32.

Page 133: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 121

Oldroyd-B system

To ensure the stability boundaries for the full Oldroyd-B system, numerical tests were made

for different values of the Weißenberg number, by keeping the other parameter constant at

the values Re = 1.0 and α = 0.89.

Tables 7.5 and 7.6 contained the results of the EOC tests for We = 0.3 and We = 0.4,

respectively, which show very good achievement also for the supercritical value 0.4. The

L2 and H1 norm of pressure and velocity fields and the L2 and dg norm (3.9) for the

stress components together with the corresponding EOC at time step width ∆t = 0.01 were

registered in these tables for different levels of mesh refinement.

Table 7.5: EOC test example 1 for the Oldroyd-B system at We = 0.3

Ref. ‖p− ph‖L2 EOC ‖p− ph‖H1 EOC

4 4.548e-04 2.01 5.267e-02 1.00

5 1.128e-04 1.97 2.627e-02 0.99

6 2.864e-05 1.81 1.318e-02 1.00

7 8.166e-06 - 6.573e-03 -

Ref. ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

4 2.743e-05 2.66 2.545e-03 1.64

5 4.325e-06 2.63 8.131e-04 1.59

6 6.960e-07 3.17 2.690e-04 2.18

7 7.686e-08 - 5.908e-05 -

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

4 5.167e-04 2.04 3.958e-04 1.95 4.305e-04 1.94

5 1.249e-04 2.01 1.027e-04 1.94 1.113e-04 1.92

6 3.084e-05 1.96 2.702e-05 1.93 2.888e-05 1.89

7 7.875e-06 - 7.276e-06 - 7.576e-06 -

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

4 5.363e-03 1.53 5.154e-03 1.49 5.094e-03 1.51

5 1.849e-03 1.51 1.799e-03 1.49 1.810e-03 1.50

6 6.487e-04 1.49 6.434e-04 1.48 6.362e-04 1.47

7 2.309e-04 - 2.284e-04 - 2.299e-04 -

Similarly as in the case of the pure stress equation with given velocity field, (7.3), the

full Oldroyd system treated as an initial-boundary value problem. In section 5.2 it was

Page 134: Stability Analysis and Numerical Simulation of Non-Newtonian

122 Chapter 7. Numerical examples

Table 7.6: EOC test example 1 for the Oldroyd-B system at We = 0.4

Ref. ‖p− ph‖L2 EOC ‖p− ph‖H1 EOC

4 4.545e-04 2.01 5.275e-02 1.00

5 1.128e-04 1.97 2.632e-02 0.99

6 2.869e-05 1.75 1.320e-02 0.98

7 8.541e-06 6.667e-03

Ref. ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

4 2.932e-05 2.69 2.724e-03 1.66

5 4.552e-06 2.65 8.609e-04 1.61

6 7.242e-07 2.48 2.812e-04 1.49

7 1.292e-07 9.973e-05

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

4 5.233e-04 2.04 3.992e-04 1.93 4.333e-04 1.94

5 1.273e-04 2.02 1.047e-04 1.92 1.124e-04 1.94

6 3.144e-05 2.01 2.760e-05 1.93 2.916e-05 1.95

7 7.806e-06 7.520e-06 7.520e-06

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

4 5.295e-03 1.52 5.111e-03 1.50 5.089e-03 1.49

5 1.841e-03 1.50 1.797e-03 1.49 1.811e-03 1.49

6 6.484e-04 1.50 6.378e-04 1.49 6.444e-04 1.49

7 2.291e-04 2.266e-04 2.289e-04

prooved that if the computational domain contains a stagnation point in the region where

the real part of any eigenvalue of matrix A5, (5.83), becomes negative, instability arises along

the streamline which comprises the stagnation point. At small supercritical Weißenberg

numbers the instability is not visible, because the convective terms dominate the linear and

β-term and convergence was achieved before the negative eigenvalue has time to produce

visible perturbations. However, as soon as the influence of the linear and β-terms grows,

the instability becomes visible and the computations blow up. For example, at We = 0.5

the instability appears along the streamline comprising the stagnation point (0.0, 1.0). In

Fig. 7.30 the time evolution of the perturbation in the stress component τxx in the upper

element layer is represented.

The Oldroyd system (7.1) is a nonlinear coupled system for the unknown fields, so, the

Page 135: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 123

Figure 7.30: Propagation of the τxx perturbation in the Oldroyd system, at We = 0.5 for

mesh size h = 2−6 for (left) t = 50, (middle) t = 70 and (right) t = 100.

instability arising in the stress component τxx will influence also the other unknown fields, as

one can see in Fig. 7.31 and Fig. 7.32. Here, the time evolution of the norms ‖τxx− τhxx‖L2 ,

‖τxy − τhxy‖L2 , ‖τyy − τhyy‖L2 and ‖u− uh‖L2 were represented at We = 0.5 and We = 0.6.

For both supercritical Weißenberg numbers the norms grow exponentially.

Neither further mesh refining nor scaling down the time step leads to better convergence

results. In Fig. 7.33 the time evolution of the norm ‖τn+1xx − τnxx‖L2 is plotted at We = 0.5

on different mesh sizes h ∈ 2−5, 2−6, 2−7 and time step widths ∆t ∈ 10−2, 10−3. At

the same parameters in Fig. 7.34 the time evolutions of the L2-error norms for the stress

component ‖τxx were represented.

In Fig. 7.35 and Fig. 7.36 one can observe the deviation of the τxx numerical values

from the exact solution near the streamline comprising the stagnation point. These two

diagrams show the deviation from the exact solution along the horizontal lines y = 0.993

and y = 0.9961, respectively, at We = 0.6 on mesh with size h = 2−7.

Oldroyd-B system in a domain comprising the stagnation point

To show that the perturbation that appears in the upper element layer is caused only by

the instabilities arising at the stagnation point (x, y) = (0.0, 1.0), and is not influenced by

the boundary, calculations were made also in the domain Ω = [−1, 1] × [0, 2]. In Fig. 7.37

the velocity field and the time evolution of the stress component τxx were represented for

We = 0.6, α = 0.89, Re = 1 on mesh with size h = 2−5 and time step width ∆t = 0.01.

From this figure it becomes clear that the instability was generated at the stagnation point

and it will be transported along the streamlines which comprise the stagnation point.

Page 136: Stability Analysis and Numerical Simulation of Non-Newtonian

124 Chapter 7. Numerical examples

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−4

t −time

erro

r no

rm

||τxx

− τxxh ||

L2

||τxy

− τxyh ||

L2

||τyy

− τyyh ||

L2

||u − uh||L

2

Figure 7.31: Time evolution of the L2-error norms for the Oldroyd-B system, for mesh size

h = 2−6 at α = 0.89 and We = 0.5.

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

t −time

erro

r no

rm

||τxx

− τxxh ||

L2

||τxy

− τxyh ||

L2

||τyy

− τyyh ||

L2

||u − uh||L

2

Figure 7.32: Time evolution of the L2-error norms for the Oldroyd-B system, for mesh size

h = 2−6 at α = 0.89 and We = 0.6.

Page 137: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 125

0 10 20 30 40 50 6010

−10

10−9

10−8

10−7

10−6

10−5

t −time

||τxxn+

1 − τ

xxn|| L 2 −

norm

h = 2−5, ∆ t = 0.01h = 2−6, ∆ t = 0.01

h = 2−7, ∆ t = 0.01h = 2−6, ∆ t = 0.001

Figure 7.33: Time evolution of the norm ‖τn+1xx −τnxx‖L2 for the Oldroyd-B system at α = 0.89

and We = 0.5.

0 10 20 30 40 50 6010

−5

10−4

10−3

10−2

t −time

||τxx

− τ

xxh|| L 2 −

erro

r no

rm

h = 2−5, ∆ t = 0.01h = 2−6, ∆ t = 0.01

h = 2−6, ∆ t = 0.001

h = 2−7, ∆ t = 0.01

Figure 7.34: Time evolution of the error norm ‖τxx − τhxx‖L2 for the Oldroyd-B system at

α = 0.89 and We = 0.5.

Page 138: Stability Analysis and Numerical Simulation of Non-Newtonian

126 Chapter 7. Numerical examples

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

x −position along y = 0.993

τ xx −

stre

ss c

ompo

nent

t = 15

t = 25

exact solution

Figure 7.35: Stress component τxx for the Oldroyd-B system, on the line y = 0.993 at

α = 0.89 and We = 0.6 for mesh size h = 2−7.

0 0.2 0.4 0.6 0.8 1−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

x −position along y = 0.9961

τ xx −

stre

ss c

ompo

nent

t = 15

t = 25

exact solution

Figure 7.36: Stress component τxx for the Oldroyd-B system, on the line y = 0.9961 at

α = 0.89 and We = 0.6 for mesh size h = 2−7.

Page 139: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 127

u t = 20

t = 30 t = 100

Figure 7.37: Velocity field and propagation of the τxx perturbation at We = 0.6 for the

Oldroyd-B system, in the domain [−1, 1]× [0, 2] for mesh size h = 2−5.

Page 140: Stability Analysis and Numerical Simulation of Non-Newtonian

128 Chapter 7. Numerical examples

Stress transport problem for a = 0

Now, let us clarify what happens in the case of a = 0. First, the stress transport prob-

lem 7.3 was analyzed. Here, the stability requirement (5.66) is fulfilled for each value of the

Weißenberg number. So, no stability boundary exists in this case. This fact is reflected in

Table 7.7, which contains EOC tests for We = 1000, Re = 1 and α = 0.89. The program

converges and gives very good results for any value of the Weißenberg number.

Table 7.7: Example 1 for a = 0 and We = 1000

Ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

4 6.272069e-04 2.10 5.865086e-04 2.06 5.677649e-04 2.04

5 1.464079e-04 2.07 1.410928e-04 2.04 1.382838e-04 2.03

6 3.497572e-05 2.04 3.429843e-05 2.03 3.388190e-05 2.02

7 8.504148e-06 - 8.415914e-06 - 8.354178e-06 -

Ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

4 5.256437e-03 1.50 5.225596e-03 1.50 5.203120e-03 1.49

5 1.858340e-03 1.50 1.853140e-03 1.50 1.848884e-03 1.50

6 6.569645e-04 1.50 6.560988e-04 1.50 6.553307e-04 1.50

7 2.322613e-04 - 2.321171e-04 - 2.319823e-04 -

Oldroyd system without the convective terms for a = 0

Although, the stress equation for a = 0 is unconditionally stable, in the Oldroyd system

instabilities still arise, as it can be shown by the eigenvalues analysis of matrix A5, (5.83).

As hitherto, let us looking for what happens at the corner (x, y) = (0, 1). Equations (5.88)

and (5.89), corresponding to the wave vector ξ = (ξ1, 0) or ξ = (0, ξ2) and ξ = (ξ1,±ξ1),respectively, have a solution equal to 1/We and the other two satisfy

(7.9) λ2 − λ(

1

We+

1− αRe

‖ξ‖2)

+1

We

‖ξ‖2Re

= 0,

and

(7.10) λ2 − λ(

1

We+

1− αRe

‖ξ‖2)

+

(

1

We∓ 1

) ‖ξ‖2Re

= 0,

respectively. By denoting

b =1

2

(

1

We+

1− αRe

‖ξ‖2)

,

Page 141: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 129

the solutions of equation (7.9)

λ± = b±√

b2 − 1

We

‖ξ‖2Re

,

are always positive. The solutions of equation (7.10)

(7.11) λ+ = b+

b2 −(

1

We∓ 1

) ‖ξ‖2Re

, λ− = b−√

b2 −(

1

We∓ 1

) ‖ξ‖2Re

,

corresponding to the wave vector ξ = (ξ1,−ξ1) are also always positive, but those corre-

sponding to the wave vector ξ = (ξ1, ξ1) are both positive only as long as

We ≤ 1.0.

The eigenvector of matrix A5 corresponding to the eigenvalue λ− from (7.11) for the wave

vector ξ = (ξ1, ξ1), is given by

(7.12) v− =

[

2iξ1

( α

We− 1)

(

1

We− λ−

)−1

, 0, −2iξ1

( α

We− 1)

(

1

We− λ−

)−1

, −1, 1

]

.

