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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
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
To the man who pleases him,
God gives wisdom, knowledge and happiness...
(the Bible)
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
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
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
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
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.
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
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.
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.
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
.
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.
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),
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,
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.
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.
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.
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
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).
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.
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
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
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.
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
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.
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.
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
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
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).
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.
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.
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).
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))
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
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
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:
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−.
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−.
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),
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]).
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.
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.
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]).
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.
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).
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
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
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‖∞,
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
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
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,
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,
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).
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,
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.
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
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)
]
,
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,
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.
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,
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)
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
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.
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.
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∗].
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.
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.
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.
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
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.
5.1. Spectral analysis of the linearized Oldroyd system 67
10−5
100
105
1010
−0.5
0
0.5
1λ
3
α = 0α = 0.2α = 0.5α = 0.99
10−5
100
105
1010
−0.5
0
0.5
1λ
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 .
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 =
2α
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
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.
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 =
2α
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,
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
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:
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,
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 −
4α
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 −
4α
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.
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.
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
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
,
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.
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.
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
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
,
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
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).
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).
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).
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).
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
4ε
[
(1 + b)‖fns ‖2 + 2‖fn+1s ‖2
]
.
Summing up the above inequalities for n = 0, 1, 2, ..., N , the desired stability estimation
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.
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.
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,
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).
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
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.
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,
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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 -
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 -
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.
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
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
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.
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.
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.
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
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
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.
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.
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.
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.
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.
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)
,
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.
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.
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.
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.
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.
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
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.
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.
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 (∗).
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.
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.
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.
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.
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.
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
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.
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.
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.
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
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
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
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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]
.
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|,
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|,
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.
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.
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.
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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.
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
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).