Chapter 14 The Use of Mixed Finite Element Methods for ... · Viscoelastic Fluid Flow Analysis Frank P.T. Baaijens, Martien A. Hulsen and Patrick D. Anderson in Encyclopedia Of Computational

Chapter 14

The Use of Mixed Finite Element Methods forViscoelastic Fluid Flow Analysis

Frank P.T. Baaijens, Martien A. Hulsen and Patrick D. Anderson

in

Encyclopedia Of Computational Mechanics

Editors Erwin Stein, René de Borst and Thomas J.R. Hughes

Volume 3 Fluids

pp. 481–498

John Wiley & Sons, Ltd, Chichester, 2004

Chapter 14The Use of Mixed Finite Element Methods forViscoelastic Fluid Flow Analysis

Frank P. T. Baaijens, Martien A. Hulsen and Patrick D. AndersonEindhoven University of Technology, Eindhoven, The Netherlands

1 Introduction 481

2 Mathematical Formulation 481

3 Steady Flow: Variational Formulations 482

4 Time Dependent Flows 488

5 Integral and Stochastic Constitutive Models 490

6 Governing Equations 491

7 The Deformation Fields Method 492

8 Brownian Configuration Fields 493

9 Numerical Methods 494

10 Results 494

11 Conclusions and Discussion 495

12 Related Chapters 496

References 496

1 INTRODUCTION

The development of accurate, stable, efficient, and robustnumerical methods for the analysis of viscoelastic flowshas proven a challenging task. Nevertheless, significantprogress has been made in the finite element formula-tions for the analysis of viscoelastic flows. The difficultiesconcern the two major aspects: first, the development ofsuitable methods to deal with the hyperbolic nature ofthe constitutive equation (CE); second, the developmentof appropriate mixed formations for the set of governing

Encyclopedia of Computational Mechanics, Edited by ErwinStein, René de Borst and Thomas J.R. Hughes. Volume 3: Fluids. 2004 John Wiley & Sons, Ltd. ISBN: 0-470-84699-2.

equations. Even for seemingly ‘simple’ CEs like the upper-convected Maxwell (UCM) model severe limits appear onthe level of elasticity (measured by the dimensionless Weis-senberg number) that can be obtained, in particular whengeometrical singularities are present.

For a few representative benchmark problems for steadyflows, agreement between a number of different formula-tions has been demonstrated at ever increasing values of theWeissenberg number see for example, Brown and McKinley(1994), Caswell (1996), and more recently Alves, Oliveiraand Pinho (2003). And, more importantly, mesh convergentresults are achieved here without lowering the maximumachievable Weissenberg number. However, limits in themaximum attainable Weissenberg number still exist forthese relative simple flows. The review paper of Baaijens,1998 discusses various mixed finite element schemes andtheir applicability for different types of viscoelastic flowproblems. A review of the field of computational rheologythat covers also other techniques, such as spectral methods,is given in the recent book by Owens and Phillips, 2002.A review focusing particular on integral methods can befound in Keunings (2003).

Similar to the review paper of Baaijens (1998), thiswork will be restricted to mixed finite element methods,however the emphasis will be on recent developments,such as application to polymer processing, time-dependentsimulations, and the application to methods for integral andstochastic models.

2 MATHEMATICAL FORMULATION

The analysis of viscoelastic flows involves the solution ofa coupled set of partial differential equations: equations

482 The Use of Mixed Finite Element Methods for Viscoelastic Fluid Flow Analysis

representing the conservation of mass, momentum, andenergy, and CEs for a number of physical quantities presentin the conservation equations such as density, internalenergy, heat flux, stress and so on. For incompressible,isothermal flow in absence of volume forces, the balanceequations of momentum, mass, and the CE for the stressread

ρ∂u

∂t+ ρu · ∇u = −∇p + ∇ · (2ηeD + τ) (1)

∇ · u = 0 (2)λ

∇τ + τ − 2ηD + g(τ) = 0 (3)

where the∇τ denotes the upper-convected time derivative

of the stress tensor τ defined as

∇τ ≡ ∂τ

∂t+ u · ∇τ − (∇u)T · τ − τ · ∇u (4)

The velocity is depicted by u, τ denotes the viscoelasticcontribution to the extra-stress tensor, p the pressure, and ρdenotes the density, while D denotes the rate of deformationtensor defined as

D = 12 (∇u + (∇u)T) (5)

Frequently, the extra-stress tensor is defined in terms of aviscous and a viscoelastic contribution, in which ηe denotesan effective viscosity as appearing in equation (1). Theeffective viscosity can represent the viscosity of the solvent(for a polymer solution) or an apparent viscosity modelingpart of stress by a Newtonian viscosity.

A large variety of approaches exists to define a modelfor the extra-stress tensor τ (Bird, Armstrong and Has-sager, 1987a; Bird et al., 1987b). Most recent models arederived from kinetic theory. However, from an implemen-tation point of view, a distinction can be made betweenclosed-form constitutive models of the integral and differen-tial type and models that cannot be written as a closed-formconstitutive model. First, constitutive models of the dif-ferential type will be discussed. Integral and stochastic(nonclosed) models will be the subject of Sections 5 andfurther. The discussion here will be limited to differentialmodels having the structure as in equation (3), where λrepresents a characteristic relaxation time for the polymersystem and η represents the polymer viscosity. The func-tion g(τ) in equation (3) represents a nonlinear function ofthe stress, that goes to zero at least as fast as quadraticallyas τ approaches zero. A number of well-known constitutivemodels are represented by equation (3), including the upper-convected Maxwell, FENE-P [finitely extensible nonlinearelastic (Peterlin)], Phan Thien-Tanner (PTT), Giesekus, andthe eXtended pom-pom model. In general both λ and η

can be considered as functions of the stress or other statevariables, but since this does not seem to drastically influ-ence the behavior of the various numerical schemes to bediscussed below, they will be assumed constants here forsimplicity.

Clearly, equation (3) may not describe the actual mechan-ical behavior of many viscoelastic fluids with sufficientaccuracy since even in the limit of small deformationsa spectrum of relaxation times is necessary to accuratelydescribe the rheology of most viscoelastic fluids. Such aspectrum frequently may be represented by a finite set ofrelaxation times. For each relaxation time, a CE of theform of equation (3) may be used for a substress τi , suchthat τ = ∑i τi . However, to discuss the algorithmic devel-opments, a single relaxation time with a CE that obeysequation (3) is sufficient.

3 STEADY FLOW: VARIATIONALFORMULATIONS

Consider the steady, incompressible, inertialess flow of anOldroyd-B fluid, that is, equations (1)–(3) with ρ = 0 andg(τ) = 0. As a natural extension of the common velocity-pressure formulation for Stokes type problems, the classicalthree field mixed formulation is chosen, in which, besidesthe momentum and continuity equation, also the CE is castin a weighted residuals form.

Problem 1 (MIX) Find u, p, and τ such that for alladmissible weighting functions v, q, and S

((∇v)T, 2ηeD + τ) − (∇ · v, p) = 0 (6)(q, ∇ · u) = 0 (7)(S, λ

∇τ + τ − 2ηD) = 0 (8)

where (., .) denotes the appropriate inner product. Most ofthe early work on viscoelastic flow analysis is based onthis formulation, see for example (Kawahara and Takeuchi,1977; Crochet and Keunings, 1982).

Assuming that suitable approximation spaces have beenselected for the triple stress-velocity-pressure in Prob-lem1 (MIX), computational difficulties exist with increas-ing levels of elasticity. The importance of the convectiveterm u · ∇τ grows and a Galerkin discretization as appliedin equation (8) is not optimal (King et al., 1988).

The most widespread method to account for the con-vective term in the constitutive equation is the so-calledstreamline-upwind/Petrov Galerkin (SUPG) method ofBrooks and Hughes (1982), first applied to viscoelasticflows by Marchal and Crochet (1987):

(S + αu · ∇S, λ∇τ + τ − 2ηD) = 0 (9)

The Use of Mixed Finite Element Methods for Viscoelastic Fluid Flow Analysis 483

Several choices for the SUPG parameter α have beenproposed in the literature, but are all of the form

α = hU

(10)

where h is a characteristic element size, that is, the elementsize along the local flow direction and U is a characteristicvelocity. A possible choice for U is to use the velocityin the direction of the local coordinates at the midpointof a biquadratic element (Marchal and Crochet, 1987), orat each integration point (Baaijens, 1992). Other choicesare the norm of the velocity u (King et al., 1988), or acharacteristic velocity of the flow (Lunsmann et al., 1993).Owens, Chauviére and Philips (2002) introduced a locallyupwinded spectral technique (LUST) for viscoelastic flows,in which locally within each member of the partition of aspectral element is formed by the quadrature grid. Marchaland Crochet (1987) demonstrated the limitations of theMIX/SUPG formulation for problems with a geometricalsingularity, such as the stick-slip problem or contractionflows. On the basis of these findings, they suggested toapply the upwind term to the convective part of the CE only,yielding the so-called streamline-upwind (SU) formulation:

(S, λ∇τ + τ − 2ηD) + (αu · ∇S, λu · ∇τ) = 0 (11)

The obvious limitation of this formulation is its inconsis-tency, in the sense that if the exact solution is insertedinto equation (11), that is, λ

∇τ + τ − 2ηD = 0, a resid-

ual (αu · ∇S, λu · ∇τ) remains. Moreover, the method isreported to be first-order accurate only such that achievingmesh converged results is extremely difficult.