In the case of the Oldroyd system without the convective terms for a = 0, numerical

test were carried out for different values of the Weißenberg number while keeping the other

parameters constant at the values Re = 1.0 and α = 0.89. Also in this case, instabilities

appears when We was increased. As one can see from Fig. 7.39, convergence of the program

was obtained until We = 1.0. For We = 1.1 the norm ‖τn+1xx − τnxx‖L2 was descend only

for large mesh sizes, Fig. 7.40. Looking at the time evolution of the stress components and

pressure at We = 1.1 for mesh size h = 2−6, in Fig. 7.38 one observes the apparition and

propagation of instabilities in all stress components. Following the blue curves in Fig. 7.41

- Fig. 7.43, one can see that the perturbation first appears in the τxx and τyy stress com-

ponents, and slightly later in the τxy component. The apparition of the instability first in

the τxx and τyy stress components is traceable by the eigenvector v− given in (7.12), which

corresponds to the eigenvalue λ− that has negative real part for Weissenberg larger that

1. Until the time t = 60 for We = 1.1 the velocity field was not visible affected by the

instability, but however the pressure field, as one can see in Fig. 7.38 and Fig. 7.44.

Page 142: Stability Analysis and Numerical Simulation of Non-Newtonian

130 Chapter 7. Numerical examples

τxx

τxy

τyy

p

Figure 7.38: Propagation of the perturbation in the stress components and pressure for the

Oldroyd system without the convective terms at a = 0, We = 1.1, Re = 1, α = 0.89, h = 2−6

at time (left) t = 15; (middle) t = 25; (right) t = 60.

Page 143: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 131

0 5 10 15 20 2510

−14

10−12

10−10

10−8

10−6

10−4

t −time

||τxxn+

1 − τ

xxn|| L 2 −

norm

We = 1.0

We = 1.1

We = 1.2

Figure 7.39: Time evolution of the norm ‖τn+1xx −τnxx‖L2 in the Oldroyd problem without the

convective terms for a = 0 at different We for mesh size h = 2−6 and time step ∆t = 10−2.

0 10 20 30 40 50

10−10

10−8

10−6

10−4

t −time

||τn+

1 −τn || L 2 −

norm

h = 2−6 ∆ t = 10−2

h = 2−5 ∆ t = 10−3

h = 2−4 ∆ t = 10−2

h = 2−3 ∆ t = 10−2

Figure 7.40: Time evolution of the norm ‖τn+1xx − τnxx‖L2 in the Oldroyd problem without

the convective terms for a = 0 at We = 1.1 for different mesh sizes.

Page 144: Stability Analysis and Numerical Simulation of Non-Newtonian

132 Chapter 7. Numerical examples

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

diagonal position

τ xx −

stre

ss c

ompo

nent

t = 15t = 25t = 60

Figure 7.41: Time evolution of the perturbation in the τxx solution of the Oldroyd system

without the convective terms for a = 0, along the line y = 1 − x, at We = 1.1, Re = 1,

α = 0.89, h = 2−6.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

diagonal position

τ xy −

stre

ss c

ompo

nent

t = 15t = 25t = 60

Figure 7.42: Time evolution of the perturbation in the τxy solution of the Oldroyd system

without the convective terms for a = 0, along the line y = 1 − x, at We = 1.1, Re = 1,

α = 0.89, h = 2−6.

Page 145: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 133

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

diagonal position

τ yy −

stre

ss c

ompo

nent

t = 15t = 25t = 60

Figure 7.43: Time evolution of the perturbation in the τyy solution of the Oldroyd system

without the convective terms for a = 0, along the line y = 1 − x, at We = 1.1, Re = 1,

α = 0.89, h = 2−6.

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.4

−0.2

0

0.2

0.4

0.6

0.8

diagonal position

p −

pres

sure

t = 15t = 25t = 60

Figure 7.44: Time evolution of the perturbation in the p solution of the Oldroyd system

without the convective terms for a = 0, along the line y = 1 − x, at We = 1.1, Re = 1,

α = 0.89, h = 2−6.

Page 146: Stability Analysis and Numerical Simulation of Non-Newtonian

134 Chapter 7. Numerical examples

Oldroyd system for a = 0

For the full Oldroyd system in the case a = 0 convergence was obtained until approximatively

We = 1.4 as one can see in Fig. 7.49 and Fig. 7.50, where the time evolution of the error

norms ‖τn+1xx − τnxx‖L2 and ‖τxx − τhxx‖L2 , respectively, were plotted for ∆t = 10−2. By

scaling down the time step width to 10−3 for We = 1.5 one can not obtain better results.

Likewise, as for the full Oldroyd-B problem, for the present case due to the convective

terms, the velocity and stress fields were transported along the streamlines. Thus, along the

streamlines which leave the computational domain no instability arises but on the streamline

x = 0 and y = 1 which comprises the stagnation point (x, y) = (0.0, 1.0), the negative

eigenvalue λ− (7.11), corresponding to the wave vector ξ = (ξ1, ξ1), would give rise to

instabilities. In Fig. 7.48 the growth of the perturbation in the stress components and

pressure field at We = 1.5 was shown at different times. It is obvious that the perturbation

appears first in the upper element layer which contains the streamline y = 1 behind the

stagnation point. The time growth of the instability in the upper element layer along the

line1 y = 0.985 at We = 1.5 was plotted in Fig. 7.45-Fig. 7.47. By means of this figures one

can see that the instability was propagated stronger in the τxx and τyy stress components

accordingly to the eigenvector (7.12) corresponding to λ−.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x −position

τ xx −

stre

ss c

ompo

nent

t = 35t = 40t = 60exact solution

Figure 7.45: Deviation of τxx from the exact solution in the upper element layer along the

line y = 0.985, for We = 1.5 and mesh size h = 2−6.

1the line y = 0.985 is for the mesh size h = 2−6 in the middle of the upper element layer

Page 147: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 135

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x −position

τ yy −

stre

ss c

ompo

nent

t = 35t = 40t = 60exact solution

Figure 7.46: Deviation of τyy from the exact solution in the upper element layer along the

line y = 0.985, for We = 1.5 and mesh size h = 2−6.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x −position

τ xy −

stre

ss c

ompo

nent

t = 35t = 40t = 60exact solution

Figure 7.47: Deviation of τxy from the exact solution in the upper element layer along the

line y = 0.985, for We = 1.5 and mesh size h = 2−6.

Page 148: Stability Analysis and Numerical Simulation of Non-Newtonian

136 Chapter 7. Numerical examples

τxx

τxy

τyy

p

Figure 7.48: Propagation of the perturbation in the stress components and pressure for the

Oldroyd system at a = 0, We = 1.5, α = 0.89, Re = 1 at time (left) t = 10; (middle) t = 40;

(right) t = 60.

Page 149: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 137

0 5 10 15 20 25 30 3510

−10

10−9

10−8

10−7

10−6

10−5

10−4

t −time

||τxxn+

1 −τ xxn

|| L 2 −no

rm

We = 1.6We = 1.5We = 1.4We = 1.0

Figure 7.49: Time evolution of the norm ‖τn+1xx − τnxx‖L2 in the Oldroyd problem with a = 0,

for mesh size h = 2−6 and time step width ∆t = 10−2 (−) and ∆t = 10−3 (∗).

0 5 10 15 20 25 30 3510

−5

10−4

10−3

10−2

10−1

t −time

||τxx

−τ xxh

|| L 2 −er

ror

norm

We = 1.6We = 1.5We = 1.4We = 1.0

Figure 7.50: Time evolution of the error norm ‖τxx − τhxx‖L2 , in the Oldroyd problem with

a = 0 for different We, mesh size h = 2−6 and time step ∆t = 10−2 (−) and ∆t = 10−3 (∗).

Page 150: Stability Analysis and Numerical Simulation of Non-Newtonian

138 Chapter 7. Numerical examples

7.1.2 EOC tests for example 2

A second example for EOC tests was taken in the square domain Ω = [−1, 1]2. Here the

following exact solutions for the Oldroyd-B system was considered

(7.13)

uex = (4 (2− r2) y , −4 (2− r2)x) , r =√

x2 + y2,

τxx, xy, yyex = 1 + exp(−2)− exp(−r2),pex = exp(−r2).

Numerical tests were presented in this subsection for the following fixed values of the pa-

rameters: Re = 1, α = 0.89 and a = 1. The velocity gradient for this example is

∇u =

[ −8xy 4(2− x2 − 3y2)

−4(2− y2 − 3x2) 8xy

]

.

From the stability requirement (5.75) we found that the critical value for the Weißenberg

number is approximatively 0.109. In Fig. 7.51 the real part of the eigenvalue λ−, defined

in (5.75), was plotted in the radial direction for different values of We. The streamlines of

the velocity field uex in the domain Ω are circles or parts of circles. When considering the

stress constitutive equation only for We greater as approximatively 0.125 one can expect

instability apparition, because only after that value there exist closed streamlines in the

computational domain which comprise points with negative real part of the eigenvalue λ−.

However, for this test example, numerical simulations only for the full Oldroyd-B problem

were presented.

For this example, the Oldroyd-B system (7.1) still converge at We = 0.3, and gives very

good EOC values as it is shown in Table 7.8.

Increasing the Weißenberg number to 0.5 no convergence and so no steady state solution

was achieved. For We = 0.5 in Fig. 7.52 the isolines of the real part of the eigenvalue λ−

are plotted in the computational domain Ω.

In Fig. 7.53 one can see the time evolution of the perturbation in the stress component

τxx. The perturbation grows and is propagates along the streamlines, in the region of the

negative eigenvalue (compare with Fig. 7.52) where the streamlines are closed curves. On

the streamlines which leave the computational domain no perturbation appear, even though

they lie in the region of negative eigenvalue.

The values of τxx on the lines y = x and y = −x were represented in Fig. 7.54 for

We = 0.3, the steady state solution is in good agreement with the exact solution (7.13). But

for We = 0.5, one can observe in Fig. 7.55 the disagreement at time t = 11.5 in the region

where, accordingly to Fig. 7.52, the real part of λ− is negative.

Page 151: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 139

0 0.2 0.4 0.6 0.8 1 1.2 1.4−8

−6

−4

−2

0

2

4

6

8

10

We = 0.5We = 0.2We = 0.125We = 0.109We = 0.1

radial position

real

(λ−)

Figure 7.51: Eigenvalue λ− in the radial direction for EOC test example 2 at different We.

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

0

00

0

0

0

0

0

−3

−3

−3

−3

−3

−3

−3

−3

−5

−5

−5

−5

−5

−5

−5

−5

−5

−5

−5

−5

−5

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−6

−7

−7

−7

−7

−7

−7

−7

−7

−7

−7

−7

Figure 7.52: Isolines of the real part of the eigenvalue λ− for EOC test example 2 at We = 0.5,

in the computational domain [−1, 1]2.

Page 152: Stability Analysis and Numerical Simulation of Non-Newtonian

140 Chapter 7. Numerical examples

Table 7.8: EOC tests for example 2 applied to the Oldroyd-B system at We = 0.3, Re = 1

and α = 0.89

ref. ‖p− ph‖H1 EOC ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

2 2.530e-01 1.13 9.840e-03 3.02 2.296e-01 2.06

3 1.149e-01 1.00 1.211e-03 2.95 5.476e-02 1.96

4 5.720e-02 1.00 1.562e-04 3.04 1.402e-02 2.06

5 2.841e-02 1.886e-05 3.356e-03

ref. ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

2 1.355e-02 2.36 1.240e-02 2.39 1.427e-02 2.52

3 2.625e-03 2.10 2.359e-03 2.08 2.484e-03 2.13

4 6.107e-04 1.99 5.562e-04 1.93 5.652e-04 1.97

5 1.531e-04 1.450e-04 1.439e-04

ref. ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

2 2.209e-01 1.61 2.055e-01 1.55 2.174e-01 1.60

3 7.235e-02 1.53 7.000e-02 1.51 7.135e-02 1.52

4 2.495e-02 1.48 2.455e-02 1.47 2.475e-02 1.48

5 8.885e-03 8.833e-03 8.840e-03

u t = 9.0 t = 11.5

Figure 7.53: Velocity field and time propagation of the τxx perturbation in the Oldroyd-B

system at We = 0.5 for mesh size h = 2−5.

Page 153: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 141

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

y = x

y = −x

diagonal position

τ xx −

stre

ss c

ompo

nent

Figure 7.54: Stationary solution τxx of the Oldroyd-B problem on the lines y = x and y = −xat We = 0.3 for mesh size h = 2−5.

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

y = x

y = −x

diagonal position

τ xx −

stre

ss c

ompo

nent

Figure 7.55: Instability of the τxx stress component on the lines y = x and y = −x for

We = 0.5 at time t = 11.5 for mesh size h = 2−5.

