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A three-field formulation for incompressible viscoelastic fluids

JaeHyuk Kwack, Arif Masud ⇑Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

a r t i c l e i n f o

Article history:Received 9 April 2010Accepted 3 September 2010Available online 27 October 2010

Keywords:Viscoelastic fluidsOldroyd-B modelVariational multiscale methodsStabilized methodsMixed finite elementsThree-field formulationEqual-order elements

a b s t r a c t

This paper presents a new stabilized finite element method for incompressible viscoelasticfluids. A three-field formulation is developed wherein Oldroyd-B model is coupled with themass and momentum conservation equations for an incompressible viscous fluid. The var-iational multiscale (VMS) framework is employed to develop a stabilized formulation forthe coupled momentum, continuity and stress equations. Based on the new stabilizedmethod a family of linear and higher-order triangle and quadrilateral elements withequal-order velocity–pressure–stress fields is developed. Stability and convergence prop-erty of the various elements is studied and optimal rates are attained in the norms consid-ered. The method is applied to some benchmark problems and accuracy and computationaleconomy of the formulation is investigated for various flow conditions.

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1. Introduction

Viscoelastic behavior of fluids is a dominant feature in the flow of polymeric fluids, and in injection molding flows inindustrial applications. Biological fluids like blood and plasma also show a significant viscoelastic and shear-rate dependentresponse [1,2]. From the viewpoint of their composition these fluids are classified as complex fluids as they possess intricatemicrostructures. For example, blood is a mixture of a variety of constituents, namely, red and white blood cells, platelets,proteins, and it is the elastic deformations of the red blood cells that contribute to its viscoelastic behavior [3]. Invariablyhomogenization techniques are applied to the microstructures of these fluids, resulting in continuum models with eithera differential or an integral form of additional constitutive equations with one or more conformation tensors. Mathematicalmodeling of incompressible viscoelastic fluids therefore involves conservation equations for mass and momentum, coupledwith constitutive equations for the viscoelastic stress. A variety of viscoelastic constitutive models have been proposed in theliterature and a good review on the topic can be found in [4]. A series of important theoretical developments of models fornon-Newtonian fluids have been proposed by Rajagopal and coworkers [4–8].

Development of mixed finite element methods for numerical simulation of viscoelastic fluid flows has also been an activearea of research [9–16]. Relatively recently there has been a surge in the application of computational fluid dynamics (CFD)techniques to the modeling of cardio-vascular blood flow [17–24]. Among the various numerical techniques proposed forthe modeling of viscoelastic behavior are the Elastic Viscous Stress Splitting (EVSS) method [25] and the Discrete Elastic ViscousStress Splitting (DEVSS) method [12], the Explicitly Elliptic Momentum Equation (EEME) method [26], the sub-element method[27], and the Discontinuous Galerkin (DG) method [11]. Since the standard finite element methods for incompressible flowssuffer from many drawbacks, therefore Streamline Upwind/Petrov–Galerkin (SUPG) [12] and the Galerkin Least-Square

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⇑ Corresponding author. Tel.: +1 217 244 2832; fax: +1 217 265 8039.E-mail address: [email protected] (A. Masud).

International Journal of Engineering Science 48 (2010) 1413–1432

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International Journal of Engineering Science

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(GLS) [10] methods have been developed that yield stable results for a range of Weissenberg number flows. The fundamentalidea in stabilized methods is to strengthen the underlying variational structure of the problem so that discrete approximationsremain stable and convergent for arbitrary combinations of interpolation functions. A general discussion on stabilized and mul-tiscale methods is presented in the works of Hughes and coworkers [28–31]. Interested reader is directed to Masud and Franca[32] for recent developments in hierarchical multiscale methods, and to Masud and Scovazzi [33] for new developments in het-erogeneous multiscale methods.

This work is an extension of our earlier works on VMS formulations for the Navier–Stokes equations [34,35] and theshear-rate dependent fluids [36], to non-Newtonian viscoelastic fluids. Since computational cost in the modeling of visco-elastic fluids increases with the introduction of additional unknown fields in the formulation, one consideration in the devel-opment of present method has been to keep the number of unknown fields to a minimum. This translates into minimumnodal degrees of freedom at the element level, thereby substantially increasing the computational efficiency. Specifically,we have endeavored to develop a three-field formulation and not introduce the rate-of-deformation as an additional field.The weak forms of the momentum, continuity and constitutive equations are cast in the VMS framework that leads to a two-level description of the problem. Consistent linearization of the fine-scale problem is performed with respect to the fine-scale field. Using bubble functions to expand the fine-scale trial and test functions and solving for the fine-scale coefficientsleads to an expression for the fine-scale field together with a definition of the stabilization tensor. The ensuing nonlinearstabilized form for the mixed velocity–pressure–stress formulation is presented and the consistent tangent is derived. A sig-nificant attribute of the proposed stabilized method is that it is computationally economic as compared to the competingmethods that invariably employ four-field formulations wherein rate-of-deformation tensor is the fourth field.