An alternative to the SUPG method is the discontinuousGalerkin (DG) or Lesaint–Raviart method. Here, the extra-stress tensor is approximated discontinuously from oneelement to the next, and upwind stabilization is obtainedas follows:

(S, λ∇τ + τ − 2ηD) −

N∑e=1

∫�ine

S: λu · n(τ − τext) d� = 0(12)

with n the unit outward normal on the boundary of elemente, �ine the part of the boundary of element e, where u · n <0, and τext the extra-stress tensor in the neighboring upwindelement. In the context of viscoelastic flows, this methodwas first introduced by Fortin and Fortin (1989) on the basisof ideas of Lesaint and Raviart (1974), who proposed themethod to solve the neutron transport equation. Comparedto the SUPG formulation, the implementation of the DGmethod in a standard finite element code is more involved.This is due to the boundary integral along the inflowboundary of each element in which stress information of

the neighboring, upwind, element is needed. This drawbackis circumvented for unsteady flows by Baaijens (1994b) byusing an implicit/explicit implementation.

Several methods have been proposed in the literature toretain an elliptic contribution of the form ((∇v)T, D) in theweak form of the momentum equation, equation (6), whichis particularly important if a purely viscous contribution isabsent or small compared to the viscoelastic contribution.One way to achieve this is the application of a changeof variables, known as the elastic-viscous stress splitting(EVSS) formulation, first introduced by Perera and Walters(1977) and Mendelson et al. (1982) for the flow of asecond-order fluid and later extended to viscoelastic flowsby Beris, Armstrong and Brown (1984, 1986):

� = τ − 2ηD (13)Substitution of this into equations (8) and (6), respectively,yields

Problem 2 (EVSS/SUPG) Find u, p, τ, and H such thatfor all admissible weighting functions v, q, S, and E

((∇v)T, 2(η + ηe)D + �) − (∇ · v, p) = 0 (14)(q, ∇ · u) = 0 (15)(S + αu · ∇S, λ ∇� + � + 2ηλ ∇H ) = 0 (16)(E, 12 (∇u + (∇u)T) − H

) = 0 (17)The tensor H is introduced as an additional variableobtained by an L2-projection of D in equation (17), whereE denotes a suitable weighting function. The purpose ofthis projection is to facilitate evaluation of the upper-convected derivative of H as appearing in equation (16).This derivative contains the gradient of H , and use of D inthis expression instead of the projection H would requiresecond-order derivatives of the velocity field, and thereforecontinuity of the gradient of the velocity field u. This exten-sion is due to Rajagopalan, Armstrong and Brown (1990)and is used by many others.

One may proceed one step further by performing aprojection of (E, G − (∇u)T) = 0, and using G as anadditional unknown rather than D, and subsequently usethe projection of G in the CEs as well. The latter leads tothe so-called EVSS-G method, introduced by Brown et al.(1993) and Szady et al. (1995).

A major disadvantage of EVSS is that the change ofvariables (see, equation (13)) is not possible for all consti-tutive relations, in particular it cannot be applied to integralmodels and models that cannot be written in a closed-formclosed expression. Guènette and Fortin (1995a) introduceda modification of the EVSS formulation, known as thediscrete EVSS method (DEVSS). Here, a stabilizing elliptic


operator is introduced in the discrete version of momentumequation, which is similar to the EVSS method, but theobjective derivative of the rate of strain tensor is avoided.Moreover, the method is not restricted to a particular classof CEs and has the same computational costs as EVSS.Using the Oldroyd-B model and SUPG, we introduce theso-called DEVSS-G/SUPG form.

Problem 3 (DEVSS-G/SUPG) Find u, p, G, and τ suchthat for all admissible weighting functions v, q, E, and S

((∇v)T, αη(2D − (G + GT)) + τ) − (∇ · v, p) = 0 (18)(q, ∇ · u) = 0 (19)(

S + αu · ∇S, λ(

∂τ

∂t+ u · ∇τ − G · τ − τ · GT

)

− η (G + GT))

= 0 (20)

(E, ∇u − GT) = 0 (21)

In the discrete momentum equation (18), an elliptic oper-ator αη(2D − (G + GT)) is introduced, where G is a dis-crete approximation of the velocity gradient tensor (∇u)Tobtained from equation (21). The auxiliary parameter α isgenerally set to 1. Notice that if the exact solution is recov-ered, the elliptic operator αη(2D − (G + GT)) vanishes.However, in a finite element calculation, this is generallynot the case because D is derived from the velocity field,and G results from the projection equation (21), and there-fore generally belong to a different approximation space.The first application of discrete elastic-viscous split stress(DEVSS) to the DG method is reported by Baaijens et al.(1997).

The acronym DEVSS is not really an appropriate namefor the method because the stress splitting is not presentin either the momentum or CE. As has been observedby Fan, Tanner and Phan-Thien (1999b), Bogaerds, Ver-beeten and Baaijens (2000), and Bogaerds (2002), mostof the above-mentioned formulations can be interpretedas being a member of the family of the Galerkin/Least-square concept. It is well known that, for the Navier–Stokesequations, various stabilizing algorithms can be derivedby introducing perturbation terms to the potential func-tional of the variational problem. For viscoelastic flows,a thorough analysis of the addition of such terms is stilllacking.

3.1 Spatial discretization; finite elements

In the well-known velocity-pressure formulation of theStokes problem, recovered by omitting τ from equations

(6) to (7), the so-called inf–sup or Ladyshenskaja-Babuška-Brezzi (LBB) compatibility condition between the velocityand pressure interpolation needs to be satisfied (Brezzi andFortin, 1991). Likewise, the addition of the weak form ofthe CE, equation (8), imposes compatibility constraints onthe interpolation of the triple stress-velocity-pressure. Fortinand Pierre (1987) have shown that in the absence of apurely viscous contribution, for example, ηe = 0 and usinga regular Lagrangian interpolation, the following conditionsmust hold:

1. The velocity-pressure interpolation must satisfy theusual LBB condition to prevent for locking and spuri-ous oscillation phenomena.

2. If a discontinuous interpolation of the extra-stress ten-sor τ is used, the space of the strain rate tensor Das obtained after differentiation of the velocity field umust be a member of the interpolation space of theextra-stress tensor τ, that is, D ⊂ τ.

3. If a continuous interpolation of τ is used, the numberof internal nodes must be larger than the number ofnodes on the side of an element used for the velocityinterpolation.

Condition (2) is relatively easily satisfied using a DGmethod; see, for example, Fortin and Fortin (1989) andBaaijens (1994a). Condition (3) is confirmed by the earlierwork of Marchal and Crochet (1987), who introduced afour-by-four bilinear subdivision of the extra stresses on abiquadratic velocity element. Baranger and Sandri (1992)have shown that the third condition need not be imposedif a purely viscous contribution is present (ηe �= 0), whichallows a much larger class of discretization schemes.

Prior to the introduction of (D)EVSS, it was customaryto use an equal-order interpolation of the velocity andstress field. Exceptions to this are the so-called 4 × 4element of Marchal and Crochet (1987), and discontinuousinterpolations of the extra stresses by Fortin and Fortin(1989). Because of the introduction of (D)EVSS, a largerselection is possible, yet the most common scheme is touse a stress and strain rate discretization that is one orderlower than the velocity interpolation.

If (D)EVSS(-G) and a continuous interpolation of theextra-stress tensor are used, the strain rate tensor (or thevelocity gradient tensor) is interpolated in the same way asthe extra-stress tensor. The most commonly applied elementfor the (D)EVSS(-G) method has a biquadratic velocity,bilinear pressure, stress and strain rate (or velocity gradient)interpolation; see Debae, Legat and Crochet (1994).

In case of the DG method, hence with a discontinuousinterpolation of the extra-stress tensor, a variety of choiceshave been experimented with by Fortin and Fortin (1989).Baaijens et al. (1997) shows that with a biquadratic velocity


interpolation and a bilinear discontinuous interpolation ofthe stresses, the most stable results are obtained with abilinear continuous interpolation of the strain rate tensor.This differs from the DEVSS/SUPG formulation, in whichthe rate of strain tensor is interpolated equal to the extra-stress tensor.

A special class of methods is formed by so-called(pseudo-) spectral collocation formulations; see Pilitsis andBeris (1989), Xu, Davies and Phillips (1993), and Phillips(1994). These may generally be viewed as a special formof MIX and EVSS in the sense that the discretization isnot of the Galerkin type since the weighting function arechosen unity at the collocation points. A different classof discretization models are the higher-order finite vol-ume methods, which recently have been used to obtainsecond-order accuracy for a number of benchmark prob-lems (Alves, Oliveira and Pinho, 2003). Earlier modelsbased on finite volumes were not able to provide accu-rate results. Fan (2003) and others have applied h–p typeof finite elements, which provide stable discretizations toviscoelastic flow problems in smooth geometries with anexponential convergence rate. A drawback of these higher-order discretizations is that they are limited for problemswith singularities.