Page 154: Stability Analysis and Numerical Simulation of Non-Newtonian

142 Chapter 7. Numerical examples

7.1.3 EOC tests for example 3

The third example was mentioned in [23]. In this example, the computational domain is

an L-shaped domain given by Ω = [−1, 1]2 \ [0, 1]2. Here, the following velocity, stress and

pressure fields were used

uex =

(

y − 0.1

r, −x− 0.1

r

)

, r =√

(x− 0.1)2 + (y − 0.1)2,

τex = 2αDex,

pex = (2− x− y)2.

The velocity and stress fields have singularities just exterior to the domain Ω near the point

(0, 0). It is interesting that for this choice the velocity field uex satisfies the relation

det(∇uex) = 0.

Accordingly to (5.75) the real part of the eigenvalue λ−, corresponding to the stress equa-

tion without the convective term, is positive in the whole computational domain, and so

no stability limitation for the Weißenberg number exists. Moreover for this example, the

streamlines neither comprise a stagnation point nor are closed curves (see Fig. 7.56) in the

whole domain Ω. Hence, theoretically, one expects not any stability difficulties for the full

Oldroyd problem.

For this example the Oldroyd-B model was simulated for the parameter values Re = 1.0,

α ∈ 0.41, 0.89 and different values of We. The simulations converge to the steady state

for every value of the Weißenberg number. In Table 7.9 and Table 7.10 the numerical well

EOC results were presented for the Oldroyd-B model at Re = 1.0, We = 100 and α = 0.41

and α = 0.89, respectively, on meshes with different refinement degrees and with time step

width ∆t = 10−3.

In Fig. 7.56 and Fig. 7.57 the velocity, pressure fields and stress fields were plotted for mesh

size h = 2−5. One observes a lightly perturbation of the pressure field, in Fig. 7.56(b), which

is caused by the singularity near the point (0, 0) outside to the domain Ω. This singularity

in the velocity field produces a numerical perturbation in the field det(∇u), which becomes

weaker on finer meshes. In Fig. 7.58 and Fig. 7.59 the determinant of the velocity gradient

is represented along the line y = −0.01 for We = 100, two values of α ∈ 0.41, 0.89 and

different mesh refinements degrees. The finer the mesh is, the smaller is the deviation of

the field det(∇u) from zero. Likewise, the finer the mesh is, the smaller is the deviation in

the pressure field from the exact solution pex in the vicinity of the corner (0, 0). This notice

gives sense to refine the mesh locally near the corner (0, 0) as it was recommended in [23].

However, one observes in figures Fig. 7.58-Fig. 7.61 that the deviations in the determinant

of the velocity gradient and pressure field from the exact corresponding solutions are quite

larger at α = 0.89 as for α = 0.41.

Page 155: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 143

Figure 7.56: (left) Velocity vectors; (right) pressure isolines for the Oldroyd-B model at

Re = 1 , α = 0.41, We = 100 for mesh size h = 2−5.

Table 7.9: EOC tests for example 3 at a = 1, Re = 1, α = 0.41, We = 100 and ∆t = 10−3

ref ‖p− ph‖H1 EOC ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

3 3.116e+00 1.21 1.153e-02 3.15 2.761e-01 1.80

4 1.345e+00 0.66 1.298e-03 3.27 7.894e-02 1.92

5 8.529e-01 0.74 1.345e-04 3.00 2.084e-02 1.95

6 5.094e-01 1.683e-05 5.371e-03

ref ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

3 1.463e-01 1.92 1.400e-01 2.23 1.568e-01 2.48

4 3.867e-02 2.37 2.980e-02 2.01 2.806e-02 1.96

5 7.468e-03 2.08 7.391e-03 2.05 7.178e-03 2.04

6 1.760e-03 1.777e-03 1.744e-03

ref ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

3 1.323e+00 1.34 1.391e+00 1.40 1.347e+00 1.35

4 5.237e-01 1.46 5.260e-01 1.37 5.264e-01 1.46

5 1.903e-01 1.46 2.024e-01 1.49 1.906e-01 1.47

6 6.887e-02 7.215e-02 6.891e-02

Page 156: Stability Analysis and Numerical Simulation of Non-Newtonian

144 Chapter 7. Numerical examples

Table 7.10: EOC tests for example 3 at a = 1, Re = 1, α = 0.89, We = 100 and ∆t = 10−3

ref ‖p− ph‖H1 EOC ‖u− uh‖L2 EOC ‖u− uh‖H1 EOC

3 6.836e+00 1.25 6.476e-02 2.68 1.240e+00 1.61

4 2.877e+00 1.49 1.009e-02 2.42 4.077e-01 1.93

5 1.020e+00 1.37 1.882e-03 2.78 1.071e-01 2.03

6 3.932e-01 1.20 2.738e-04 2.99 2.620e-02 1.96

7 1.709e-01 3.444e-05 6.711e-03

ref ‖τxx − τxxh ‖L2 EOC ‖τxy − τxyh ‖L2 EOC ‖τyy − τyyh ‖L2 EOC

3 7.378e-01 2.77 4.268e-01 2.09 6.474e-01 2.10

4 1.078e-01 2.28 9.977e-02 2.49 1.507e-01 2.81

5 2.222e-02 2.41 1.773e-02 2.14 2.148e-02 2.32

6 4.158e-03 2.10 4.012e-03 2.04 4.316e-03 2.09

7 9.685e-04 1.009e-03 9.730e-04

ref ‖τxx − τxxh ‖dg EOC ‖τxy − τxyh ‖dg EOC ‖τyy − τyyh ‖dg EOC

3 3.216e+00 1.31 3.122e+00 1.43 3.342e+00 1.43

4 1.290e+00 1.58 1.156e+00 1.39 1.235e+00 1.54

5 4.296e-01 1.51 4.409e-01 1.49 4.241e-01 1.49

6 1.507e-01 1.50 1.567e-01 1.49 1.511e-01 1.50

7 5.318e-02 5.562e-02 5.326e-02

Figure 7.57: Isolines of the stress component (left) τxx and (right) τxy for the Oldroyd-B

model at Re = 1 , α = 0.41, We = 100 for mesh size h = 2−5.

Page 157: Stability Analysis and Numerical Simulation of Non-Newtonian

7.1. Experimental order of convergence 145

−1 −0.5 0 0.5 1−12

−10

−8

−6

−4

−2

0

2

x −position along y = −0.01

det (

∇ u

)

h = 2−4

h = 2−5

h = 2−6

α = 0.41

Figure 7.58: Variation of det(∇u) along the line y = −0.01 for the Oldroyd-B fluid at

We = 100 and α = 0.41 for different meshes.

−1 −0.5 0 0.5 1−20

−15

−10

−5

0

5

x −position along y = −0.01

det (

∇ u

)

h = 2−4

h = 2−5

h = 2−6

h = 2−7

α = 0.89

Figure 7.59: Variation of det(∇u) along the line y = −0.01 for the Oldroyd-B fluid at

We = 100 and α = 0.89 for different meshes.

Page 158: Stability Analysis and Numerical Simulation of Non-Newtonian

146 Chapter 7. Numerical examples

−1 −0.5 0 0.5 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

x −position along y = −0.01

p −

pres

sure

h = 2−4

h = 2−5

h = 2−6

exact solution

α = 0.41

Figure 7.60: Variation of the pressure field along the line y = −0.01 for the Oldroyd-B fluid

at We = 100 and α = 0.41 on different meshes.

−1 −0.5 0 0.5 10.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

x −position along y = −0.01

p −

pres

sure

h = 2−4

h = 2−5

h = 2−6

exact solution

α = 0.89

Figure 7.61: Variation of the pressure field along the line y = −0.01 for the Oldroyd-B fluid

at We = 100 and α = 0.89 on different meshes.

Page 159: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 147

7.2 Benchmark problems

7.2.1 Lid-driven cavity

In this subsection, a test problem will be considered for the Oldroyd system, in the square

duct Ω = [0, 1]2 with zero body forces and with nonzero velocity field prescribed on the lid

u|y=1= 16x2(1− x)2.

This benchmark problem is taken from [41] where a nonzero body force is considered.

Since the velocity is set to be zero at the corners it is a type of a regularized cavity problem.

Because the flow domain Ω has no inflow boundary, no boundary conditions are needed for

the stress field.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

Figure 7.62: Streamlines in the (left) hole cavity; (right) right bottom corner.

First numerical calculations were carried out for the Oldroyd-B model in the parameter

area α = 0.89, Re ∈ 1, 10, 100 and We ≤ 0.45. In Fig. 7.63 the time evolution of the norm

||τn+1xx − τnxx||L2

corresponding to the flow of an Oldroyd-B fluid at Re = 1, α = 0.89 was

plotted. Convergence of the simulations were obtained for mesh size h = 2−6 and time step

∆t = 10−3 until We ≈ 0.4. By further increasing of the Weißenberg number to We = 0.45

the simulation was blowing up even when one scales down the time step width.

Figure 7.62(left) shows the streamlines in the hole cavity for Re = 1, We = 0.4, a = 1 and

α = 0.89. Also in the corners vortices will be generate. The vortex in the right bottom corner

can be observed in Fig. 7.62(right). In Table 7.11 the values together with the location of the

maximal vortex intensity of the Oldroyd-B fluid flow in the lid-driven cavity were registered

for α = 0.89 and different values of Re and We. The corresponding curves were plotted

in Fig. 7.66. At constant Re and by increasing We, the vortex intensity decreases and the

vortex core was displaced toward left and upstairs. One can observe that at constant We

Page 160: Stability Analysis and Numerical Simulation of Non-Newtonian

148 Chapter 7. Numerical examples

Table 7.11: Maximum value of the stream function for the Oldroyd-B model at α = 0.89

Re = 1 Re = 10 Re = 100

We Ψmax × 102 (x, y) Ψmax × 102 (x, y) Ψmax × 102 (x, y)

NSt 8.365 (0.515, 0.781) 8.364 (0.601, 0.757) 8.710 (0.562, 0.593)

0.05 8.257 (0.500, 0.781) 8.244 (0.515, 0.781) 8.121 (0.609, 0.765)

0.1 8.002 (0.500, 0.781) 7.978 (0.507, 0.781) 7.742 (0.609, 0.781)

0.15 7.692 (0.492, 0.789) 7.663 (0.507, 0.789) 7.327 (0.609, 0.781)

0.2 7.254 (0.492, 0.789) 7.341 (0.500, 0.789) 6.927 (0.601, 0.796)

0.25 6.957 (0.484, 0.789) 7.033 (0.500, 0.796) 6.559 (0.593, 0.804)

0.3 6.789 (0.484, 0.796) 6.748 (0.492, 0.796) 6.221 (0.585, 0.812)

0.35 6.530 (0.476, 0.796) 6.485 (0.484, 0.796) 5.922 (0.585, 0.820)

0.4 6.291 (0.468, 0.796) 6.245 (0.484, 0.804) 5.658 (0.578, 0.828)

0 2 4 6 810

−8

10−6

10−4

10−2

100

We = 0.2We = 0.3We = 0.4We = 0.45

t −time

||τxxn+

1 −τ xxn

|| L 2 −no

rm

Figure 7.63: Time evolution of the norm ||τn+1xx − τnxx||L2

for the Oldroyd-B fluid at Re = 1,

α = 0.89 and different We for mesh size h = 2−6 and time step ∆t = 10−3.

Page 161: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 149

ψ

τxx

τxy

τyy

Figure 7.64: Streamlines and stress isolines for Oldroyd-B fluid at α = 0.89, We = 0.05 and

(left) Re = 1; (middle) Re = 10; (right) Re = 100.

Page 162: Stability Analysis and Numerical Simulation of Non-Newtonian

150 Chapter 7. Numerical examples

ψ

τxx

τxy

τyy

Figure 7.65: Streamlines and stress isolines for a = 1, α = 0.89 and We = 0.4 and

(left) Re = 1; (middle) Re = 10; (right) Re = 100.

Page 163: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 151

0 0.1 0.2 0.3 0.45.5

6

6.5

7

7.5

8

8.5

9

Re = 1Re = 10Re = 100

We

Ψm

ax ×

102

Figure 7.66: Dependence of the vortex intensity on We for the Oldroyd-B fluid at α = 0.89

and Re ∈ 1, 10, 100.

0 0.5 1 1.5 24.5

5

5.5

6

6.5

7

7.5

8

8.5

a = 1, α = 0.41

a = 1, α = 0.89

a = 0, α = 0.89

We

Ψm

ax ×

102

Figure 7.67: Dependence of the vortex intensity on We for the Oldroyd fluid at Re = 1 and

different values of a and α.