An outline of the paper is as follows: Section 2 presents the Oldroyd-B model for viscoelastic fluids. Section 3 presents thesystem of partial differential equations that comprises the incompressible generalized Navier–Stokes equations and the vis-coelastic constitutive equation. The VMS framework is described in Section 4 and the three-field velocity–pressure–stressformulation is derived. Numerical results are presented in Section 5, and conclusions are drawn in Section 6.

2. The stress–strain relation for the Oldroyd-B model

This section presents constitutive equations for the Oldroyd-B model [19]. The stress tensor R for the viscous fluid is as-sumed decomposed into the following two parts.

R ¼ �pI þ r; ð1Þ

where r is the viscoelastic stress tensor and p is the Lagrange multiplier. The viscoelastic stress follows the stress relaxationprocess inherent in the Oldroyd-B model and is given by the following equation.

rþ krpr¼ 2g eðvÞ þ ks eðvÞ

r� �; ð2Þ

where v is the velocity field, e (v) = 1/2(rv + (rv)T) is the strain rate tensor, k is the stress relaxation time, ks(=kgs/g) is the

retardation time, g is viscosity, gs is solvent viscosity and ð�Þr

is the upper-convected time-derivative defined as

ð�Þr¼ @ð

�Þ@tþ v � rð�Þ � ð�Þ � rv � ðrvÞT � ð�Þ: ð3Þ

The viscoelastic stress tensor can be decomposed into the non-Newtonian stress tensor rp and the Newtonian stress tensor rs.

r ¼ rp þ rs: ð4Þ

The corresponding constitutive relations are:

rp þ krp

r¼ 2ð1� bÞgeðvÞ; rs ¼ 2bgeðvÞ; ð5-a;bÞ

where b(=gs/g) is ratio of solvent viscosity gs and total viscosity g.

Remark 1. This constitutive model is compatible with the Upper-Convected Maxwell model [19] for b = 0. It reduces to theNewtonian model for b = 1. Likewise, for k = 0 and b – 1, equation (5-a) degenerates to the Newtonian model.

3. The incompressible generalized Navier–Stokes equations for viscoelastic fluids

3.1. The strong form

Let X � Rnsd be an open bounded region with piecewise smooth boundary C. The number of space dimensions, nsd is equalto 2 or 3. Strong forms of the governing equations for an incompressible viscoelastic fluid are given by the generalized Na-vier–Stokes equations, augmented by the non-Newtonian constitutive model represented by the nonlinear PDE (5-a).

1414 J. Kwack, A. Masud / International Journal of Engineering Science 48 (2010) 1413–1432


qv ;t þ qv � rv � 2bgr � eðvÞ þ rp�r � rp ¼qf in X��0; T½; ð6Þr � v ¼0 in X��0; T½; ð7Þ

krp;t þ rp þ kv � rrp � krp � rv � kðrvÞT � rp � 2ð1� bÞgeðvÞ ¼0 in X��0; T½; ð8Þv ¼gv on Cg��0; T½; ð9Þrp ¼gr on Cg��0; T½; ð10Þ

R � n ¼ ðrp þ 2bgeðvÞ � pIÞ � n ¼h on Ch��0; T½; ð11Þvðx;0Þ ¼v0 on X� f0g; ð12Þrpðx;0Þ ¼rp 0 on X� f0g; ð13Þ

where v ,t is the time derivative of the velocity field v , p is the Lagrange multiplier, rp,t is the time derivative of the additionalviscoelastic stress tensor rp, q is the density, f is the body force vector, gv represents the prescribed boundary velocities, gr

represents the prescribed inflow viscoelastic stresses, h is the vector of prescribed boundary tractions, v0 is the prescribedinitial velocity conditions, rp0 represents the prescribed initial viscoelastic stress conditions, n is the unit normal to theboundary C, and I is the identity tensor. Eqs. (6)–(13) represent balance of momentum, the continuity equation, the visco-elastic constitutive equation, the Dirichlet and Neumann boundary conditions, and the initial conditions, respectively.