An illustration of the effect of the formulation and asso-ciated discretization is obtained by examining the planarstick-slip problem assuming Newtonian flow, but in amixed setting. Hence, in equations (1) and (3), the viscosityηe = 0 and the relaxation time λ = 0. The stick-slip prob-lem is schematically represented in Figure 1. Along �st thevelocity is set to zero, while along �sl only the normalcomponent of the velocity is zero. Along the inflow bound-ary, �i a parabolic inflow velocity profile is prescribed,and along the symmetry line �s , the velocity in the verticaldirection is also suppressed.

Employing the MIX method, the horizontal velocity com-ponent along �st

⋃�sl is shown in Figure 2. In Figure 2(a),

the velocity and stress are discretized with a biquadratic(Q2) interpolation, while the pressure is discretized with abilinear Q1 polynomial. The use of a continuous bilinearapproximation of the stress yields a singular set of equa-tions. Using a bilinear but discontinuous interpolation (Qd1 )of the stress field results in the velocity field of Figure 2(b).

ΓstΓsl

ΓoΓi

Γs

Figure 1. Geometry of the stick-slip problem.

(a) (b)

Figure 2. Horizontal velocity along �st⋃

�sl . (a) (u, τ, p) :Q2Q2Q1, (b) (u, τ, p) : Q2Qd1Q1. A color version of this imageis available at http://www.mrw.interscience.wiley.com/ecm

(b)(a)

Figure 3. Horizontal velocity along �st⋃

�sl . (a) (u, τ, p,H ) :Q2Q1Q1Q1, (b) (u, τ, p,H ) : Q2Qd1Q1Q1. A color version ofthis image is available at http://www.mrw.interscience.wiley.com/ecm

The oscillations in the velocity field along the slip boundary�sl clearly demonstrate the limitation of this formulation.

Analysis of the same problem, but now with the DEVSSformulation, yields the velocity in Figure 3. Now, both dis-cretizations, (u, τ, p, H ) : Q2Q1Q1Q1 and (u, τ, p, H ) :Q2Q

d1Q1Q1, yield smooth and accurate velocity curves.

Notice that in the latter case, the stress approximation isdiscontinuous, while a continuous approximation of H isused. Other discretizations may also be used; for example,Baaijens et al. (1997) and Baaijens (1998).

3.2 Solution technology

The resulting set of nonlinear equations is often solvedusing a Newton–Raphson scheme in combination with afirst-order continuation in the Weissenberg number Wi. Theresulting linear set of equations is generally solved using adirect LU factorization and a frontal solver. This is referredto as a fully coupled approach, in which the full set ofequations is solved simultaneously.

Even for two-dimensional problems with modest geo-metrical complexity and a single relaxation time, memorylimitations prohibit the use of direct solvers. The need forfine meshes maintains to resolve the steep stress boundarylayers near curved boundary and singularities that occurwith many of the existing CEs. Consequently the coupledapproach requires a (frequently too) large amount of mem-ory and may lead to excessive CPU time consumption. Inparticular for three-dimensional computations, the coupledapproach with direct solvers is not feasible.


0 0.5 1 1.5 2 2.5 3 3.53.5

4

4.5

5

5.5

6

Weissenberg number

Dra

g co

rrec

tion

fact

or

Jin et al. (EEME)Legat (adaptive hp)Baaijens et al. (DEVSS/DG)Sun et al. (AVSS/SI)Fan et al. (DEVSS)Fan et al. (MIX1)

V V

V

V

r

z

R

Rc

(b)(a)

Figure 4. Falling-sphere configuration and drag correction factor. Only those results have been included that exceed a Wi number of2 and are obtained on the finest mesh reported.

One way to reduce memory requirements is to decouplethe viscoelastic extra stresses from the momentum andcontinuity equation. Coupling between the two sets ofequations is then achieved iteratively by means of a Picarditeration. However, convergence of this scheme is slowand the maximum attainable Weissenberg number is usuallysignificantly lower than with a coupled solver.

The use of iterative solvers for the set of equationsresulting from methods like DEVSS is not that trivial. Oftenthe matrix is ill conditioned and the number of iterationsto reach convergence is too large. The development ofappropriate preconditioners for these iterative methods isstill an active field of research. Advantages of the decoupledset of equations are that simpler matrix systems need to beinverted where iterative solvers converge much faster. Theapplication of domain decomposition based techniques incombination with parallel iterative solvers appears to beparticularly attractive for future developments.

3.3 Performance evaluation

The falling-sphere-in-a-tube problem, using the UCM mo-del, is by far the most cited benchmark problem for numer-ical methods for viscoelastic flow computations. It is wellknown that the major difficulty for this problem results fromthe development of stress boundary layers on the spherewall and the increase of the stress wake after the rear stag-nation point as the Weissenberg number increases. Recentanalysis of Fan (2003) show that the numerical simulationof the falling-sphere problem for an Oldroyd-B fluid is evenmore complicated than for the UCM fluid, which is a con-tradiction with earlier beliefs. For most other problems, the

maximum achievable Weissenberg number is minimal forthe UCM model.

The falling-sphere benchmark problem, graphically de-picted in Figure 4(a), is defined as follows. The spherewith radius R is located at the centerline of the tube withradius Rc. The tube wall moves parallel to the centerlinewith a velocity V in the positive z-direction. The ratioof the cylinder radius Rc and the sphere radius R isβ = (Rc/R) = 2. The Weissenberg number is defined as

Wi = λVR

(22)

The drag F0 on a sphere falling in an unbounded Newtonianmedium is given by

F0 = 6πηRV (23)

It is customary to compare the so-called drag correctionfactor given by

K(Wi) = F(Wi)F0

(24)

where F is the drag on the cylinder as a function of theWeissenberg number.

Figure 4(b) depicts an overview of currently availableresults, with the restriction that only reference is made tostudies that report drag correction factors beyond a Weis-senberg number of 2, while results on the finest meshavailable are included. The τzz stress distribution is depictedin Figure 5 and illustrates the difficulty of computing vis-coelastic flows even for a geometry having no geometricalsingularity. Steep stress boundary layers are formed near


0 0.5 1 1.5−2

−1

0

1

2

3

4

r/R

z/R

De = 0.5

0 0.5 1 1.5−2

−1

0

1

2

3

4

r/R

z/R

De = 1

0 0.5 1 1.5−2

−1

0

1

2

3

4

r/R

z/R

De = 1.5

0 0.5 1 1.5−2

−1

0

1

2

3

4

r/R

z/R

De = 2

Min: −1.07Max: 42.84

Min: −0.7075Max: 69.17

Min: −0.533Max: 92.35

Min: −0.4271Max: 113

Figure 5. Distribution of the τzz stress component as a function of the Weissenberg number.

curved boundaries. Moreover, rapid changes in stress fieldsare observed at the wake of the sphere, which emanate fromthe wake of the sphere. These steep stress gradients requirethe use of highly refined meshes.

Convergent results up to reasonably high values of theWeissenberg number (larger than 2) have been obtainedfor the falling-sphere-in-a-tube benchmark problem using avariety of methods, including EVSS, DEVSS, and explicitlyelliptic momentum equation (EEME) based methods, asa well as hp methods, provided that sufficiently refinedmeshes have been used.

It is unclear at this point whether results at even highervalues of the Weissenberg number can be obtained withsecond or higher-order methods when more refined meshesare used, although this may indeed be expected on the basisof previous experience. Nevertheless, it is fair to state thatsignificant progress has been made over the past decade.Petera (2002), who also studied the falling-sphere bench-mark problem, has reached Weissenberg numbers up to 6.6.However, a careful analysis of the stress levels obtainedby Petera show that the values obtained differ a factor 4with the results obtained by other authors, explaining thedifference in the maximum achieved Weissenberg number(Fan, 2003). Comparing the drag correction factor for dif-ferent numerical methods only is insufficient to judge thequality of the numerical solution. It is more appropriate tocompare the stress in the wake of the sphere for the differ-ent models. Moreover, if a time-marching scheme is used,special attention has to be given to demonstrate that thesteady state has been reached; in particular, for these high

Weissenberg numbers (Hulsen, Peters and van den Brule,2001).

3.4 Application to polymer processing

The viscoelastic flow of polymer melts has been inves-tigated in several nominally two-dimensional geometries,such as contraction flows (Schoonen, 1998; Verbeeten,2001), flows past a confined cylinder and the cross-slot flow(Peters et al., 1999; Verbeeten, Peters and Baaijens, 2002).Using the stress-optical rule, the computed stress fields maybe compared to birefringence measurements. Clearly, accu-racy, stability, and robustness of the computational schemeis a prerequisite for such an exercise, but the quality of thepredictions largely depends on the ability of the constitutivemodel to predict the rheological behavior of the polymermelt. In recent years, a number of constitutive models havebeen developed that are able to quantitatively describe rhe-ological measurements in both shear and elongation. Onthe basis of the pom-pom model (McLeish and Larson,1998), the XPP model was developed by Verbeeten, Petersand Baaijens (2001) that quantitatively describes rheolog-ical data of a number of commercial polymer melts. Thismodel is applied to the analysis of the flow past a confinedcylinder and the cross-slot flow.