Page 164: Stability Analysis and Numerical Simulation of Non-Newtonian

152 Chapter 7. Numerical examples

0 0.2 0.4 0.6 0.8 115

15.5

16

16.5

17

17.5

18

18.5

19

19.5

20

x −position

real

(λ −

)

y = 0.8

y = 0.9

y = 0.95

y = 0.96

We = 0.05

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x −position

real

(λ −

)

y = 0.8

y = 0.9

y = 0.95

y = 0.96

We = 0.4

Figure 7.68: Variation of the real part of eigenvalue λ−, (5.75), along different horizontal

lines in the lid vicinity for the Oldroyd-B fluid at Re = 1, α = 0.89 and (up) We = 0.05;

(down) We = 0.4 for mesh size h = 2−6.

Page 165: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 153

the vortex intensity decreases when the Reynolds number increases, on the other hand, the

position of the vortex displaces towards right, as one can see also in the streamlines plots

from Fig. 7.65 and Fig. 7.64.

An overview of changing in the streamlines and stress fields with the Reynolds number,

for the Oldroyd-B fluid at α = 0.89, Re ∈ 1, 10, 100, We = 0.05 and We = 0.4, was given

in Fig. 7.64 and Fig. 7.65 respectively.

In order to give an impression what happens for other values of α or when a = 0, numerical

tests were done for the Oldroyd-B model at α = 0.41 and for the Oldroyd model with a = 0

at α = 0.89, when keeping the Reynolds number fixed at the value Re = 1. Convergence of

the Oldroyd-B model with α = 0.41 was obtained until We = 0.75. The time step width ∆t

was taken to be 10−2 until We = 0.3 and for higher values of We to be 10−3. In Fig. 7.67 the

dependence of the vortex intensity on We for Re = 1, α ∈ 0.41, 0.89 and a ∈ 0, 1 was

represented for comparison. Figs. 7.69-7.74 show the changing in the pressure and stress

fields, for different values of a, α and We. For a = 0 by means of the isoline plots in figures

7.73 and 7.74 an equilibrium between the stress components τxx and τyy can be observed.

For the Oldroyd model with a = 0 and α = 0.89 convergence to the steady state was

achieved until We = 1 when the time step width was taken to be ∆t = 10−2 and the mesh

size h = 2−6, as one can see in Fig. 7.75. By further increasing the Weißenberg number

the flow becomes nonstationary. By scaling down the time step to ∆t = 10−3 one gets

for We ∈ 1.1, 2.0 likewise a periodically flow. The corresponding evolution curves were

represented in Fig. 7.76.

This periodically flow is treated for computations with the time step width ∆t = 10−2.

In Fig. 7.77 and Fig. 7.78 the periodical time evolution of the stress L2 error norms can

be viewed for We = 1.1 and We = 2.0, respectively. The time evolution of the vortex in-

tensity at We = 1.1 and We = 2.0 was plotted in Fig. 7.79 and Fig. 7.80, respectively.

For We = 1.1 the maximum of the stream function was reached at the same position

(x, y) = (0.515625, 0.828125), but for We = 2.0 the maximum change periodically between

(0.507812, 0.804688) and (0.5, 0.804688).

Let us analyse now the behavior of the eigenvalues of matrix A5, (5.83) at some points

in the flow domain, on the basis of the numerical data of the velocity and stress fields.

This analysis was accomplished on a structured mesh with mesh size h = 2−6. For such

a structured triangular mesh each inner point is vertex of eight triangles, and so at each

interior point eight degrees of freedom exists for the stress field. At the boundary points

there exists four and at the corner point of the square domain two degrees of freedom for

the stress field and also for the velocity gradient field.

A comparison between the real part of the smallest solution of equation (5.88) and equation

(5.89) for the wave vector ξ = (ξ1,−ξ1) at boundary points for We = 2.0 and ‖ξ‖2/Re = 1

was presented in Fig. 7.81 and Fig. 7.82, respectively. These figures show passage in the

negative region at one of the two degrees of freedom at the corner (x, y) = (1.0, 1.0). At all

other test points on the upper and right boundary parts the real parts of the eigenvalues

lies in the stable region. It seems that the vicinity of the corner (x, y) = (1.0, 1.0) is most

Page 166: Stability Analysis and Numerical Simulation of Non-Newtonian

154 Chapter 7. Numerical examples

Figure 7.69: Isolines of (left) stream function; (right) pressure for Oldroyd-B fluid at We =

0.4 and α = 0.41.

Figure 7.70: Isolines of (left) stream function; (right) pressure for Oldroyd fluid with a = 0

at α = 0.89 and We = 0.4.

Figure 7.71: Isolines of (left) stream function; (right) pressure for Oldroyd fluid with a = 0

at α = 0.89 and We = 1.

Page 167: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 155

Figure 7.72: Isolines of the stress components (left) τxx; (middle) τxy; (right) τyy for

Oldroyd-B fluid at We = 0.4 and α = 0.41.

Figure 7.73: Isolines of the stress components (left) τxx; (middle) τxy; (right) τyy for Oldroyd

fluid with a = 0 at α = 0.89 and We = 0.4.

Figure 7.74: Isolines of the stress components (left) τxx; (middle) τxy; (right) τyy for Oldroyd

fluid with a = 0 at α = 0.89 and We = 1.

Page 168: Stability Analysis and Numerical Simulation of Non-Newtonian

156 Chapter 7. Numerical examples

0 10 20 30 40 50 60

10−8

10−6

10−4

10−2

We = 0.6We = 0.8We = 1.0We = 1.1We = 2.0

t −time

||τxxn+

1 −τ xxn

|| L 2 −no

rm

Figure 7.75: Time evolution of the norm ||τn+1xx − τnxx||L2

for the Oldroyd fluid at a = 0,

Re = 1, α = 0.89 and different We for mesh size h = 2−6 and time step ∆t = 10−2.

25 30 35 40 45 5010

−5

10−4

10−3 We = 2 ∆ t = 10−2

We = 1.1 ∆ t = 10−2

We = 2 ∆ t = 10−3

We = 1.1 ∆ t = 10−3

t −time

||τxxn+

1 −τ xxn

|| L 2 −no

rm

Figure 7.76: Time evolution of the norm ||τn+1xx − τnxx||L2

for the Oldroyd fluid at a = 0,

Re = 1, α = 0.89 and We ∈ 1.1, 2.0 for mesh size h = 2−6 and time step ∆t ∈ 10−2, 10−3.

Page 169: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 157

40 42 44 46 48 50

0.3475

0.348

0.3485

0.349

0.3495

||τxxh ||

L2 ||τ

yyh ||

L2

t −time

erro

r no

rm

Figure 7.77: Time periodicity of the stress L2-error norms for Oldroyd fluid with a = 0,

α = 0.89 and We = 1.1 for h = 2−6 and ∆t = 10−2.

40 42 44 46 48 500.2845

0.285

0.2855

0.286

0.2865

0.287

||τxxh ||

L2 ||τ

yyh ||

L2

t −time

erro

r no

rm

Figure 7.78: Time periodicity of the stress L2-error norms for Oldroyd fluid with a = 0,

α = 0.89 and We = 2.0 for h = 2−6 and ∆t = 10−2.

Page 170: Stability Analysis and Numerical Simulation of Non-Newtonian

158 Chapter 7. Numerical examples

40 42 44 46 48 50

4.838

4.839

4.84

4.841

4.842

4.843

4.844

4.845

t −time

Ψm

ax ×

102

Figure 7.79: Time periodicity of the vortex intensity for Oldroyd fluid with a = 0, α = 0.89

and We = 1.1 for h = 2−6 and ∆t = 10−2.

40 42 44 46 48 505.5

5.52

5.54

5.56

5.58

5.6

5.62

t −time

Ψm

ax ×

102

Figure 7.80: Time periodicity of the vortex intensity for Oldroyd fluid with a = 0, α = 0.89

and We = 2.0 for h = 2−6 and ∆t = 10−2.

Page 171: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 159

40 42 44 46 48 50−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t −time

λ −

eige

nval

ues

(x,y) = (0.5, 1.0)

(x,y) = (0.75,1.0)

(x,y) = (1.0, 1.0)

(x,y) = (1.0, 0.75)

Figure 7.81: Time periodicity of the smallest solution of equation (5.88) at boundary points

for a = 0, α = 0.89, We = 2.0 and ‖ξ‖2/Re = 1.

40 42 44 46 48 50−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t −time

λ −

eige

nval

ues

(x,y) = (0.5, 1.0)

(x,y) = (0.75,1.0)

(x,y) = (1.0, 1.0)

(x,y) = (1.0, 0.75)

Figure 7.82: Time periodicity of the smallest solution of equation (5.89) for the wave vector

ξ = (ξ1,−ξ1) at boundary points for a = 0, α = 0.89, We = 2.0 and ‖ξ‖2/Re = 1.

Page 172: Stability Analysis and Numerical Simulation of Non-Newtonian

160 Chapter 7. Numerical examples

problematically. Therefore, the adjacent diagonal points were analyzed, namely (0.984, 0.984),

(0.968, 0.968) and (0.953, 0.953). From these three points at (x, y) = (0.984, 0.984) the mini-

mal real parts of the eigenvalues were reached but also for the other points the smallest eigen-

values becomes negative. The time evolution of the smallest solution of equation (5.88), cor-

responding to the three smallest of the eight eigenvalues at the point (x, y) = (0.984, 0.984)

were plotted in Fig. 7.83-Fig. 7.85 for ‖ξ‖2/Re = 1 at We ∈ 1.0, 1.1, 2.0, respectively. For

We = 0.9 or smaller values the real parts of these eigenvalues are all positive.

40 42 44 46 48 50−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

t −time

λ −

eige

nval

ues

Figure 7.83: Time uniformity of the smallest solution of equation (5.88), corresponding to the

three smallest eigenvalues of the eight degrees of freedom at the point (x, y) = (0.984, 0.984)

for a = 0, α = 0.89, We = 1.0 and ‖ξ‖2/Re = 1.

Page 173: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 161

40 42 44 46 48 50

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t −time

λ −

eige

nval

ues

Figure 7.84: Time periodicity of the smallest solution of equation (5.88), corresponding to the

three smallest eigenvalues of the eight degrees of freedom at the point (x, y) = (0.984, 0.984)

for a = 0, α = 0.89, We = 1.1 and ‖ξ‖2/Re = 1.

40 42 44 46 48 50−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t −time

λ −

eige

nval

ues

Figure 7.85: Time periodicity of the smallest solution of equation (5.88), corresponding to the

three smallest eigenvalues of the eight degrees of freedom at the point (x, y) = (0.984, 0.984)

for a = 0, α = 0.89, We = 2.0 and ‖ξ‖2/Re = 1.

Page 174: Stability Analysis and Numerical Simulation of Non-Newtonian

162 Chapter 7. Numerical examples

7.2.2 Four-to-one planar contraction

Many flow problems of interest involve stress singularities at corners of the flow domain.

For instance, the contraction flow which is shown in Fig. 7.86. At the point where the flow

domain narrows, there is a re-entrant θ = 32π corner. The resolution of the stress singularity

associated with this corner is a major numerical problem.

L1 ≥ 40H2

L2 ≥ 100H2xR

U1

inletH1

U2

outletH2

y

xaxis of symmetry

re-entrant corner

+

Figure 7.86: A schematic of the four-to-one planar contraction device.

Renardy [59] pointed out the difference between the Newtonian and non-Newtonian sit-

uations of flows with corner singularities. Suppose r is the distance from the corner. If

the velocity is proportional with rθ , then the viscous stresses behave like rθ−1, while the

Reynolds stresses behave like r2θ. As long as θ is positive, the viscous stresses will always

dominate over the Reynolds stresses, and consequently the local flow near the corner is

described by the Stokes problem

∆u−∇ p = 0, div u = 0.

This leads for Newtonian flows to the separable form Ψ(r, φ) ≈ rλf(φ), 0 ≤ φ ≤ θ of the

stream function around the corner. The function f(φ) is found as the eigenfunction of an

eigenvalue problem

λ(λ− 2)− (λ− 1)2 cos(2θ) + cos(2θ(λ− 1)) = 0,

with eigenvalue λ. The smallest eigenvalue determined in [59] is less than 2 if θ > π and has

real part greater than 2 if θ < π. Renardy concluded that for θ < π the velocity gradient,

and hence the viscous stresses at the corner, is zero. In the presence of non-Newtonian

effects, this makes the local Weißenberg number at the corner equal to zero, and one can

expect a corner behavior which is dominated by the Newtonian part. If, on the other hand,

θ > π, then the velocity gradient and viscous stresses are infinite at the corner. In that

Page 175: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 163

case, the local Weißenberg number is infinite and the non-Newtonian corner behavior is

fundamentally different from the Newtonian case.

