PAMM · Proc. Appl. Math. Mech. 13, 515 – 516 (2013) / DOI 10.1002/pamm.201310250

Asymptotics and numerics for non-Newtonian jet dynamics

Maike Lorenz1, Nicole Marheineke1,∗, and Raimund Wegener2

1 Friedrich-Alexander-Universität Nürnberg-Erlangen, Department Mathematik, Cauerstr. 11, D-91058 Erlangen2 Fraunhofer Institut für Techno- und Wirtschaftsmathematik, Fraunhofer Platz 1, D-67663 Kaiserslautern

This work deals with the modeling and simulation of non-Newtonian jet dynamics. Proceeding from a 3d boundary valueproblem of upper-convected Maxwell equations, a 1d viscoelastic string model can be derived asymptotically. The resultingsystem of PDEs has a hyperbolic-elliptic character with an additional differential constraint. Its applicability regime is limiteddepending on physical parameters and boundary conditions. Numerical results are shown for gravitational in-/outflow set-ups.

c© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Asymptotic model and its solution regime

The dynamics of a viscoelastic jet can be described as a 3d boundary value problem in terms of the upper-convected Maxwellequations. The slender geometry motivates the reduction to a 1d model where the jet is represented by a time-dependent,arc-length parameterized curve (e.g. center-line) γ : Q → E3,Q = {(s, t) | s ∈ I(t) ⊂ R, t ∈ R+

0 } in the Euclidean space. Astrict systematic derivation using asymptotic expansions in the slenderness parameter yields particularly the following stringmodel as leading order system [1] – supplemented with respectively deduced initial and boundary conditions:

∂tA+ ∂s(uA) = 0, Re(∂t(Av) + ∂s(uAv)) = ∂s(Aσ∂sγ) +Af

We(∂tσ + u∂sσ − (3p+ 2σ)∂su) + σ = 3∂su, We(∂tp+ u∂sp+ p∂su) + p = −∂su∂tγ + u∂sγ = v, ‖∂sγ‖ = 1.

(1)

Neglecting surface tension and temperature effects, the dimensionless system (1) is characterized by the Reynolds Re andWeissenberg We numbers that denote the ratio of inertial and viscous forces as well as the ratio of relaxation and processtimes, respectively. Apart from the curve γ, the unknowns are the cross-sectional area A, the momentum-associated velocityv, the stress component σ, the pressure p and the intrinsic speed u. Hereby, u can be considered as Lagrangian multiplierto the arclength-constraint. The external forces f are given, e.g. in case of gravity f = Re/Fr2eg with Froude number Frand gravitational direction eg. The string model allows for the unrestricted motion and shape of γ and also covers the purelyviscous case [2] for We = 0. For an uniaxial straight jet of fixed length and Re = 0 the existence of unique solutions is provedunder certain assumptions in [3]. A general solution theory is not available so far. It depends crucially on the terms

q1 = u− 1

Re

σ

u, q2 = 3 +We(σ + 3p)− ReWeu2.

Introducing χ = ∂sγ, the system (1) can be rewritten as

M · ∂tφ+C(φ) · ∂sφ+ l(φ) + g = 0 with φ = (A,v, σ, p,γ,χ, u), M = diag(112, 0) (2)

0.51

1.52

0

1

2

30

5

10

15

Re

Limiting surfaces for q1(0)=0.1, q

2(1)=0.1

Fr

We

solutions

Limiting surface (q1)

Limiting surface (q2)

no solutions

Fig. 1: Solution regime for stationary problem.

whose classification is determined by the eigenvalues λi of C, i.e. λ1 = 0(multiplicity 4), λ2 = u > 0 (multiplicity 3), λ3,4 = u ±

√w (mul-

tiplicity 3 each) with w = u − q1u. Hence, depending on the sign ofw the system (2) changes its character. It is hyperbolic if w > 0 andmixed elliptic-hyperbolic if w < 0. The case w = 0 yields an eigenvalueλ2 = u of multiplicity 9, but with an eigenspace of dimension 5, involv-ing a parabolic deficiency. Investigating the most relevant hyperbolic casemore deeply, q1 = 0 separates two regimes with respect to the run of thecharacteristics, i.e. q1 > 0: λ3 > 0, λ4 < 0 versus q1 < 0: λ3,4 > 0.Note that an analogous classification and hyperbolic specification holdsfor the reduced uniaxial system, here q2 = 0 separates the characteris-tic hyperbolic regimes. The possible existence of these two hyperbolicregimes depending on the physical parameters (Re,We,Fr) requires thecareful consideration of the asymptotically deduced boundary conditionsin view of consistency and well-posedness of the problem. This issue is