To characterize the strength of the different flows, theWeissenberg number is defined as

Wi = λ̄ū2Dh

(25)


Experiment

Extended Pompon 1

x/h

y/h

0 1 2 3 4 5 6 7 8 9 10

−2

−1

0

1

We = 4.3

Figure 6. Comparison of the predicted and measured birefringence pattern in the cross-slot flow device.

Here, λ̄ denotes the viscosity averaged relaxation time forthe material and ū2D is the two-dimensional mean velocityand h a characteristic length of the flow geometry. Aspectrum of relaxation time is necessary to quantitativelycapture the rheological data of polymer melts.

The cross-slot geometry is particularly challenging be-cause regions with steady shear and steady elongation andcombination thereof are present. A typical example of acomparison of predicted and experimental birefringencepatterns is found in Figure 6. Steady shear is observed inthe upstream and down-stream channel, while steady pla-nar elongation is observed in a finite region near the centralstagnation point. In Figure 6, one half of the total geometryis shown, the top quarter depicts the experimental result.The bottom half of the image depicts the predicted bire-fringence profile. Excellent agreement between computedand measured birefringence is found. Verbeeten, Peters andBaaijens (2002) have shown that for this particular flow, theXPP model outperforms other models that are frequentlyused to analyze viscoelastic flows of polymer melts.

4 TIME DEPENDENT FLOWS

The most direct extension to unsteady flows of the mixedformulations discussed above are based on an implicit tem-poral discretization as first used by Northey, Armstrongand Brown (1992) and later by many others. By introduc-ing a selective implicit/explicit treatment of various parts ofthe equations, a certain decoupling at each time step of theset of equations may be achieved to improve computationalefficiency. For instance, Singh and Leal (1993) first appliedthe three-step operator splitting methodology developed byGlowinski and Pironneau (1992) to viscoelastic flows, laterfollowed by Saratimo and Piau (1994) and Luo (1996). Inthe first and third step, a generalized Stokes problem isobtained, while in the second step a convection–diffusion

type of problem needs to be solved. This offers the pos-sibility to apply dedicated solvers to subproblems of eachfractional time step.

In order to obtain an efficient time-marching scheme, weconsider an operator splitting method to perform the tem-poral integration. The major advantage of operator splittingmethods is the decoupling of the viscoelastic operator intoparts that are ‘simpler’ and can be solved more easily thanthe full problem. Hence, if we write equation (20) to (21) as

∂x

∂t= A(x) = A1(x) + A2(x) (26)

the θ-scheme is defined following Glowinski and Pironneau(1992):

xn+θ − xnθ�t

= A1(xn+θ) + A2(xn) (27)xn+1−θ − xn+θ(1 − 2θ)�t = A1(x

n+θ) + A2(xn+1−θ) (28)

xn+1 − xn+1−θθ�t

= A1(xn+1) + A2(xn+1−θ) (29)

with time step �t and θ = 1 − (1/√2) in order to retainsecond-order accuracy. The remaining problem is to definethe separate operators A1 and A2. In essence, we like tochoose A1 and A2 in such a way that solving equation (27)to (29) requires far less computational effort as compared tosolving the implicit problem while the stability envelope ofthe time integrator remains sufficiently large. If the simpli-fied problem A1 = βA and A2 = (1 − β)A is considered,the stability envelopes are plotted in Figure 7 for differentvalues of β. The arrows point towards the region of thecomplex plane for stable time integration. Obviously, set-ting β = 0 yields only a small portion of the complex planewhereas 0.5 ≤ β ≤ 1.0 results in a scheme that is uncondi-tionally stable. On the basis of the argument that we split the


−15 −10 −5 0 5 10 15−15

−10

−5

0

5

10

15 β = 0.50

β = 0.75

β = 1.00β = 0.00

β = 0.25

µ i∆t

µr ∆t

Figure 7. Stability envelope of the θ-scheme for A1 = βA andA2 = (1 − β)A with µ the spectrum of A. Regions for stabletime integration are indicated by the arrows for different valuesof β and it can be seen that for 0.5 ≤ β ≤ 1.0 this θ-scheme isunconditionally stable for this special choice of A1 and A2.

viscoelastic operator into a kinematic problem and a trans-port problem for the advection of polymer stress, we candefine A1 and A2 from the approximate location of the gov-erning eigenvalues. For instance, the viscous (Stokes) prob-lem has eigenvalues that are essentially real and negative.The absolute value has the tendency to grow very fast withmesh refinement (for a one-dimensional diffusion prob-lem using a low-order finite element method, max(|µ|) =O(N2) with N the number of grid points) and it is con-venient to define A1 as the kinematic problem for givenpolymer stress. On the other hand, the eigenspectrum of theremaining advection operator in the CE is located close tothe imaginary axis (however not on the imaginary axis dueto the introduction of Petrov–Galerkin weighting functionslater on) and we define A2 as the transport of extra stress.One possible definition of the θ-scheme (A1 and A2) is

A1 = −

0−∇ ·

(τ + αη (D − (G + GT))) + ∇p

∇ · uGT − ∇u

(30)

and

A2 = −

λ(u · ∇τ − G · τ − τ · GT ) + τ − η(G + GT)

000

(31)On the basis of the above definitions for A1 and A2, thekinematic (elliptic saddle point) problem for u, p, G is

updated implicitly in the first (equation (27)) and last (equa-tion (29)) step of the θ-scheme, whereas the transport ofpolymeric stress is updated explicitly.

A similar strategy is employed in the implicit/explicitNewton-like implementation of the DG method by Baaijens(1994b), which allows the elimination of the extra stresseson the element level at each time step. The resulting set ofequations has the size of a Stokes problem in the regularvelocity-pressure setting.

Notice that, rather than solving the steady flow prob-lem as such, one may use a time-marching procedure toapproach steady state as a limiting case. In particular, thesplitting of the set of equations as is achieved in the θ-scheme is of interest to reduce memory requirements aseach of the subproblems is significantly smaller than thefull coupled set of equations.

4.1 Linear stability analysis of complex flows

The computational analysis of the stability of viscoelas-tic flows has proven to be a major challenge. This isamply demonstrated by Brown et al. (1993) and Szadyet al. (1995), who computed the linear stability of a pla-nar Couette flow of the upper-convected Maxwell to accessthe numerical stability of a number of mixed finite ele-ment formulations. Theoretical results have shown thatthis inertialess flow is stable for any value of the Weis-senberg number. Methods based on a Galerkin, EEME,and EVSS formulation in combination with either SUPGor SU demonstrated a limiting Weissenberg beyond whichthe numerical solution became unstable. It is for this reasonthat the EVSS-G/SUPG and EVSS-G/SU formulations inwhich introduced for which no limiting Weissenberg num-ber was found within the range of Weissenberg numbersand mesh resolutions examined.

Simulation and prediction of transient viscoelastic flowphenomena probably represent the most challenging taskof computational rheology today. For instance, the study ofthe stability of polymer solutions in complex flows, suchas the flow around a cylinder (Sureshkumar et al., 1999)or the corrugated channel flow (Sureshkumar, 2001), hasgained attention.

The development of numerical tools that are able tohandle these dynamic flow phenomena (i.e. the transitionfrom steady to transient flows as dictated by the stabil-ity problem) requires a number of important issues tobe addressed properly. First of all, the CE that relatesthe polymer stress to the fluid deformation (and defor-mation history) needs to be defined. While most presentresearch focused on the stability behavior of the UCMmodel (e.g. Gorodtsov and Leonov, 1967; Renardy, 2000),


there has been much less interest on how the choice ofconstitutive model affects the predicted stability of a givenpolymer melt flow (Ho and Denn, 1977; Ganpule andKhomami, 1999; Grillet et al., 2002). This is an impor-tant issue when the dynamics of polymer melt flows arepredicted using numerical simulations. Grillet et al. (2002)and Bogaerds et al. (2002) have shown that this is not atrivial task since generally accepted models for polymermelts can behave very different in terms of their stabilitycharacteristics.

Bogaerds et al. (2002) introduced an extension of thecommonly used three-step θ-scheme described above, whichshows superior stability behavior. The idea is to eliminatethe stress from the hyperbolic equation and to substitute itin the momentum equation yielding a modified momentumequation:

Problem 4 (θ-Bogaerds 1a) Given the base flow (u, τ) attime t = tn the solution is found at t = tn + θ�t by

τtn+θ�t − τnθ�t

+ utn+θ�t · ∇τn − Gtn+θ�t · τn

− τn · (Gtn+θ�t)T + 1λ

(ωτtn+θ�t + (1 − ω)τn)

− η (Gtn+θ�t + (Gtn+θ�t)T) = 0 (32)− ∇ · (τ tn+θ�t + βηe(2D − Gtn+θ�t

− (Gtn+θ�t)T)) + ∇p = 0 (33)∇ · utn+θ�t = 0 (34)Gtn+θ�t − (∇utn+θ�t)T = 0 (35)

Problem 5 (θ-Bogaerds 1b) The second step of the θ-scheme using the quantities at time level t = tn+θ�t todetermine intermediate values at t = tn+(1−θ)�t by

τtn+(1−θ)�t − τtn+θ�t1 − 2θ�t + u

tn+θ�t · ∇τ tn+(1−θ)�t

− Gtn+θ�t · τ tn+(1−θ)�t − τ tn+(1−θ)�t · (Gtn+θ�t)T

+ 1λ

(ωτ tn+θ�t + (1 − ω)τ tn+(1−θ)�t)

− η (Gtn+θ�t + (Gtn+θ�t)T) = 0 (36)The third step of the θ-scheme is equal to the first step,

but now using the quantities at t = tn+(1−θ)�t , which areupdated to t = tn+1.