For flow in the four-to-one contraction (θ = 32π), it is found that the smallest eigenvalue

for the reentrant corner is λ = 1.544 (see e.g.[59]), so that the velocity gradient and the stress

components grow as r−0.456 as the corner is approach, i.e. as r → 0. In the non-Newtonian

case the stress near the corner is also singular and the singularity is stronger than for a

Newtonian liquid.

Alcocer-Gallo in [4] was starting from the idea that the corner singularity is integrable

and thus the flow away from the corner is not effected when the flow around the corner

is resolved by using a radial mesh with sufficient resolution in the tangential direction at

the corner. Since the velocity gradient field retains angular dependence even in the limit

r → 0, in [4] the behaviour of viscoelastic fluids into a four-to-one contraction on radial

meshes around the corner was investigated. Here, computations of the FENE fluid model

were carried out in the range of values Re = 3.3 · 10−3 and We ≤ 1. The numerical results

were showing that by using radial meshes, convergend solutions both near and away from

the corner can be obtained and that there is a region near the corner in which the velocity

gradient tensor for the viscoelastic fluids is of generalized power law form.

The purpose of this chapter is to provide quantitative data for the flow through a four-

to-one planar contraction of viscoelastic liquids obeying the constant viscosity Oldroyd-B

constitutive model. The data given comprises the size of the corner vortex formed upstream

of the contraction, its intensity in terms of entrapped flow rate, longitudinal profiles of

velocity and normal stress along the centerline, and variation of stress near the singular

re-entrant corner.

Comparisons with experiments are at best qualitative because of the poor representation

of the real polymeric fluids by the Oldroyd-B model. The deficiencies of this model are

firstly the absence of the shear thinning property (see relations (1.18)) of the polymeric

materials. Then, the second normal stress difference is zero whereas for real polymeric

fluids it is negative. Thirdly, the flow near the contraction plane is a complex mixture of

shear and elongation and the Oldroyd-B model predicts an infinite value for the steady state

elongational viscosity (see relation (1.19)).

Experimental results

The planar four-to-one contraction is equally relevant to engineering flows in extrusion

dies. It is suited for visualization studies through birefringence strand techniques and laser-

Doppler velocimetry, as in the works [3, 42, 56, 57].

The fluids for experiment used by Ahmed et al. in [3] were two high-density polyethylenes,

Natene and Rigidex and one low-density polyethylene. Here the experimental centerline and

global stress distributions, velocity distributions and vortex recirculation were compared with

numerical simulations using the commercial software Polyflow applied to the K-BKZ inte-

gral constitutive equation. Self consistency in stress data between experiment and simulation

Page 176: Stability Analysis and Numerical Simulation of Non-Newtonian

164 Chapter 7. Numerical examples

was obtained for Natene. In the case of Rigidex, it was found that numerical extensional

flow behavior in the re-entrant flow could be matched with experiment. In the case of low

density polyethylene a significant difference was seen in the re-entrant flow behavior.

Quinzani et al. in [56, 57] used in their experiments, a solution of polyisobutilene in

tetradecane that exhibits shear thinning of both the viscosity and the first normal stress

coefficient. In this experiment, the shear-rate-dependent Weißenberg and Reynolds numbers

for the contraction flow were accessible in the range 0.25 ≤We ≤ 0.77 and 0.08 ≤ Re ≤ 1.43.

Experimental results concerning the extrusion instabilities were presented by Legrand and

Piau in [42], where a polymer melt of high molecular weight, polydimethylsiloxane, is used.

Geometry and computational meshes

A sketch of the contraction geometry is given in Fig. 7.86. Since the two-dimensional flow

domain is symmetric, only half of the domain is used for the computations. To be consistent

with previous work, the half-width of the shorter downstream channel of the contraction

H2 was taken as the characteristic length scale and the average velocity in that channel U2

was the characteristic velocity scale. Stress and pressure were normalized with µ0U2/H2.

The length of the upstream and downstream channel were assumed to be L1 = 40H2 and

L2 = 40H2, respectively.

The computational domain is assumed to be long enough to verify Poiseuille velocity

profiles at inlet and outlet sections Γ1 and Γ2 respectively. Thus, for velocity the following

boundary conditions were taken

u|Γ1=

3

8

(

1− y

4

2)

, u|Γ2=

3

2

(

1− y2)

.

Symmetry conditions were imposed at the centerline y = 0 as follows

uy = 0, 2(1− α)∂ux∂y− p+ τxx = 0 at y = 0.

At the upstream section the inflow boundary conditions for the stress field are taken from the

analytical solution of the Oldroyd-B problem corresponding to the fully developed Poiseuille

flow. Thus the inflow boundary conditions for the stress components are given by

τxx|Γ1= 18αWe

y2

163,

τxy |Γ1= −3α

y

43,

τyy |Γ1= 0.

Computations has been carried out on two consecutively refined meshes. The mesh M1

represented in Fig. 7.87 has the refinement size h = 2−3 in the contraction region, and the

mesh M2 was obtained by globally refining M1. The mesh data in Table 7.12 comprises the

total number of elements in the mesh (NE), the degrees of freedom for velocity (DOFv), for

Page 177: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 165

Figure 7.87: Section of the structured mesh M1 in the contraction vicinity.

Table 7.12: Major characteristics of the computational meshes

Mesh NE DOFv DOFp DOFτ h

M1 7740 16069 4165 23220 2−3

M2 30960 63097 16069 92880 2−4

pressure (DOFp) and for the stress components (DOFτ ) and the mesh width h normalized

with H2. To allow comparison with existing results, the Reynolds number is taken small

Re = 10−3. The fraction of viscoelastic viscosities was kept constant at the value α = 0.89,

often used in the literature for this problem.

Numerical results for the Oldroyd-B model

Simulation of the flow into a four-to-one contraction of the Oldroyd-B fluid were done on the

meshes M1 and M2 at α = 0.89 and Re = 10−3. On mesh M1 the computations with time

step width ∆t = 10−2 converge until We = 0.4. On the finer mesh M2 the computations

with time step width ∆t = 10−2 converge until We = 0.3 and for We = 0.4 the time step

must be scaled down to ∆t = 10−3 for achieveing convergence.

The reason of blowing up of the simulations for higher Weißenberg numbers is the violation

of the stability condition (5.66) in combination with the stagnation point at the vicinity of

the re-entrant corner on the streamline comprised in the boundary part Γ∗ = x ≤ 0, y =

4∪x = 0, y ∈ [1, 4]∪x ≥ 0, y = 1. In Fig. 7.90 the variation of the eigenvalue λ−, from

(5.75), along the line x = 0 was plotted. One can observe that the finer the mesh, the larger

the negative real part of λ−, because the more accurate the velocity field is computed. Also

the real part of λ− decreases by increasing the Weißenberg number and becomes negative

Page 178: Stability Analysis and Numerical Simulation of Non-Newtonian

166 Chapter 7. Numerical examples

at We = 0.4. After that value the computations blows up.

The accuracy of the computations can be asserted by considering the full-developed

Poiseuille profiles for velocity and stress at the downstream section. In Fig. 7.88 the ve-

locity component ux were plotted in the section x = 20 in comparison to the analytical

solutions of the Poiseuille profiles at We = 0.4. For We = 0.2 the numerical solution for

the stress component τxx shows, in Fig. 7.89, very good agreement with the the analytical

solution. For higher values of the Weißenberg number, We = 0.4, the computations on mesh

M1 agree well with the analytical solution but the computations on the finer mesh M2 are

quite deviated in the vicinity of the upper downstream boundary where the influence of the

negative real part of the eigenvalue λ− becomes noticeable.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

y −position

u x −ve

loci

ty c

ompo

nent

mesh M1

mesh M2

analytical solution

Figure 7.88: Comparison of the numerical to the analytical distribution of the velocity

component ux along the line x = 20 at We = 0.4.

Numerical data from literature for this problem, as concluded in [6], illustrate a high

level of scatter in these data. The results of all these computations may be divided into

two classes: qualitative and quantitative. The qualitative results to be given essentially

comprise streamline plots, an effective way of illustrating the effect of elasticity on vortex

enchancement. However the quantitative results are more relevant. These results comprise

tables and figures for the recirculating corner vortex size and intensity Ψmax.

For the purpose of comparison with data for the vortex intensity Ψmax reported in the

literature, the recent data of Meng et al. [48] and Alves et al. [6] were included in Fig. 7.91.

In this work, convergence could be achieved up to a level We ≈ 0.4. In the range of We for

Page 179: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 167

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

7

y −position

τ xx −

stre

ss c

ompo

nent

We = 0.4 mesh M1

We = 0.4 mesh M2

We = 0.4 analytical solution

We = 0.2 mesh M1

We = 0.2 mesh M2

We = 0.2 analytical solution

Figure 7.89: Comparison of the numerical to the analytical distribution of the stress com-

ponent τxx along the line x = 20 at We = 0.2 and We = 0.4.

which the program converges there could be shown very good agreement with the results of

Alves et al. [6]. It is not surprising that the data of Meng et al. [48] are different, because also

their qualitative results covering the corner vortex are quite different. The vortex intensity

is seen to decrease with increasing Weißenberg number.

A view of the streamlines, the various stress component fields and the pressure field in

the entrance region was provided in Fig. 7.93 for two values of the Weißenberg number, i.e.

We = 0.01 and We = 0.4. All fields are smooth, with highly localized stress concentration

at the walls adjacent to the re-entrant corner.

In Fig. 7.94, one can see the profile ux velocity component along the axis of symmetry

close to the re-entrant corner. A velocity overshoot may be observed, in comparison with the

downstream Poiseuille velocity profile, which for Newtonian fluid is absent. This overshoot

was found to increase versus We and was reported also in [63] and [64]. In Fig. 7.95 we

observed the influence of the Weißenberg number on the stress component τxx along the line

y = 1 close to the re-entrant corner. The peak, which is theoretically infinite at the corner

singularity (see [59]), becomes sharp and higher as the Weißenberg number increases. At each

Weißenberg number the stress at the downstream section tends to the stress corresponding to

the full-developed Poiseuille flow at that section. Similar phenomena could be also observed

for the other stress components τxy and τyy.

Page 180: Stability Analysis and Numerical Simulation of Non-Newtonian

168 Chapter 7. Numerical examples

0 1 2 3 40.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

M1

M2

We = 0.2

y −position along the line x = 0

real

(λ −

)

0 1 2 3 4−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

M1M2

We = 0.4

y −position along the line x = 0

real

(λ −

)

Figure 7.90: Real parts of the eigenvalue λ− along the line x = 0 for (left) We = 0.2;

(right) We = 0.4.

Page 181: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 169

0 0.2 0.4 0.6 0.8 10.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Meng et al. (2002)

Alves et al. (2003)

this work

We

Ψm

ax ×

103

Figure 7.91: Dependence of the corner vortex intensity on the Weißenberg number.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

Figure 7.92: Streamlines and corner vortex for (left) We = 0.01; (right) We = 0.4.

Page 182: Stability Analysis and Numerical Simulation of Non-Newtonian

170 Chapter 7. Numerical examples

Ψ

p

τxx

τxy

τyy

Ψ

p

τxx

τxy

τyy

Figure 7.93: Sections of the contraction geometry. Streamlines, pressure and stress isolines

for (left) We = 0.01; (right) We = 0.4.

Page 183: Stability Analysis and Numerical Simulation of Non-Newtonian

7.2. Benchmark problems 171

−20 −10 0 10 20

0.4

0.6

0.8

1

1.2

1.4

1.6

x −position

u x −ve

loci

ty c

ompo

nent

−20 −10 0 10 20

0.4

0.6

0.8

1

1.2

1.4

1.6

x −positionu x −

velo

city

com

pone

nt

Figure 7.94: Velocity profile along the axis of symmetry of the planar contraction on mesh

M2 for (left) We = 0.01; (right) We = 0.4.

−20 −10 0 10 20−5

−4

−3

−2

−1

0

1

We = 0.01

x −position

τ xx −

stre

ss c

ompo

nent

−20 −10 0 10 20

−2

0

2

4

6

8

We = 0.1

x −position

τ xx −

stre

ss c

ompo

nent

−20 −10 0 10 20

−2

0

2

4

6

8

10

We = 0.2

x −position

τ xx −

stre

ss c

ompo

nent

−20 −10 0 10 20

0

5

10

15

We = 0.4

x −position

τ xx −

stre

ss c

ompo

nent

Figure 7.95: Stress component τxx along the line y = 1 in the planar contraction problem

on mesh M2 for different We.