∗ Corresponding author: e-mail marheineke@math.fau.de, phone +49 (0)9131 85-67214, fax +49 (0)9131 85-67225

c© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

516 Section 22: Scientific computing

addressed by a numerical study for the stationary problem of a jet with fixed length under gravity. Using the monotonicity ofq1 (i.e. ∂sq1 = f ·χ/(Reu) > 0 because of the jet dynamics due to gravity) and empirical knowledge about the run and rootsof q2, a hyperplane limiting the solution regime can be computed for the parameter space, Fig. 1. The loss of solutions arisesas the asymptotic model is singularly perturbed (inconsistency of degenerated equations and boundary conditions).

2 Transient simulation results

Applying a semidiscretization in space, various methods might be used, e.g. finite volume scheme with up- and downwindeddifferences, central scheme for Hamilton-Jacobi equations with artificial diffusion [4]. The last one has a stabilizing effect,but leads to artefacts in the constraint. Hence, both variants are combined to a first order scheme, involving a slight numericaldiffusion except for curve γ and tangent χ. The resulting temporal differential-algebraic system of index 2 is solved by astiffly accurate Radau IIa method (stage s=1, i.e. implicit Euler), providing also first order convergence in time – in differentialand algebraic variables.

0 0.2 0.4 0.6 0.8 1−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

γ1

γ 2

t = 0t = 0.31626t = 0.82229t = 1.6319t = 2.9273t = 5stationary

Fig. 2: Jet curve under gravity (Re,Fr,We) = (1, 2, 1)(similar results for We = 0.1 and 1.8).

Consider a viscoelastic jet of fixed length s ∈ I = [0, 1] under gravityin a 2d scenario where its position, tangent, cross-section and both veloc-ities are prescribed at the inlet (s = 0). At the outlet (s = 1) the intrinsicspeed is kept constant which implies (σ + We/Fr2uχ · eg)(1, t) = 0.Moreover, in agreement with the stationary investigations q2(0, t) = 1 isposed yielding a boundary condition for p. Figures 2 and 3 illustrate ex-emplarily the jet dynamics and behavior in t ∈ [0, 5] for (Re,Fr) = (1, 2)and varying We ∈ {0.1, 1}. Whereas the viscoelasticity has almost noinfluence on γ, a clear impact on the other quantities is observed. For aviscous jet (We = 0), A decreases and u increases monotonically overtime to approach quickly the stationary state [2], this holds also true forWe � 1. For higher We the monotonicity vanishes, instead an under-shoot and overshoot, respectively, form out, before the equilibrium isslowly reached in long-time simulations. Increasing We up to 2, the pres-sure profile p additionally changes from concave to convex. In this cho-sen parameter set-up significantly larger We have no stationary solutions(cf. Fig. 1). Hence, the respective transient simulations show the arisingof a boundary layer that steepens due to the inconsistency of equationsand boundary conditions, for further details see [5].

0 0.5 10.99

0.995

1

1.005

s

A

Area

0 0.5 11

1.005

1.01

1.015

s

u

Velocity

0 0.5 10

0.01

0.02

0.03

0.04

s

σ

Stress

0 0.5 1−8

−6

−4

−2

0

s

p

Pressure

0 0.5 10.94

0.96

0.98

1

1.02

s

A

Area

0 0.5 11

1.02

1.04

1.06

1.08

s

u

Velocity

0 0.5 1−0.05

0

0.05

0.1

s

σ

Stress

0 0.5 1−0.4

−0.3

−0.2

−0.1

0

s

p

Pressure

Fig. 3: Viscoelastic effect on temporal evolution of A, u, σ and p. (Re,Fr) = (1, 2), cf. Fig. 2. Left: We = 0.1. Right: We = 1.

References[1] M. Lorenz, N. Marheineke, and R. Wegener, PAMM 12, 601–602 (2012).[2] W. Arne, N. Marheineke, and R. Wegener, Math. Mod. Meth. Appl. Sci. 21(10), 1987–2018 (2011).[3] T. C. Hagen, Advances in Mathematics Research 1, 187–205 (2002).[4] S. Bryson and D. Levy, SIAM J. Sci. Comp. 25, 767–791 (2004).[5] M. Lorenz, PhD thesis, TU Kaiserslautern, Germany (Verlag Dr. Hut, Mathematik, 2013).

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