The above decoupling of the constitutive relation fromthe remaining equations provides a very efficient time inte-gration technique that is second-order accurate for linearstability problems. The efficiency becomes even more evi-dent when real viscoelastic fluids are modeled for which the

spectrum of relaxation times is approximated by a discretenumber of viscoelastic modes. For simplicity, the procedureis described for the UCM model. However, if nonlinearmodels like the PTT, Giesekus, or the XPP model are con-sidered, as is common for polymer melts, a generalizationof the θ-scheme is readily obtained.

An illustration of the effectiveness of the DEVSS-G/SUPG(θ) formulation relative to, for instance, theDEVSS-G/DG formulation is demonstrated in Figure 8.This shows the L2 norm of the perturbation of the velocityfield in the stability analysis of the planar Couette flowemploying the UCM model. This flow is known to bestable and the L2 norm of the perturbation should decay ata rate proportional to 1/(2We); see Gorodtsov and Leonov(1967). Clearly, the instability predicted by the DEVSS-G/DG method is nonphysical and a numerical artifact.Details of this analysis may, for example, be found in Grilletet al. (2002).

5 INTEGRAL AND STOCHASTICCONSTITUTIVE MODELS

The conventional approach in the simulation of viscoelasticfluid flow has been to start from a closed-form CE. In theearly days, the CE was based on continuum mechanics only(see Figure 9). These models contain limited informationon the structure of the polymer and are thus insufficient toultimately base polymer processing designs directly on thepolymer architecture. Therefore, the microstructural model-ing became popular in the last two decades. The focus hasbeen on finding models that contain microstructural infor-mation of the polymer but remain simple enough that theystill can be written as a closed-form CE. Examples are theFENE-P model for dilute polymer solutions (a differen-tial model) and the Doi–Edwards integral model with theso-called independent alignment approximation (IAA). Acomprehensive overview of both approaches can be foundin Bird, Armstrong and Hassager (1987a) and Bird et al.(1987b).

A basic problem with finding a closed-form CE is theapproximations that have to be made in the kinetic theoryto arrive at such an equation. Therefore, a technique hasbeen introduced by Laso and Öttinger (1993), to avoid theproblems described above, by simply bypassing the needfor a CE. The essential idea of this so-called Calculationof Non-Newtonian Flow: Finite Elements & StochasticSimulation Technique (CONNFFESSIT) approach is tocombine traditional finite element techniques and Browniandynamics simulations that solve the kinetic theory equationsof the polymer. In contrast to a conventional finite elementapproach, however, the polymer contribution to the stress,


0 5 10 15 20−4

−3

−2

−1

0

1

2

−40 100 200 300 400

−2

0

2

4

6

8Lo

g(|L

2|)

ε

Log(

|L2|

)ε

Time (s)

DG DEVSS-G/SUPG(θ)

Time (s)

We = 1.3We = 1.2We = 1.1

We = 80We = 60We = 40We = 20

Figure 8. Stability analysis of the planar Couette flow using the DG method and the SUPG formulation using the θ-scheme. A colorversion of this image is available at http://www.mrw.interscience.wiley.com/ecm

Flow calculations using FEM, FVM, ...

Continuummechanics

Microstructuralmodeling

Differentialmodels

Integralmodels

“Stochastic”models

Deformationfields

(Brownian)configuration

fields

Figure 9. Overview of the modeling of the flow of viscoelas-tic fluids. The traditional approach starts from pure continuummechanics whereas the more modern approach uses modelsbased on microstructural theory. Closed-form constitutive equa-tions from both approaches are either in differential or integralform. The latest simulations that avoid closed-form constitutiveequations are based on stochastic integration techniques, such asBrownian dynamics.

needed in the finite element calculation, is not calculatedfrom a CE, but instead from the configuration of a largeensemble of model polymers. The time evolution of thisensemble is calculated using Brownian dynamics. Since inthis stochastic approach the stress field is calculated on thebasis of the configurations of the polymers, microstructuralinformation (such as distribution functions) remains fullyavailable during the flow calculation.

In this text, we will review two basic techniques forimplementing integral and stochastic models: the defor-mation fields (DF) method for integral models and the

Brownian configuration fields (BCF) method for stochasticmethods. The BCF method is basically a much-improvedimplementation of the original CONNFFESSIT idea.

6 GOVERNING EQUATIONS

The governing momentum equations are as stipulated byequation (2) and (1). As before, we will use the Oldroyd-Bmodel.

As an example of an integral model derived frommicrostructural theory, we will use the well-knownDoi–Edwards model with (IAA) by Doi and Edwards(1986), in which only the longest relaxation time of theoriginal Doi–Edwards spectrum, the reptation time, heredenoted by λ, is retained. The polymer stress τ in thismodel is proportional to the orientation tensor S,

τ = 5G0S (37)

The constant G0 is the elastic modulus, G0 = nkBT , wheren is the number of tube segments per unit volume, kBdenotes Boltzmann’s constant, and T denotes the absolutetemperature. The orientation tensor S is an integral over thedeformation history:

S(t) =∫ t

−∞µ(t; t ′)Q̂(t; t ′) dt ′ (38)

where the weight factor µ(t; t ′) = exp[−(t − t ′)/λ]/λ andQ̂(x, t; t ′) is a tensor representing the orientation of tubesegments that were created at time t ′ and surviving atthe current time t . Using Currie’s approximation (Currie,


1982), the tensor Q̂(x, t; t ′) can be fully expressed in termsof the Finger tensor B(x, t; t ′):

Q̂ = 1(IB − 1 + 2

√IIB + 13/4)

[B − B

−1√IIB + 13/4

](39)

In this expression IB denotes the trace of B and IIB denotesthe second invariant of B,

IIB = 12(I 2B − IB2

)(40)

As example of a stochastic model, we use the Hookeandumbbell. In this model, a polymer solution is considered asa suspension of noninteracting elastic dumbbells consistingof two Brownian beads with friction coefficient ζ connectedby a linear spring. The configuration of a dumbbell, thatis, the length and orientation of the spring connectingthe two beads, is indicated by a vector Q. The springforce can thus be written as F (c) = HQ, where H is thespring constant. It is possible to describe this model interms of a distribution function and a diffusion equation(the Fokker–Planck equation); see Bird et al. (1987b). Itturns out that the model can be closed and the stress isgiven by the Oldroyd-B equation with a relaxation timeλ = ζ/4H and η = nkT λ. An alternative, but equivalent,approach for generating the configuration distribution isto make use of a stochastic differential equation. Thismeans that the dynamics described by the Fokker–Planckequation can be considered as the result of a stochasticprocess acting on the individual dumbbells. For the detailson the formal relationship between the Fokker–Planckequation and stochastic differential equations, the readeris referred to Öttinger (1996). For the Hookean dumbbellmodel considered here, the stochastic equation reads

dQ(t) =(L(t) · Q(t) − 2H

ζQ(t)

)dt +

√4kT

ζdW (t)

(41)where W (t) is a Wiener process, that accounts for therandom displacements of the beads due to thermal motion.The Wiener process is a Gaussian process with zero meanand covariance 〈W (t)W (t ′)〉 = min(t, t ′)I . In a BrownianDynamics simulation, this equation is integrated for a largenumber of dumbbells. A typical algorithm, based upon anexplicit Euler integration, reads

Q(t + �t) = Q(t) +(L(t) · Q(t) − 2H

ζQ(t)

)�t

+√

4kT

ζ�W (t) (42)

where the components of the random vector �W (t) areindependent Gaussian variables with zero mean and vari-ance �t . Once the configurations are known, the stress canbe estimated by

τ ≈ −nkT I + nH 1Nd

Nd∑i=1

QiQi (43)

where Nd is the number of dumbbells and Qi representsthe ith dumbbell in the ensemble.

7 THE DEFORMATION FIELDSMETHOD

In integral type models, such the Doi–Edwards model givenby equations (37) to (39), we need to find the Finger tensorB t ′(t) at all past times t

′. The time evolution of the Fingertensor B t ′(t) for a fixed particle and fixed reference time t

′is governed by

Ḃ t ′ = L · B t ′ + B t ′ · LT (44)

where L = (∇u)T. The initial condition is B t ′(t ′) = I ,where I is the unity tensor. In principle, equation (44) mustbe solved for all particles and all reference times t ′ < tin order to compute the extra stress in every point of thedomain.