Page 184: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 185: Stability Analysis and Numerical Simulation of Non-Newtonian

Chapter 8

Concluding remarks

The equations governing the flow of viscoelastic non-Newtonian fluids have been intensively

studied due to their industrial applications in polymer processing. In addition to the conser-

vation of mass and momentum, a material constitutive equation was necessary for taking into

account the memory effects of such polymeric materials. In 1950, J. G. Oldroyd proposed

a differential equation for modeling viscoelastic fluids [53]. During the last few decades,

significant progress has been made in the development of numerical algorithms for stable

and accurate solution of viscoelastic flow problems. For a number of benchmark problems

in case of steady flows, the reviews e.g. [19, 10, 39] showed that the limits in the maximum

attainable Weißenberg number still exist independently of the numerical method used for

solving the Oldroyd problem.

In this thesis, a numerical method was introduced for solving non-Newtonian viscoelastic

fluid models of Oldroyd type, based on the finite element spatial discretization and on the

fractional step θ-scheme time discretization used as operator splitting method. For the

velocity and pressure fields, the stable Taylor-Hood finite element was used, whereas the

stress field was discretized using discontinuous elements which satisfy an inf-sup condition

in relation to the velocity space. Due to the mixed hyperbolic-parabolic character of the

time-dependent system of equations governing the motion of an Oldroyd fluid, the basic

idea in the present numerical approach was to decouple the calculation of the velocity and

pressure fields from that of the stress field. By the operator splitting algorithm, one reduces

the Oldroyd system to three simple subproblems: a Stokes like problem, a Burgers like one

and a stress transport problem.

A comprehensive stability analysis was given in chapter 5, for the Oldroyd system of

equations starting from the continuous case, where already stability limits occurred. Further

on, the semi-discretized in time Oldroyd system was analyzed and also the stability of

the fractional step θ-scheme coupled with the finite element approximation applied to the

linearized Oldroyd problem was investigated. The spectral analysis of the θ-scheme applied

to the linearized Oldroyd system, by neglecting the nonlinear terms, was showing good

Page 186: Stability Analysis and Numerical Simulation of Non-Newtonian

174 Chapter 8. Concluding remarks

stability properties and second order accuracy of the time discretization scheme. For the

linear Oldroyd system no restrictions for the problem parameters were found.

Considering the pure stress equation without the stress transport term and with a given

stationary velocity field, in the case of a 6= 0, a stability limit can exist. More precisely, if

their exist a point or a region in the computational domain where the velocity field is so that

det(∇u) < 0 then for a = ±1 the stress equation is stable until Wecr = (2√

−det(∇u))−1.

In the case of a = 0, no stability limit was found.

Considering the stress constitutive equation with given velocity field, the stress will be

transported along the characteristics which were the streamlines. Along streamlines which

leave the computational domain, no perturbation arise. But, if there exist a stagnation point

of the flow in the region where det(∇u) < 0 (in the case a = ±1), or if the streamlines are

closed curves which intersect such a region, then instability of the stress components along

the streamlines arises for Weißenberg numbers over the critical value. For a = 0 the pure

stress equation was not affected, but in the full Oldroyd system stability limits exists.

For the Oldroyd-B system at least the stability limits arising in the pure stress consti-

tutive equation exists, if the flow domain comprises a stagnation point in the region where

det(∇u) < 0, or if the streamlines are closed curves which intersect such a region. The

eigenvalue analysis was showing that even in the case of a = 0 instability regime can occur.

The numerical tests were confirming this stability limits.

Two benchmark problems were studied: the lid driven cavity and the four-to-one planar

contraction problem. For the lid driven cavity benchmark problem, the numerical behavior

of the Oldroyd fluids in the cases a = 1 and a = 0 was analyzed. For the Oldroyd-B fluid

convergence of the numerical program for α = 0.89 was arrived until We ≈ 0.4. In the

case of a = 0 convergence to steady solutions was reach until We ≈ 1.0. Also, the reached

Weißenberg number depends on the rate of viscoelastic viscosities α; smaller values of α lead

to higher Weißenberg limits of stability. Our numerical results confirmed that the Reynolds

number (as long as the flow is laminar) has no influence on the stability of the stress field.

For the Oldroyd-B fluid flow into a four-to-one contraction, in the limits of convergence of

our numerical code, We ≈ 0.4 for α = 0.89, very good agreement with the results from the

literature were obtained.

The numerical implementation of the Oldroyd system was based on the Navier-Stokes

solver incorporated in the program package Albert. The principal personal contribution

was the implementation of the stress tensor field with the corresponding routines for assem-

bling the Oldroyd system and solving the stress transport equation by the discontinuous

Galerkin method.

Although the author largely focus in this work on the Oldroyd fluid model, the numerical

algorithm can readily be extended to other differential and rate type non-Newtonian fluid

models.

Page 187: Stability Analysis and Numerical Simulation of Non-Newtonian

Appendix A

Stability analysis

A.1 A priori stability estimates of the fractional θ-scheme

Schwarz’s equality 2(x, y) = ‖x‖2 + ‖y‖2 − ‖x− y‖2

Young’s inquality 2(x, y) ≤ ε‖x‖2 + ε−1‖y‖2

A.2 Spectral analysis of the Oldroyd System

A.2.1 Spectral analysis of the θ-scheme for the linearized Oldroyd

system

10−5

100

105

1010−0.5

0

0.5

1

1.5

2

2.5

10−5

100

105

10100

0.5

1

1.5

w w

φ ϕ

Figure A.1: Dependence of (left) φ; (right) ϕ on w.

Page 188: Stability Analysis and Numerical Simulation of Non-Newtonian

176 Appendix A. Stability analysis

10−5

100

105

10100.01

0.015

0.02

0.025

10−5

100

105

1010

0.01

0.015

0.02

0.025

10−5

100

105

10100.26

0.27

0.28

0.29

0.3

0.31

10−5

100

105

10100.09

0.1

0.11

0.12

0.13

0.14

10−5

100

105

1010

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

10−5

100

105

10100.8

1

1.2

1.4

1.6

1.8

w w

w w

ww

c11 c11

c21 c21

c31

c31

Figure A.2: Dependence of coefficients c31, c21, c

11 from the numerator of (left) (1 + k2);

(right) (1− k2) on w.

Page 189: Stability Analysis and Numerical Simulation of Non-Newtonian

A.2. Spectral analysis of the Oldroyd System 177

10−5

100

105

10100.04

0.045

0.05

0.055

0.06

0.065

10−5

100

105

1010

0.005

0.01

0.015

0.02

10−5

100

105

10100.05

0.1

0.15

0.2

10−5

100

105

10100.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

10−5

100

105

10100.5

0.6

0.7

0.8

0.9

1

1.1

10−5

100

105

10100

0.2

0.4

0.6

0.8

1

w w

w w

ww

c11 c10

c21c20

c31c30

Figure A.3: Dependence of the coefficients from the numerator of (1 + P0) on w.

Page 190: Stability Analysis and Numerical Simulation of Non-Newtonian

178 Appendix A. Stability analysis

10−5

100

105

10100.008

0.01

0.012

0.014

0.016

10−5

100

105

10102

3

4

5

6

7

8

x 10−3

10−5

100

105

10100.2

0.25

0.3

10−5

100

105

10100.2

0.25

0.3

0.35

10−5

100

105

10100

0.1

0.2

0.3

0.4

0.5

10−5

100

105

1010

0.2

0.4

0.6

0.8

1

w w

w w

ww

c11 c10

c21 c20

c31c30

Figure A.4: Dependence of the coefficients from the numerator of of (1− P0) on w.

Page 191: Stability Analysis and Numerical Simulation of Non-Newtonian

A.2. Spectral analysis of the Oldroyd System 179

10−5

100

105

1010

0.01

0.015

0.02

10−5

100

105

10100.022

0.024

0.026

0.028

0.03

0.032

0.034

10−5

100

105

10100.05

0.1

0.15

0.2

0.25

0.3

10−5

100

105

1010

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

10−5

100

105

1010

−1

−0.5

0

0.5

10−5

100

105

1010

1

1.5

2

2.5

3

3.5

w w

w w

ww

c11c11

c21c21

c31 c31

Figure A.5: Dependence of c31, c21, c

11 from the numerator of (left) (1 + P0 + P1);

(right) (1 + P0 − P1) on w.

Page 192: Stability Analysis and Numerical Simulation of Non-Newtonian

180 Appendix A. Stability analysis

10−5

100

105

10102.15

2.2

2.25

2.3

2.35

2.4

2.45

10−5

100

105

1010

0.4

0.45

0.5

0.55

0.6

0.65

w w

c1c2

Figure A.6: Dependence of the (left) first; (right) second order coefficient from (5.41) on w.

10−5

100

105

10100

1

2

3

4

5

6

10−5

100

105

10100

0.5

1

1.5

2

2.5

w w

c0

c1

Figure A.7: Dependence of the (left) zero; (right) first order coefficients from (5.46) on w.

10−5

100

105

1010−0.4

−0.2

0

0.2

0.4

0.6

0.8

w

c2

Figure A.8: Dependence of the second order coefficients from (5.46) on w.

Page 193: Stability Analysis and Numerical Simulation of Non-Newtonian

Accuracy

For the second order accuracy of the θ-scheme the following expansions around x = 0 are

used, a and b are constants.

(A.1)

x

1 + ax= x− ax2 + a2x3 + O(x4),

1

1 + ax= 1− ax+ a2x2 + O(x3),

1

1 + ax+ bx2= 1− ax+ (a2 − b)x2 + O(x3).

Some helpful approximations around ∆t = 0

(A.2)

a1

a′≈ θ′

θ

[

1− (1− ω)(θ + θ′)∆t

We+ (1− ω)2θ′(θ + θ′)

(

∆t

We

)2]

,

a1a′1

aa′≈ 1− (θ + θ′)

∆t

We+ (ωθ2 + (1− ω)θ′2 + θθ′)

(

∆t

We

)2

,

a1

a2a′≈ θθ′

(

∆t

We

)2

,

d1

d′≈ θ′

θ

[

1− (1− k)l(θ + θ′)∆t

Re+ ((1− k)l)2θ′(θ + θ′)

(

∆t

Re

)2]

,

m ≈ θ∆t

Re− kl

(

θ∆t

Re

)2

+

(

(kl)2 − α l

1− αRe

We

)(

θ∆t

Re

)3

,

ma1

a≈ θ

∆t

Re−(

kl +Re

We

)(

θ∆t

Re

)2

,

ma1

a2≈ θ2

∆t2

ReWe,

md1 ≈ 1− lθ∆t

Re+

(

kl2 − α l

1− αRe

We

)(

θ∆t

Re

)2

,

md1

a≈ θ

∆t

We−(

ω + lWe

Re

)(

θ∆t

We

)2

,

(A.3)

y ≈ θ + θ′

θ

[

−1 + l(kθ + (1− k)(θ + θ′))∆t

Re

+

(

2Re

We

α l

1− αθ2 − l2

[

(1− k)2θ′(θ + θ′) + k(1− k)θ(θ + θ′) + kθ2]

)(

∆t

Re

)2]

.

Page 194: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 195: Stability Analysis and Numerical Simulation of Non-Newtonian

Appendix B

Implementation aspects

Here, the results of the calculation of the element matrices corresponding to several terms of

the Oldroyd system are given. Using the finite element discretization (6.11) of the unknown

fields, one obtains the following discretization of terms. The unknown fields at the actual

time step were denoted by τ, u, and the fields calculated at the step before by τ , u. The

integrals below have to be calculated on the standard simplex and by |det| the Jacobian of

the transformation from the actual element to the standard simplex was denoted.

First, the discretization of the terms in the Stokes and Burgers subproblems is given

(u, v)→[

C 0

0 C

][

uxuy

]

, Cji =

ψjψi|det|,

2(Du,Dv)→[

Axx AxyAxy Ayy

] [

uxuy

]

,

(Axx)ji =

ψj,λsψi,λp

(2λs,xλp,x + λs,yλp,y)|det|,

(Axy)ji =

ψj,λsψi,λp

λs,xλp,y|det|,

(Ayy)ji =

ψj,λsψi,λp

(λs,xλp,x + 2λs,yλp,y)|det|,

((u.∇)u, v)→[

N 0

0 N

] [

uxuy

]

, Nji =

ψjψi,λs(uyλs,x + uλs,y)|det|.

For terms which appear only on the right side the discretization holds

(τ,Dv)→[

T0 τxx + T1 τxyT0 τxy + T1 τyy

]

,(T0)jı =

ψj,λsξı λs,x|det|,

(T1)jı =

ψj,λsξıλs,y|det|,

Page 196: Stability Analysis and Numerical Simulation of Non-Newtonian

184 Appendix B. Implementation aspects

((u.∇)τ,Dv)→[

G0 τxx +G1 τxyG0 τxy +G1 τyy

]

,(G0)jı =

ψj,λsξı,λp

(uxλp,x + uyλp,y)λs,x|det|,

(G1)jı =

ψj,λsξı,λp

(uxλp,x + uyλp,y)λs,y |det|,

(βa(τ,∇u), Dv)→[ −(2aB00 +B21 + aB11)τxx + 2(B20 − aB10)τxy + (B21 − aB11)τyy

−(B20 + aB10)τxx − 2(B21 + aB11)τxy + (2aB01 +B20 − aB10)τyy

]

,

(B00)jı =

ψj,λsξı (∇u)xxλs,x|det|,

(B01)jı =

ψj,λsξı (∇u)xxλs,y|det|,

(B10)jı =1

2

ψj,λsξı ((∇u)xy + (∇u)yx)λs,x|det|,

(B11)jı =1

2

ψj,λsξı ((∇u)xy + (∇u)yx)λs,y |det|,

(B20)jı =1

2

ψj,λsξı ((∇u)yx − (∇u)xy)λs,x|det|,

(B21)jı =1

2

ψj,λsξı ((∇u)yx − (∇u)xy)λs,y |det|.

Likewise for the terms containing the pressure or the right-hand sides f and fs, a similar

discretization is formulated. For terms which appears in the stress transport subproblem

one has

(τ, σ)→

M 0 0

0 M 0

0 0 M

τxxτxyτyy

, Mı =

ξξı|det|,

((u · ∇)τ, σ)→

P 0 0

0 P 0

0 0 P

τxxτxyτyy

, Pı =

ξξı,λs(uxλs,x + uyλs,y)|det|,

(βa(τ,∇u), σ)→

−2aQ0 2(Q2 − aQ1) 0

−(Q2 + aQ1) 0 Q2 − aQ1

0 −2(Q2 + aQ1) 2aQ0

τxxτxyτyy

,

(Q0)ı =

ξξı (∇u)xx|det|,

(Q1)ı =1

2

ξξı ((∇u)xy + (∇u)yx)|det|,

(Q2)ı =1

2

ξξı ((∇u)yx − (∇u)xy)|det|,

Page 197: Stability Analysis and Numerical Simulation of Non-Newtonian

185

and for terms which appears only on the right-hand side one uses

(Du, σ)→

S0 ux

12 (S1 ux + S0 uy)

S1 uy

,(S0)i =

ξψi,λsλs,x|det|,

(S1)i =

ξψi,λsλs,y|det|.

Here, the discretization only in the two-dimensional case was given, but this can be easily

generalized to the three-dimensional case.

Page 198: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 199: Stability Analysis and Numerical Simulation of Non-Newtonian

Appendix C

Numerical Examples

C.1 Experimental order of convergence

Let be λ ∈ R, λ 6= 0 and a scalar field ϕex(x) = f(x)/λ, x ∈ R be the solution of the

problem

(C.1) λϕ = f.

Further let us consider the initial-value problem for a scalar field ϕ(x, t)

(C.2)

∂ϕ

∂t+ λϕ = f, t > 0,

ϕ(0, x) = ϕex(x),

where f = λϕex. By writing ϕ as

ϕ(x, t) = ϕex(x) + ϕ(x, t),

the field ϕ satisfies the problem

∂ϕ

∂t+ λϕ = 0, t > 0,

ϕ(0, x) = 0,

which has the solution

ϕ(t, x) = ϕ(0, x)e−λt.

Thus, the solution of problem (C.2) is

(C.3) ϕ(x, t) = ϕex(x) + ϕ(0, x)e−λt.

Page 200: Stability Analysis and Numerical Simulation of Non-Newtonian

188 Appendix C. Numerical Examples

Relation (C.3) shows that for λ ≥ 0 the solution ϕ of problem (C.2) is in fact the stationary

solution of problem (C.1) bounded in time. But if λ < 0 the term e−λt is blowing up in time

and there is no guaranty that the term ϕ(0, x)e−λt is zero, and therefore that the solution

of the transient problem (C.2) is equal to the stationary solution ϕex.

C.1.1 EOC test for example 1

At the corner (x, y) = (0, 1), one has the following values corresponding to the fields (7.2)

Dxx =π

2, Dxy = 0, Wyx = 0,

τxx = τxy = τyy = 1, det(∇u) = −π2

4.

Page 201: Stability Analysis and Numerical Simulation of Non-Newtonian

Bibliography

[1] Aboubacar, M., Matallah, H., Webster, M. F., High elastic solutions for

Oldroyd-B and Pan-Thien/Tanner fluids with a finite volume/element method: planar

contraction flows, J. Non-Newtonian Fluid Mech. 103 (2002), 65–103.

[2] Aboubacar, M., Webster, M. F., A cell-vertex finite volume/element method on

triangles for abrupt contraction viscoelastic flows, J. Non-Newtonian Fluid Mech. 98

(2001), 45–75.

[3] Ahmed, R., Liang, R. F., Mackley, M. R., The experimental observation and

numerical prediction of planar entry flow and die swell for molten polyethylenes, J.

Non-Newtonian Fluid Mech. 59 (1995), 129–153.

[4] Alcocer-Gallo, F., Numerical studies of newtonian and viscoelastic fluids, Disser-

tation, The New Jersey Institute of Technology, 2001.

[5] Alves, M. A., Oliveira, P. J., Pinho, F. T., A convergent and universally bounded

interpolation scheme for the treatment of advection, Int. J. Numer. Meth. Fluids 41

(2003), 47–75.

[6] , Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar con-

tractions, J. Non-Newtonian Fluid Mech. 110 (2003), 45–75.

[7] Arada, N., Sequeira, A., Strong steady solutions for a generalized Oldroyd-B model

with shear-dependent viscosity in a bounded domain, Math. Models Meth. Appl. Sciences

13 (9) (2003), 1303–1323.

[8] Baaijens, F. P. T., Application of low-order discontinuous Galerkin method to the

analysis of viscoelastic flows, J. Non-Newtonian Fluid Mech. 52 (1994), 37–57.

[9] , An iterative solver for the DEVSS/DG method with application to smooth and

non-smooth flows of the upper-convected Maxwell fluid, J. Non-Newtonian Fluid Mech.

75 (2-3) (1998), 119–138.

[10] , Mixed finite element methods for viscoelastic flow analysis: A review, J. Non-

Newtonian Fluid Mech. 79 (1998), 361–385.

Page 202: Stability Analysis and Numerical Simulation of Non-Newtonian

190 Bibliography

[11] Bansch, E., An adaptive finite-element strategy for the three-dimensional time-

dependent Navier-Stokes equations, J. Comp. Appl. Math. 36 (1991), 3–28.

[12] , Numerical methods for the instationary navier-stokes equations with a free

capillary surface, Habilitationsschrift, Albrecht-Ludwigs-Universitat Freiburg, 1998.

[13] , Simulation of instationary, incompressible flows, Acta Math. Univ. Comeni-

anae LXVII (1998), 101–114.

[14] Baranger, J., Sandri, D., Finite element approximation of viscoelastic fluid flow:

Existence of approximate solution and error bounds, Numer. Math. 63 (1992), 13–27.

[15] , Formulation of Stokes problem and the linear elasticity equations suggested by

Oldroyd model for viscoelastic flows, RAIRO Model. Math. Anal. Numer. 26 (1992),

331–345.

[16] Bohme, G., Stromungsmechanik nicht-Newtonscher Fluide, B.G.Teubner Stuttgart,

Stuttgart, 1981.

[17] Brezzi, F., Marini, L. D., Suli, E., Discontionuous Galerkin methods for first-order

hyperbolic problems., Oxford University, Computing Laboratory (2004), 1–12.

[18] Bristeau, M. O., Glowinski, R., Periaux, J., Numerical methods for the Navier-

Stokes equations. Applications to the simulation of compressible and incompressible vis-

cous flows., Computer Physics reports 6 (1987), 73–187.

[19] Caswell, B., Report on the IXth international workshop on numerical methods in

non-Newtonian flows., J. Non-Newtonian Fluid Mech. 62 (1996), 99–110.

[20] Coleman, B. D., Markovitz, H., Noll, W., Viscometric flows of non-Newtonian

fluids. Theory and experiment, Springer-Verlag, Berlin, 1966.

[21] Crochet, M. J., Davies, A. R., Walters, K., Numerical simulation of non-

Newtonian flow, Elsevier, Amsterdam, Oxford, 1984.

[22] Crochet, M. J., Legat, V., The consistent streamline-upwind/Petrov-Galerkin

method for viscoelastic flow revisited., J. Non-Newtonian Fluid Mech. 42 (1992), 283–

299.

[23] Ervin, V. J., Ntasin, L. N., A posteori error estimation and adaptive computation

of viscoelastic fluid flow, SIAM Numer. Methods and Part. Diff. Equations 21 (2004),

297–322.

[24] Fernandez-Cara, E., Guillen, F., Ortega, R. R., Mathematical modeling and

analysis of viscoelastic fluids of the oldroyd kind, Handbook of numerical analysis, Vol.

VIII, 543–661, North-Holland, Amsterdam, 2002.

Page 203: Stability Analysis and Numerical Simulation of Non-Newtonian

Bibliography 191

[25] Fietier, N., Deville, M. O., Simulations of time-dependent flows of viscoelastic

fluids with spectral element methods, Journal of Scientific Computing 17 (1-4) (2002),

649–657.

[26] Fortin, A., Guenette, R., Pierre, R., On the discrete EVSS method, Comput.

Meth. Appl. Mech. Eng. 189 (2000), 121–139.

[27] Fortin, M., Finite element solution of the Navier-Stokes equations, Acta Numerica

(1993), 239–284.

[28] Fortin, M., Fortin, A., A new approach for the FEM simulation of viscoelastic flows,

J. Non-Newtonian Fluid Mech. 32 (1989), 295–310.

[29] Fortin, M., Pierre, R., On the convergence of a mixed method of Crochet and Mar-

chal for viscoelastic flows, Comput. Methods Appl. Mech. Eng. 73 (1989), 341–350.

[30] Giesekus, H., Phanomenologische Rheologie, Springer, Berlin, Heidelberg, 1994.

[31] Girault, V., Raviart, P. A., Finite element methods for Navier-Stokes equations;

Theory and Algorithms, Springer, Berlin, Heidelberg, New York, 1986.

[32] Glowinski, R., Pironneau, O., Finite element method for Navier-Stokes equations,

Annu. Rev. Fluid Mech. 24 (1992), 167–204.

[33] Guenette, R., Fortin, M., A new mixed finite element method for computing vis-

coelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995), 27–52.

[34] Guillope, C., Saut, J. C., Resultat d’existence pour les fluides viscoelastiques a loi

de comportement de type differentiel, Compte-rendu de l’Academie des Sciences de Paris

305 (I) (1985), 489–492.

[35] , Global existence and one-dimensional nonlinear stability of shearing motions

of viscoelastic fluids of Oldroyd type, Math. Model. Num. Anal. 24 (3) (1990), 369–401.

[36] Johnson, C., Numerical solutions of partial differential equations by the finite element

method, Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne,

Sydney, 1987.

[37] Johnson, M. W., Segalman, D., A model for viscoelastic fluid behavior which allows

non-affine deformation., J. Non-Newtonian Fluid Mech. 2 (1977), 255–270.

[38] Joseph, D. D., Renardy, M., Saut, J. C., Hperbolicity and change of type in the

flow of viscoelastic fluids, Arch. Ration. Mech. Anal. 87 (1985), 213–251.

[39] Keunings, R., On the high Weißenberg number problem, J. Non-Newtonian Fluid

Mech. 20 (1986), 209–226.

Page 204: Stability Analysis and Numerical Simulation of Non-Newtonian

192 Bibliography

[40] , Advances in the computer modeling of the flow of polymeric liquids, Comp.

Fluid Dyn. J. 9 (2001), 449–458.

[41] Kloucek, P., Rys, F., Stability of the fractional step theta-scheme for the nonsta-

tionary Navier-Stokes equations, SIAM J. Numer. Anal. 31 (5) (1994), 1312–1335.

[42] Legrand, F., Piau, J. M., Spatial resolved stress birefringence and flow visualiza-

tion in flow instabilities of a polydimethylsiloxane extruded through a slit die, J. Non-

Newtonian Fluid Mech. 77 (1998), 123–150.

[43] Lesaint, P., Raviart, P. A., On a finite element method for solving the neutron

transport equation, Carl de Boor, Academic Press, 1974.