Particle tracking can be avoided by introducing a fieldvariable B t ′(x, t) and solving equation (44) in a Eulerianframe:

∂

∂tB t ′ + u · ∇B t ′ = L · B t ′ + B t ′ · LT (45)

with an initial condition

B t ′(x, t′) = I (46)

for the complete field. Note, that t ′ is still fixed andwe have to solve equation (45) for all reference timest ′ < t . The field variable B t ′(x, t) for fixed t ′ is calleda deformation field in Peters, Hulsen and van den Brule(2000a). DFs (labelled by their creation time t ′) that areactually computed are restricted to a small number (say100) of discrete values of t ′. At each time step, a newfield is created at the current time starting with the initialcondition I and an old one is removed such that the ageτ = t − t ′ of all fields is smaller than some cutoff valueτc. The discretized version of the integral expression (38)is just a weighted sum over all fields. We use a simpletrapezoidal rule for the integration.


The original deformation fields method (DFM) as des-cribed above has some serious drawbacks, such as removaland creation of fields and unsteady fields even if the flowis steady. These drawbacks are related to the labeling offields by the absolute reference (creation) time t ′. Therefore,in Hulsen, Peters and van den Brule (2001) an improvedDFMs is developed that uses the age τ = t − t ′ as indepen-dent variable instead of t ′:

B(x, t, τ) = B t−τ(x, t) (47)

The equation for B(x, t, τ) now contains an extra termcompared to the equation for B t ′(x, t):

∂B

∂t+ ∂B

∂τ+ u · ∇B = L · B + B · LT (48)

where u and L are functions of (x, t) only and thus inde-pendent of τ. Instead of solving a n-dimensional hyperbolicequation for each t ′ < t , we now have to solve a n + 1-dimensional hyperbolic equation. Also the initial conditionequation (46) for each field has now turned into the bound-ary condition

B(x, t, 0) = I (49)

We usually assume that at the start (t = 0) the fluid hasbeen at rest for a long time and the initial condition

B(x, 0, τ) = I (50)

is valid. Once we have B(x, t, τ), we can compute the stressby integrating over the τ-co-ordinate.

8 BROWNIAN CONFIGURATION FIELDS

In order to solve a flow problem, it is necessary to findan expression for the stress at a specified position x attime t . For the stochastic model, we have to convect asufficiently large number of dumbbells through the flowdomain until they arrive at x at time t . Neglecting center-of-mass diffusion, these dumbbells all experienced the samedeformation history but were subjected to different andindependent stochastic processes.

However simple in theory, a number of problems haveto be addressed in practice. For instance, if we dispersea large number of dumbbells into the flow domain we notonly have to calculate all their individual trajectories but, tocalculate the local value of the stress, we must every timestep also sort all the dumbbells into cells (or elements).Once these problems are solved, it is possible to constructa transient code to simulate a nontrivial flow problem. Weused a different approach that overcomes the problems

associated with particle tracking. Instead of convectingdiscrete particles specified by their configuration vectorQi an ensemble of Nf continuous configuration fieldsQi (x, t) is introduced. Initially, the configuration fieldsare spatially uniform and their values are independentlysampled from the equilibrium distribution function of theHookean dumbbell model. After start-up of the flow field,the configuration fields are convected by the flow and aredeformed by the action of the velocity gradient, by elasticretraction, and by Brownian motion in exactly the same wayas a discrete dumbbell. The evolution of a configurationfield is thus governed by

dQ(x, t) =(−u(x, t) · ∇Q(x, t) + L(x, t) · Q(x, t)

− 2Hζ

Q(x, t)) dt +√

4kT

ζdW (t) (51)

The first term on the RHS of equation (51) accounts forthe convection of the configuration field by the flow. Itshould be noted that dW (t) only depends on time and henceit affects the configuration fields in a spatially uniformway. For this reason, the gradients of the configurationfields are well defined and smooth functions of the spatialcoordinates. Of course, the stochastic processes acting ondifferent fields are uncorrelated.

From the point of view of the stress calculation, this pro-cedure is completely equivalent to the tracking of individualdumbbells: An ensemble of configuration vectors {Qi} withi = 1, Nf is generated at (x, t), which all went through thesame kinematical history but experienced different stochas-tic processes. This is precisely what is required in orderto determine the local value of the stress. The fields Qiare called Brownian configuration fields Hulsen, van Heeland van den Brule (1997). In Öttinger, van den Brule andHulsen (1997), it is shown that the BCF approach can beregarded as an extremely powerful extension of variancereduction techniques based on parallel process simulation.

In the remainder of this paper, we prefer to scale thelength of the configuration vector with

√(kT /H), which

is one-third of the equilibrium length of a dumbbell. Therelevant equations thus become

τ = nkT (−I + c) = ηλ

(−I + c) (52)

where c is the conformation tensor, which is dimensionlessand reduces to the unity tensor at equilibrium. The closed-form CE for the Hookean dumbbell (=Oldroyd-B equation)can now be written as

c + λ∇c = I (53)


The equation for the evolution of the configuration fieldsbecomes

dQ̃(x, t) =(−u(x, t) · ∇Q̃(x, t) + L(x, t) · Q̃(x, t)

− 12λ

Q̃(x, t))

dt +√

1

λdW (t) (54)

Finally, the conformation tensor field follows from

c(x, t) = 1Nf

Nf∑i=1

Q̃i(x, t)Q̃i(x, t) (55)

From now on the tildes are dropped and it is understoodthat Q is a dimensionless quantity.

9 NUMERICAL METHODS

9.1 Spatial discretization

For the spatial discretization of the system of equations,we will use the finite element method. In order obtain abetter stability and extend the possible stress space, we usethe DEVSS formulation of Guénette and Fortin (1995b)for the discretization of the linear momentum balanceand the continuity equation. The DG formulation will beused to discretize the closed-form CEs, the convectionequation for the DFs and the equation for the configurationfields. In the DG formulation, the interpolation functionsare discontinuous across elements, leading to a minimalcoupling between elements. This means that in our time-stepping scheme the convected variables at the next timestep can be computed at the element level. In this way, weavoid solving a large number of coupled equations.

In this work, we use quadrilateral elements with continu-ous biquadratic polynomials for the velocity, discontinuouslinear polynomials for the pressure and continuous bilin-ear polynomials for the projected velocity gradients anddiscontinuous bilinear polynomials for c, B, and Q.

9.2 Time discretization

For the time discretization of the convection equations,we use explicit schemes. For the CE and the equation forthe configuration fields we use an explicit Euler scheme,whereas for the DF we use a second-order Adams–Bashforth scheme for stability (Hulsen, Peters and van denBrule, 2001). At each step we find the convected quantityat the next time level tn+1, that is, cn+1, Bn+1 or Qn+1, by

solving the equations at the element level. From these val-ues we can compute the polymer stress τn+1. The nonlinearinertia terms are also treated in an explicit way, which leadsto a system matrix for solving (un+1, pn+1, H n+1) that issymmetrical and LU decomposition is performed at the firsttime step. Since this matrix is constant in time, solutionsat later time steps can be found by back substitution only.This results in a significant reduction of the CPU time.

10 RESULTS

We consider the planar flow past a cylinder of radius apositioned between two flat plates separated by a distance2H . The ratio a/H is equal to 2 and the total length ofthe flow domain is 30a. The flow geometry is shown inFigure 10.

Rather than specifying inflow and outflow boundaryconditions, we take the flow to be periodical. This meansthat we periodically extend the flow domain such thatcylinders are positioned 30a apart. The flow is generatedby specifying a flow rate Q that is constant in time.The required pressure gradient is computed at each instantin time. We assume no-slip boundary conditions on thecylinder and the walls of the channel. Since the problemis assumed to be symmetrical we only consider half of thedomain and use symmetry conditions on the center line,that is, zero tangential traction.

The dimensionless parameters governing the problemare the Reynolds number Re = ρUa/η, the Weissenbergnumber Wi = λU/a, and the viscosity ratio ηe/η, whereU = Q/2H is the average velocity and η is the zero-shear-rate viscosity of the fluid. For an Oldroyd-B fluid/Hookeandumbbell, we have η = ηe + η and, for the Doi–Edwardsmodel, it is given by η = ηe + G0λ. We will define thedimensionless drag coefficient by K = Fx/ηU , where Fxis the drag force per unit length on the cylinder.

First we show some results for the Oldroyd-B andHookean dumbbell model for which we take Re = 0.01 andηe/η = 1/9. The dimensionless drag coefficient at a Weis-senberg number of Wi = 0.6 until tU/a = 7 is shown in

H = 2a

L = 30a

xa

y

Figure 10. Geometry of the cylinder between two flat plates. Theflow is from left to right.


10

20

30

40

50

60

70

80

90

100

110

0 1 2 3 4 5 6 7

Dim

ensi

onle

ss d

rag

coef

ficie

nt K

t *U/a

Hookean dumbbell Nf = 2000Hookean dumbbell Nf = 4000Hookean dumbbell Nf = 8000Oldroyd-B differential model

Figure 11. The drag on the cylinder for Wi = 0.6 as a function of time for the Oldroyd-B and the Hookean dumbbell.

Figure 11. We have a good agreement between the Oldroyd-B model and the Hookean dumbbell, although the lattermodel shows fluctuations that are typical for stochasticmodels. Increasing the number of BCF Nf reduces the fluc-tuations and the drag coefficient seems to converge to acurve slightly below the macroscopic curve. The reason forthe slightly different drag coefficient is due to the coarse-ness of the mesh. In Hulsen, van Heel and van den Brule(1997), it is shown that the BCF method is convergent withmesh refinement.