[44] Lions, P. L., Masmoudi, N., Global solution for some Oldroyd models of non-

Newtonian flows, Chin. Ann. of Math. 2000 (21B:2), 131–146.

[45] Liu, I. S., On the transformation property of the deformation gradient under a change

of frame, J. of Elasticity 71 (1-3) (2003), 73–80.

[46] Marchal, J. M., Crochet, M. J., A new mixed finite element formulation for cal-

culating viscoelastic flow, J. Non-Newtonian Fluid Mech. 26 (1987), 77–114.

[47] Meincke, O., Scurtu, N., Egbers, C., Bansch, E., On the influence of boundary

conditions in taylor-couette flows, Physics of Fluid Metrology (Eds.: C. Egbers, G.

Pfister), Springer-Verlag (2000), 22–36.

[48] Meng, S., Li, X. K., Evans, G., Numerical simulation of Oldroyd-B fluid in a con-

traction channel, The J. of Supercomputing 22 (2002), 29–43.

[49] Muller-Urbaniak, S., Eine Analyse des Zwischenschritt-θ-Verfahrens zur Losung

der instationaren Navier-Stokes-Gleichungen, SFB 359 Preprint 94-01, Uni. Heidelberg,

1994.

[50] Muschik, W., Restuccia, L., Changing the observer and moving materials in con-

tinuum physics: objectivity and frame-indifference, Technische Mechanik 22 (2) (2002),

152–160.

[51] Najib, K., Sandri, D., On a decoupled algorithm for solving a finite element problem

for the approximation of viscoelastic fluid flow, Num. Math. 72 (1995), 223–238.

[52] Najib, K., Sandri, D., Zine, A. M., On a posteriori estimates for a linearized

Oldroyd’s problem, J. Comput. and Appl. Math. 167 (2004), 345–361.

[53] Oldroyd, J. G., On the formulation of rheological equations of state, Proc. Roy. Soc.

London A 200 (1950), 523–541.

Page 205: Stability Analysis and Numerical Simulation of Non-Newtonian

Bibliography 193

[54] Picasso, M., Rappaz, J., Existence, a priori and a posteriori error estimates for a

nonlinear three-field problem arising from Oldroyd-B viscoelastic flows, ESIAM, Math.

Modell. and Num. Anal. 35 (5) (2001), 879–897.

[55] Pontrjagin, L. S., Gewohnliche differentialgleichungen, Dt. Verl. der Wiss., Berlin,

1965.

[56] Quinzani, L. M., Armstrong, R. C., Brown, M. R., Birefringence and LDV

studies of viscoelastic flow through a planar contraction, J. Non-Newtonian Fluid Mech.

52 (1994), 1–36.

[57] , Use of coupled birefringence and LDV studies of flow through a planar con-

traction to test constitutive equations for concentrated polymer solutions, J. Rheol. 39

(1995), 1201–1228.

[58] Renardy, M., Recent advances in the mathematical theory of steady flow of viscoelastic

fluids, J. Non-Newtonian Fluid Mechanics 29 (1988), 11–24.

[59] , Mathematical analysis of viscoelastic flows, CBMS-NSF, Regional conference

series in applied mathematics, SIAM, 2000.

[60] Rutkevich, I. M., The propagation of small perturbations in a viscoelastic fluid, J.

Appl. Math. Mech. 34 (1970), 35–50.

[61] Saad, Y., Schultz, M. H., GMRES: A generalized minimal residual algorithm for

solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (3) (1986), 856–

869.

[62] Saramito, P., Simulation numerique d’ecoulements des fluides viscoelastiques par

elements finis incompressibles et une method de direction alternees-applications, These

de Docteur, Institute National Polytechnique de Grenoble, 1990.

[63] , A new θ-scheme algorithm and incompressible FEM for viscoelastic fluid flows,

Mathematical Modeling and Numerical Analysis 28 (1994), 1–34.

[64] , Efficient simulation of nonlinear viscoelastic fluid flows, J. Non-Newtonian

Fluid Mech. 60 (1995), 199–223.

[65] Schmidt, A., Siebert, K. G., ALBERT: An adaptive hierarchical finite element

toolbox, Institut fur Angewandte Mathematik, Albert-Ludwigs-Universitat Freiburg i.

Br., Version 1.0, 2000.

[66] Sequeira, A., Baıa, M., A finite element approximation for the steady solution of a

second-grade fluid model, J. Comput. and Appl. Math. 111 (1-2) (1999), 281–295.

Page 206: Stability Analysis and Numerical Simulation of Non-Newtonian

194 Bibliography

[67] Sun, J., Smith, M. D., Armstrong, R. C., Brown, R. A., Finite element

method for viscoelastic flows based on discrete adaptive stress splitting and discontinuous

Galerkin method: DAVSS-G/DG, J. Non-Newtonian Fluid Mech. 86 (1999), 281–307.

[68] Thiffeault, J. L., Covariant time derivatives for dynamical systems, J. of Physics A

34 (2001), 1–16.

[69] Truesdell, C., Rational continuum mechanics, Academic Press, Boston, New York,

London, 1991.

[70] Truesdell, C., Noll, W., The nonlinear field theories of mechanics, Springer, Berlin,

1992.

[71] Videman, J. H., Mathematical analysis of viscoelastic non-Newtonian fluids, Mestre,

Universidade technica de Lisboa, 1997.

[72] Yeleswarapu, K. K., Kameneva, M. V., Rajagopal, K. R., Antaki, J. F., The

flow of blood in tubes: theory and experiment, Mechanics Research Communications 25

(3) (1998), 257–262.

Page 207: Stability Analysis and Numerical Simulation of Non-Newtonian

ZusammenfassungIm Rahmen dieser Arbeit wurden Stabilitats- und numerische Untersuchungen am Oldroyd-

Gleichungssystem durchgefuhrt. Das numerische Verfahren basiert auf der Finite-Elemente-

Raumdiskretisierung und auf dem θ-Zwischenschritt-Verfahren, ausgefuhrt als Operatoren-

Splitting-Methode. Angesichts des gemischten hyperbolisch-parabolischen Charakters des

Gleichungsystems besteht die fundamentale Idee des numerischen Verfahrens in der entkop-

pelten Berechnung des Geschwindigkeits-, Druck- und Spannungsfeldes. Im Gegensatz zu

Saramito [63] wurde in dieser Arbeit auch der nichtlineare konvektive Geschwindigkeitsterm

in der Bewegungsgleichung berucksichtigt. Durch die Operatoren-Splitting-Methode wird

das Oldroyd-System auf drei Unterprobleme reduziert: ein Stokes-ahnliches Problem, eines

von der Art der Burgers-Gleichung und ein Transport-Problem fur die Spannungen.

Die Stabilitat des Oldroyd-Gleichungssystems wurde umfassend analysiert, beginnend

vom kontinuierlichen Fall, wo schon Stabilitatsgrenzen auftreten. Weiterhin wurde die

Stabilitat der Zeitapproximation und des θ-Verfahrens gekoppelt mit der Finite-Elemente-

Raumdiskretisierung untersucht. Die formale Spektralanalyse des θ-Verfahrens, angewandt

auf das linearisierte Oldroyd-System, zeigt gute Stabilitatseigenschaften und eine Genauigkeit

zweiter Ordnung der Zeitdiskretisierung. Fur das lineare Oldroyd-System wurden keine

Beschrankungen der vier Parameter Weißenberg-Zahl We, Reynolds-Zahl Re, Anteil der

viskoelastischen Viskositaten α und Gleitparameter a, gefunden.

Bei Vernachlassigung des konvektiven Spannungsterms im Materialgesetz kann aufgrund

des β-Terms fur a 6= 0 eine obere Stabilitatsgrenze Wecr fur die Weißenberg-Zahl existieren.

Im Falle a = ±1 gibt es zum Beispiel eine Stabilitatsgrenze, falls im Berechnungsgebiet

ein Bereich existiert, wo der Geschwindigkeitsgradient die Ungleichung det(∇u) < 0 erfullt.

Diese Grenze wurde von den numerischen Tests aus Kapitel 7 bestatigt.

Berucksichtigt man auch den konvektiven Spannungsterm, wird ebenfalls eine Stabilitats-

grenze gefunden. Wenn im Bereich von det(∇u) < 0 (im Falle a = ±1) ein Staupunkt ex-

istiert oder dieser Bereich von geschlossenen Stromlinien durchschnitten wird, dann entsteht

bei uberkritischen Weißenberg-Zahlen eine Instabilitat der Spannungskomponenten, die zeit-

lich entlang der Stromlinien wachst. Im Falle a = 0 bleibt die Spannungsgleichung immer

stabil, aber im vollen Oldroyd-System konnen Instabilitaten vorkommen.

Das Programm wurde an zwei Benchmark-Problemen getestet: an der getriebenen Kavitat

und an einer Kontraktion im Verhaltnis 4 : 1. Fur die Kavitat wurde die Bewegung eines

Oldroyd-Fluids in den Fallen a = 1 und a = 0 numerisch analysiert. Die Konvergenz-

grenze fur a = 0 ist hoher als fur a = 1, und ist ebenfalls vom Parameter α abhangig. Im

Rahmen der Konvergenzgrenzen des Programms zeigen die Ergebnisse dieser Arbeit fur die

Kontraktion eine sehr gute Ubereinstimmung mit neuesten Ergebnissen aus der Literatur.

Die numerische Implementierung wurde mit Hilfe des Programmpakets Albert realisiert.

Der wichtigste personliche Beitrag ist die Implementierung des Spannungstensors mit den

entsprechenden Routinen fur die Assemblierung des Oldroyd-Systems und fur die Losung

des Spannungstransport-Unterproblems mit Hilfe der Discontinuous-Galerkin-Methode.

Page 208: Stability Analysis and Numerical Simulation of Non-Newtonian
Page 209: Stability Analysis and Numerical Simulation of Non-Newtonian

Curriculum

Personliche Daten:

Name: Nicoleta Dana Scurtu

Geburtsdatum: 5. Dezember 1970

Geburtsort: Brasov, Rumanien

Wohnort: 03042 Cottbus, Elisabeth-Wolf-Straße 43

Schulausbildung:

09/1976-07/1984 Besuch der Allgemeinschule in Brasov

09/1984-07/1989 Besuch des Lyzeums “Andrei Saguna” in Brasov

(Richtung Mathematik-Physik)

07/1989 Abitur

Berufsausbildung:

10/1989-07/1994 Studium der Mathematik an der Universitat Bukarest

10/1992-07/1994 Spezialisierungsrichtung Stromungsmechanik

07/1994 Diplom

10/1994-07/1995 Vertieftes Studium – Mechanik der flussigen und festen Korper

an der Universitat Bukarest

07/1995 Examensarbeit

Page 210: Stability Analysis and Numerical Simulation of Non-Newtonian

Wissenschaftlicher Werdegang:

09/1994-05/1999 Assistentin des Professors und Doktorandin

am Lehrstuhl fur Thermotechnik und Stromungsmechanik

der Universitat Transilvania Brasov mit Durchfuhrung

von Forschungsarbeiten im Bereich Grenzschichttheorie,

Ubungen und Praktika

06/1999-11/1999 DAAD-Stipendiatin in der Forschungsgruppe Rotierende

Stromungen des Zentrums fur Angewandte Mathematik und

Raumfahrttechnik in Bremen mit Durchfuhrung von

3D-Simulationen im asymmetrischen Taylor-Couette-System

12/1999-09/2000 Doktorandin am Zentrum fur Technomathematik der

Universitat Bremen

10/2000-06/2004 Doktorandin in der Forschungsgruppe Numerische Mathematik

und Wissenschaftliches Rechnen am Weierstraß-Institut fur

Angewandte Analysis und Stochastik in Berlin, mit dem

Arbeitsgebiet Simulation, Stabilitatsuntersuchungen und

Musterbildung bei Newtonschen und nicht-Newtonschen

inkompressiblen Fluiden.

seit 07/2004 Wissenschaftliche Mitarbeiterin am Lehrstuhl fur Aerodynamik

und Stromungslehre der Fakultat Maschinenbau, Elektrotechnik

und Wirtschaftsingenieurwesen der BTU Cottbus mit Durchfuhrung

von Forschungen zur Stromungsanalyse von Triebwerkslarm durch

gekoppelte Stromungs- und akustische Sensorik mit Hilfe

chaosdynamischer Methoden sowie zur Strukturbildung, Stabilitat

und Ubergang ins Chaos in komplexen Fluiden.

29/11/2004 Bewerbung um die ausgeschriebene Stelle eines Juniorprofessors

zum Thema Modellierung und Optimierung, insbesondere im Bereich

der Aeroakustik (JP 27/04).