To show that the BCF method leads to smooth functionsin space, we have plotted contours of cxx in Figure 12 forboth the Oldroyd-B and the Hookean dumbbell model. Wesee that the agreement is excellent.

Now we show some results for the Doi–Edwards modelwith IAA and a single relaxation time. We take Re = 0 andηe = 0.05η for stability reasons (see Peters et al., 2000b).

Figure 12. Contours of cxx for Wi = 0.6. Time = 7. Upperhalf: stochastic Hookean dumbbell with Nf = 2000. Lower half:Oldroyd-B differential model.

The dimensionless drag coefficient at a Weissenberg num-ber of Wi = 0.6 until tU/a = 4 is shown in Figure 13.Also shown is the dimensionless drag coefficient for anOldroyd-B model with the same value of ηe, η, and λ. Thedrag of the Doi–Edwards model for larger times is muchlower than the Oldroyd-B model due to shear-thinning inthe Doi–Edwards model.

11 CONCLUSIONS AND DISCUSSION

We have reviewed a number of mixed finite elementmethods to solve viscoelastic flow problems. Today, themethod of choice appears either the DEVSS(-G)/DG orthe DEVSS(-G)/SUPG formulation. The use of the ‘-G’formulation appears mandatory if the stability of viscoelas-tic flows is examined, and therefore is also recommendedfor time-dependent flow analysis. The key advantage ofthe DEVSS(-G)/DG method is its relative efficiency dueto possibility to eliminate the extra stress variables onan element-by-element level. Moreover, the DG methodappears to be more robust in the presence of geometri-cal singularities (Baaijens, 1998). The limitation of thismethod, however, is its inability to correctly predict thestability of viscoelastic flows. For this purpose, the DEVSS-G/SUPG formulation is superior.

We have also reviewed two methods that are the buildingblocks for implementation in fluid mechanics codes ofmodels that start from microstructural concepts: the DFmethod and the BCF method. The DFM method is forimplementation of integral models and the BCF method


0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5 3 3.5 4

Dim

ensi

onle

ss d

rag

coef

ficie

nt K

Doi–EdwardsOldroyd-B

t*U/a

Figure 13. The drag on the cylinder for Wi = 0.6 as a function of time for the Doi–Edwards model and an Oldroyd-B model with thesame parameters.

is for implementation of stochastic models. Both methodsseem to behave excellently in a time-dependent Eulerianframe.

We have only discussed the basic techniques using themost elementary models. However, both the DF and BCFmethod can be applied to much more advanced models.For example, the DF method has been applied to complexmodels for melts such as the Mead–Larson–Doi model(Peters et al., 2000b), the Marrucci–Greco–Iannirubertomodel (Wapperom and Keunings, 2000) and the integralpom-pom model (Wapperom and Keunings, 2001). TheBCF method has been applied to models for which closureis not possible, such as the FENE (finitely extensiblenonlinear elastic) model (van Heel, Hulsen and van denBrule, 1998) and a model for fiber suspensions (Fan,Phan-Thien and Zheng, 1999a). Also, the linear stabilityof flows for a FENE model has been studied using theBCF method (Somasi and Khomami, 2000). In a recentdevelopment, the DF method and the BCF method havebeen combined to study complex flows of the stochasticmodel of Öttinger for polymer melts (Gigras and Khomami,2002). No doubt, that more developments can be expectedin the near future.

12 RELATED CHAPTERS

(See also Chapter 9 of Volume 1, Chapter 2, Chapter 4of this Volume)

REFERENCES

Alves MA, Oliveira PJ and Pinho FT. Benchmark solutions forthe flow of Oldroyd-B and PTT fluids in planar contractions.J. Non-Newtonian Fluid Mech. 2003; 110:45–75.

Baaijens FPT. Numerical analysis of unsteady viscoelastic flow.Comput. Methods Appl. Mech. Eng. 1992; 94:285–299.

Baaijens FPT. Application of low-order Discontinuous Galerkinmethods to the analysis of viscoelastic flows. J. Non-NewtonianFluid Mech. 1994a; 52:37–57.

Baaijens FPT. Numerical experiments with a DiscontinuousGalerkin method including monotonicity enforcement on thestick-slip problem. J. Non-Newtonian Fluid Mech. 1994b;51:141–159.

Baaijens FPT. Mixed finite element methods for viscoelasticflow analysis: a review. J. Non-Newtonian Fluid Mech. 1998;79:361–386.

Baaijens FPT, Selen SHA, Baaijens HPW, Peters GWM andMeijer HEH. Viscoelastic flow past a confined cylinder of aLDPE melt. J. Non-Newtonian Fluid Mech. 1997; 68:173–203.

Baranger J and Sandri D. A formulation of Stoke’s problemand the linear elasticity equations suggested by the oldroydmodel for viscoelastic flow. Math. Modell. Numer. Anal. 1992;26(2):331–345.

Beris AN, Armstrong RC and Brown RA. Finite element calcu-lation of viscoelastic flow in a Journal bearing: I, Small eccen-tricities. J. Non-Newtonian Fluid Mech. 1984; 16:141–172.

Beris AN, Armstrong RC and Brown RA. Finite element cal-culation of viscoelastic flow in a Journal bearing: II, mod-erate eccentricities. J. Non-Newtonian Fluid Mech. 1986; 19:323–347.


Bird RB, Armstrong RC and Hassager O. Dynamics of PolymericLiquids, Vol. 1: Fluid Mechanics (2nd edn). John Wiley & Sons:New York, 1987a.

Bird RB, Curtiss CF, Armstrong RC and Hassager O. Dynamicsof Polymeric Liquids, Vol. 2: Kinetic Theory (2nd edn). JohnWiley & Sons: New York, 1987b.

Bogaerds ACB. Stability Analyses of Viscoelastic Flows. PhDthesis, Eindhoven University of Technology, 2002.

Bogaerds ACB, Verbeeten WMH and Baaijens FPT. Successesand failures of discontinuous Galerkin methods in viscoelasticfluid analysis. In Discontinuous Galerkin Methods, Number 11in Lecture Notes in Computational Science and Engineering,Cockburn B, Karniadakis GE and Shu C-W (eds). Springer-Verlag, Berlin, 2000; 263–270.

Bogaerds ACB, Grillet AM, Peters GWM and Baaijens FPT. Sta-bility analysis of viscometric polymer flows using the pom-pomconstitutive equation. J. Non-Newtonian Fluid Mech. 2002;108:187–208.

Brezzi F and Fortin M. Mixed and Hybrid Finite Element Meth-ods, Series in Computational Mechanics. Springer-Verlag: NewYork, 1991.

Brooks AN and Hughes TJR. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows withparticular emphasis on the incompressible Navier–Stokes equa-tions. Comput. Methods Appl. Mech. Eng. 1982; 32:199–259.

Brown RA and McKinley GH. Report on the VIIIth internationalworkshop on numerical methods in viscoelastic flows. J. Non-Newtonian Fluid Mech. 1994; 52(3):407–413.

Brown RA, Szady MJ, Northey PJ and Armstrong RC. On thenumerical stability of mixed finite element methods for vis-coelastic flows governed by differential constitutive equations.Theor. Comput. Fluid Dyn. 1993; 5:77–106.

Caswell B. Report on the IXth international workshop on numer-ical methods in non-Newtonian flows. J. Non-Newtonian FluidMech. 1996; 62:99–110.

Crochet MJ and Keunings R. Finite element analysis of die swellat highly elastic fluids. J. Non-Newtonian Fluid Mech. 1982;10:339–356.

Currie PK. Constitutive equations for polymer melts predicted bythe Doi-Edwards and Curtis-Bird kinetic theory models. J. Non-Newtonian Fluid Mech. 1982; 11:53–68.

Debae F, Legat V and Crochet MJ. Practical evaluation of fourmixed finite element methods for viscoelastic flow. J. Rheol.1994; 38(2):421–441.

Doi M and Edwards SF. The Theory of Polymer Dynamics, Inter-national Series of Monographs on Physics, No. 73. ClarendonPress: Oxford, 1986.

Fan Y. Limiting behavior of the solutions of a falling sphere ina tube filled with viscoelastic fluids. J. Non-Newtonian FluidMech. 2003; 110:77–102.

Fan X-J, Phan-Thien N and Zheng R. Simulation of fibre sus-pension flows by the brownian configuration field method. J.Non-Newtonian Fluid Mech. 1999a; 84:257–274.

Fan Y, Tanner RI and Phan-Thien N. Galerkin/least-square finite-element methods for steady viscoelastic flows. J. Non-New-tonian Fluid Mech. 1999b; 84:233–256.

Fortin M and Fortin A. A new approach for the fem simulationof viscoelastic flows. J. Non-Newtonian Fluid Mech. 1989;32:295–310.

Fortin M and Pierre R. On the convergence of the mixed methodof Crochet and Marchal for viscoelastic flows. Comput. Meth-ods Appl. Mech. Eng. 1987; 7:1035–1052.

Ganpule HK and Khomami B. An investigation of interfacialinstabilities in the superposed channel flow of viscoelasticfluids. J. Non-Newtonian Fluid Mech. 1999; 81:27–69.

Gigras PG and Khomami B. Adaptive configuration fields: a newmultiscale simulation technique for reptation based models witha stochastic strain measure and local variations of life spandistribution. J. Non-Newtonian Fluid Mech. 2002; 108:99–122.

Glowinski R and Pironneau O. Finite element methods for theNavier–Stokes equations. Ann. Rev. Fluid Mech. 1992; 24:167–204.

Gorodtsov VA and Leonov AI. On a linear instability of a planeparallel couette flow of viscoelastic fluid. J. Appl. Math. Mech.1967; 31:310–319.

Grillet AM, Bogaerds ACB, Peters GWM and Baaijens FPT. Sta-bility analysis of constitutive equations for polymer meltsin viscometric flows. J. Non-Newtonian Fluid Mech. 2002;103:221–250.

Guènette R and Fortin M. A new mixed finite element method forcomputing viscoelastic flows. J. Non-Newtonian Fluid Mech.1995a; 60:27–52.

Guénette R and Fortin M. A new mixed finite element method forcomputing viscoelastic flows. J. Non-Newtonian Fluid Mech.1995b; 60:27–52.

Ho TC and Denn MM. Stability of plane Poiseuille flow of ahighly elastic liquid. J. Non-Newtonian Fluid Mech. 1977;3:179–195.

Hulsen MA, Peters EAJF and van den Brule BHAA. A newapproach to the deformation fields method for solving complexflows using integral constitutive equations. J. Non-NewtonianFluid Mech. 2001; 98:201–221.

Hulsen MA, van Heel APG and van den Brule BHAA. Simula-tion of viscoelastic flows using Brownian configuration fields.J. Non-Newtonian Fluid Mech. 1997; 70:79–101.

Kawahara M and Takeuchi N. Mixed finite element methodfor analysis of viscoelastic fluid flow. Comput. Fluids 1977;5:33–45.

Keunings R. Finite element methods for integral viscoelastic flu-ids. In Rheology Review 2003, Binding DM and Walters K(ed.). British Society of Rheology, Aberystwyth, UK, 2003;167–195.

King RC, Apelian MN, Armstrong RC and Brown RA. Numer-ically stable finite element techniques for viscoelastic calcu-lations in smooth and singular geometries. J. Non-NewtonianFluid Mech. 1988; 29:147–216.

Laso M and Öttinger HC. Calculation of viscoelastic flow usingmolecular models: the CONNFFESSIT approach. J. Non-Newtonian Fluid Mech. 1993; 47:1–20.

Lesaint P and Raviart PA. On a Finite Element Method for Solvingthe Neutron Transport Equation. Academic Press: New York,1974.


Lunsmann WJ, Genieser L, Armstrong RC and Brown RA. Finiteelement analysis of steady viscoelastic flow around a sphere ina tube: calculations with constant viscosity models. J. Non-Newtonian Fluid Mech. 1993; 48:63–99.

Luo X-L. Operator splitting algorithm for viscoelastic flow andnumerical analysis for the flow around a sphere in a tube. J.Non-Newtonian Fluid Mech. 1996; 63:121–140.

Marchal JM and Crochet MJ. A new mixed finite element forcalculating viscoelastic flow. J. Non-Newtonian Fluid Mech.1987; 26:77–114.

McLeish TCB and Larson RG. Molecular constitutive equationsfor a class of branched polymers: the Pom-Pom polymer. J.Rheol. 1998; 42(1):81–110.

Mendelson MA, Yeh PW, Armstrong RC and Brown RA. Ap-proximation error in finite element calculation of viscoelasticfluid flows. J. Non-Newtonian Fluid Mech. 1982; 10:31–54.

Northey PJ, Armstrong RC and Brown RA. Finite-amplitudetime-periodic states in viscoelastic taylor-couette flow describedby the ucm model. J. Non-Newtonian Fluid Mech. 1992;42:117–139.

Öttinger HC. Stochastic Processes in Polymeric Fluids. Springer-Verlag: Berlin, 1996.

Öttinger HC, van den Brule BHAA and Hulsen MA. Brownianconfiguration fields and variance reduced CONNFFESSIT. J.Non-Newtonian Fluid Mech. 1997; 70:255–261.

Owens RG and Phillips TN. Computational Rheology. ImperialCollege Press: London, 2002.

Owens RG, Chauviére C and Philips TN. A locally-upwindedspectral technique (LUST) for viscoelastic flows. J. Non-Newtonian Fluid Mech. 2002; 108:49–71.

Perera MGN and Walters K. Long-range memory effects in flowsinvolving abrupt changes in geometry, Part I. Flows associatedwith L-shaped and T-shaped geometries. J. Non-NewtonianFluid Mech. 1977; 2:49–81.

Petera J. A new finite element scheme using the Lagrangianframework for simulation of viscoelastic fluid flows. J. Non-Newtonian Fluid Mech. 2002; 103:1–43.

Peters EAJF, Hulsen MA and van den Brule BHAA. InstationaryEulerian viscoelastic flow simulations using time separableRivlin-Sawyers constitutive equations. J. Non-Newtonian FluidMech. 2000a; 89:209–228.

Peters EAJF, van Heel APG, Hulsen MA and van den BruleBHAA. Generalisation of the deformation field method tosimulate advanced reptation models in complex flow. J. Rheol.2000b; 44:811–829.

Peters GWM, Schoonen JFM, Baaijens FPT and Meijer HEH. Onthe performance of enhanced constitutive models for polymermelts in a cross-slot flow. J. Non-Newtonian Fluid Mech. 1999;82:387–427.

Phillips TN. The spectral simulation of axisymmetric non-New-tonian flows using time splitting techniques. Finite Elem. Anal.Des. 1994; 16:229–236.

Pilitsis S and Beris AN. Calculations of steady-state viscoelasticflow in an undulating tube. J. Non-Newtonian Fluid Mech.1989; 31:231–287.

Rajagopalan D, Armstrong RC and Brown RA. Calculation ofsteady, viscoelastic flow using a multimode maxwell model:application of the explicitly elliptic momentum equation (eeme)formulation. J. Non-Newtonian Fluid Mech. 1990;36:135–157.

Renardy M. Location of the continuous spectrum in complexflows of the UCM fluid. J. Non-Newtonian Fluid Mech. 2000;94:75–85.

Saratimo P and Piau JM. Flow characteristics of viscoelastic flu-ids in an abrupt contraction by using numerical modeling. J.Non-Newtonian Fluid Mech. 1994; 42:263–288.

Schoonen JFM. Determination of Rheological Constitutive Equa-tions Using Complex Flows. PhD thesis, Eindhoven Universityof Technology, 1998.

Singh P and Leal LG. Finite element simulation of the start-upproblem for a viscoelastic fluid in eccentric rotating cylindergeometry using a third order upwind scheme. Theor. Comput.Fluid Dyn. 1993; 5:107–137.

Somasi M and Khomami B. Linear stability and dynamicsof viscoelastic flows using time-dependent stochastic sim-ulation techniques. J. Non-Newtonian Fluid Mech. 2000;93:339–362.

Sureshkumar R. Local linear stability characteristics of viscoelas-tic periodic channel flow. J. Non-Newtonian Fluid Mech. 2001;97:125–148.

Sureshkumar R, Smith MD, Armstrong RC and Brown RA. Lin-ear stability and dynamics of viscoelastic flows using time-dependent numerical simulations. J. Non-Newtonian FluidMech. 1999; 82:57–104.

Szady MJ, Salomon TR, Liu AW, Bornside DE, Armstrong RCand Brown RA. A new mixed finite element method for vis-coelastic flows governed by differential constitutive equations.J. Non-Newtonian Fluid Mech. 1995; 59:215–243.

van Heel APG, Hulsen MA and van den Brule BHAA. On theselection of parameters in the fene-p model. J. Non-NewtonianFluid Mech. 1998; 75:253–271.

Verbeeten WMH. Computational Polymer Melt Rheology. PhDthesis, Eindhoven University of Technology, 2001.

Verbeeten WMH, Peters GWM and Baaijens FPT. Differentialconstitutive equations for polymer melts: the Extended Pom-Pom model. J. Rheol. 2001; 45(4):823–844.

Verbeeten WMH, Peters GWM and Baaijens FPT. Viscoelasticanalysis of complex polymer melt flows using the extendedpom-pom model. J. Non-Newtonian Fluid Mech. 2002;108(1–3):301–326.

Wapperom P and Keunings R. Simulation of linear polymer meltsin transient complex flow. J. Non-Newtonian Fluid Mech. 2000;95:67–83.

Wapperom P and Keunings R. Numerical simulation of branchedpolymer melts in transient complex flow using pom-pom mod-els. J. Non-Newtonian Fluid Mech. 2001; 97:267–281.

Xu S, Davies AR and Phillips TN. Pseudospectral method fortransient viscoelastic flow in axisymmetric channel. Numer.Methods Partial Diff. Equations 1993; 9:691–710.

Documents

Chapter 14 The Use of Mixed Finite Element Methods for ... · Viscoelastic Fluid Flow Analysis Frank P.T. Baaijens, Martien A. Hulsen and Patrick D. Anderson in Encyclopedia Of Computational