3.2. The standard weak form

The appropriate spaces of weighting functions for the velocity, pressure and viscoelastic stress fields are:

V ¼ fwjw 2 ðH1ðXÞÞnsd ; w ¼ 0 on Cgg; ð14-aÞQ ¼ fqjq 2 L2ðXÞg; ð14-bÞ

X ¼ fwjw 2 ðH1ðXÞÞnsd�nsd ; w ¼ 0 on Cgg: ð14-cÞ

The appropriate spaces of trial solutions S for velocity, P for pressure, and T for viscoelastic stress are time-dependentspaces corresponding to V, Q and X , respectively.

S ¼ fvð�; tÞjvð�; tÞ 2 ðH1ðXÞÞnsd ; vð�; tÞ ¼ gv on Cg��0; T½g; ð15-aÞP ¼ fpð�; tÞjpð�; tÞ 2 L2ðXÞg; ð15-bÞ

T ¼ frpð�; tÞjrpð�; tÞ 2 ðH1ðXÞÞnsd�nsd ; rpð�; tÞ ¼ gr on Cg��0; T½g: ð15-cÞ

The standard weak form is: Find V ¼ fv; p;rpg 2 S � P � T such that 8W ¼ fw; q;wg 2 V �Q� X ,

qðw;v ;tÞ þ qðw;v � rvÞ þ 2bgðrw; eðvÞÞ � ðr �w;pÞ þ ðrw;rpÞ ¼ qðw; f Þ þ ðw;hÞCh; ð16Þ

ðq;r � vÞ ¼ 0; ð17Þ

ðw; krp;tÞ þ ðw;rpÞ þ ðw; kv � rrpÞ � ðw; krp � rvÞ � ðw; kðrvÞT � rpÞ � 2ð1� bÞgðw; eðvÞÞ ¼ 0; ð18Þ

where ð�; �Þ ¼R

Xð�ÞdX is the L2(X) – inner product.

Remark 2. Developing mixed finite elements for the velocity–pressure–stress fields that yield stable results for arbitrarycombinations of interpolation functions has been a formidable task [10–13,15,16,25–27].

4. The variational multiscale (VMS) stabilized method

The bounded domain X is considered discretized into non-overlapping sub-regions Xe (element sub-domains) withboundaries Ce, e = 1,2 . . . numel: X ¼

Snumele¼1 Xe. The union of element interiors and element boundaries is indicated as X0

and C0 respectively, and defined as X0 ¼Snumel

e¼1 ðintÞXe, and C0 ¼Snumel

e¼1 Ce.We assume an overlapping sum decomposition of the velocity field into coarse- or resolvable-scales and fine- or subgrid-

scales.

vðx; tÞ ¼ �vðx; tÞ|fflfflffl{zfflfflffl}coarse scale

þv 0ðx; tÞ|fflfflfflffl{zfflfflfflffl}fine scale

: ð19Þ

We assume that v 0 is represented by piecewise polynomials of sufficiently high order, continuous in space but discontinuousin time. In particular, v 0 is assumed to be composed of piecewise constant-in-time functions leading to vðx; tÞ ¼�vðx; tÞ þ v 0tðxÞ. Consequently, v ;t ¼ �v ;t and v 0;t ¼ 0.

Likewise, we assume an overlapping sum decomposition of the weighting functions into coarse- and fine-scale compo-nents indicated as �w and w0, respectively.

J. Kwack, A. Masud / International Journal of Engineering Science 48 (2010) 1413–1432 1415


wðxÞ ¼ �wðxÞ|ffl{zffl}coarse scale

þ w0ðxÞ|fflffl{zfflffl}fine scale

: ð20Þ

4.1. The variational multiscale form

Substituting the additively decomposed form of the velocity field and the weighting functions in Eqs. (16)–(18) leads tothe following set of equations.

qð �wþw0; �v ;tÞ þ qð �wþw0; ð�v þ v 0Þ � rð�v þ v 0ÞÞ þ 2bgðrð �wþw0Þ; eð�v þ v 0ÞÞ � ðr � ð �wþw0Þ;pÞþ ðrð �wþw0Þ;rpÞ ¼ qð �wþw0; f Þ þ ð �wþw0;hÞCh

; ð21Þ

ðq;r � ð�v þ v 0ÞÞ ¼ 0; ð22Þ

ðw; krp;tÞ þ ðw;rpÞ þ ðw; kð�v þ v 0Þ � rrpÞ � ðw; krp � rð�v þ v 0ÞÞ � ðw; kðrð�v þ v 0ÞÞT � rpÞ � 2ð1� bÞgðw; eð�v þ v 0ÞÞ ¼ 0:

ð23Þ

Because of the overlapping additive decomposition of the velocity field, the rate of deformation tensor can be written interms of its coarse- and fine-scale components as follows.

eð�v þ v 0Þ ¼ eð�vÞ þ eðv 0Þ ¼ 12ðr�v þ ðr�vÞTÞ þ 1

2ðrv 0 þ ðrv 0ÞTÞ: ð24Þ

4.2. Coarse-scale sub-problem

Employing the linearity of the weighting function slot we can split (21)–(23) into coarse-scale and fine-scale sub-prob-lems that can be written in a residual form as follows:

R1ð �w; �v ;v 0;p;rpÞ ¼def qð �w; �v ;tÞ þ qð �w; ð�v þ v 0Þ � rð�v þ v 0ÞÞ þ 2bgðr �w; eð�v þ v 0ÞÞ � ðr � �w; pÞ þ ðr �w;rpÞ

� qð �w; f Þ � ð �w;hÞCh¼ 0; ð25Þ

R2ðq; �v ;v 0Þ ¼defðq;r � ð�v þ v 0ÞÞ ¼ 0; ð26Þ

R3ðw; �v ;v 0;rpÞ ¼defðw; krp;tÞ þ ðw;rpÞ þ ðw; kð�v þ v 0Þ � rrpÞ � ðw; krp � rð�v þ v 0ÞÞ � ðw; kðrð�v þ v 0ÞÞT � rpÞ

� 2ð1� bÞgðw; eð�v þ v 0ÞÞ ¼ 0: ð27Þ

Eqs. (25)–(27) represent the weak forms of the balance of momentum, the continuity equation, and the constitutive equationfor the coarse scales, respectively.

4.3. Fine-scale sub-problem

The weak form of the balance of momentum equation for the fine-scales is:

R4ðw0; �v ;v 0;p;rpÞ ¼def qðw0; �v ;tÞ þ qðw0; ð�v þ v 0Þ � rð�v þ v 0ÞÞ þ 2bgðrw0; eð�v þ v 0ÞÞ � ðr �w0;pÞ þ ðrw0;rpÞ

� qðw0; f Þ � ðw0;hÞCh¼ 0: ð28Þ

4.4. Linearization with respect to the fine scale velocity field

Both coarse- and fine-scale sub-problems are nonlinear with respect to the coarse-scale velocity �v and fine-scale velocityv 0. We linearize these equations only with respect to the fine-scale velocity field. The linearization operators are defined asfollows:

LðR1ð �w; �v ;v 0;p;rpÞÞ ¼def d

deR1ð �w; �v;v 0 þ edv 0;p;rpÞ

��e¼0; ð29-aÞ

LðR2ðq; �v ;v 0ÞÞ ¼def dde

R2ðq; �v ;v 0 þ edv 0Þ��e¼0; ð29-bÞ

LðR3ðw; �v ;v 0;p;rpÞÞ ¼def d

deR3ðw; �v ;v 0 þ edv 0; p;rpÞ

��e¼0; ð29-cÞ

LðR4ðw0; �v ;v 0; p;rpÞÞ ¼def d

deR4ðw0; �v ;v 0 þ edv 0;p;rpÞ

��e¼0; ð29-dÞ



Applying the linearization operators (29-a),(29-b),(29-c),(29-d) to the weak forms for momentum, continuity and constitu-tive equations, we get the following linearized equations.

qð �w; dv 0 � rð�v þ v 0Þ þ ð�v þ v 0Þ � rdv 0Þ þ ðr �w;2bgeðdv 0ÞÞ ¼ � R1ð �w; �v;v 0;p;rpÞ; ð30Þ

ðq;r � dv 0Þ ¼ � R2ðq; �v;v 0Þ; ð31Þ

ðw; kðdv 0 � rrp � rp � rdv 0 � ðrdv 0ÞT � rpÞ � 2ð1� bÞgeðdv 0ÞÞ ¼ � R3ðw; �v ;v 0;rpÞ; ð32Þ

qðw0; dv 0 � rð�v þ v 0Þ þ ð�v þ v 0Þ � rdv 0Þ þ ðrw0;2bgeðdv 0ÞÞ ¼ � R4ðw0; �v ;v 0;p;rpÞ: ð33Þ

4.5. Solution of the fine-scale sub-problem

We rearrange (33) by keeping the dv 0 terms on the left hand side and taking all the other terms onto the right hand side.

qðw0; dv 0 � rvÞ þ qðw0;v � rdv 0Þ þ ðrw0;2bgeðdv 0ÞÞ ¼ ðw0; rÞ; ð34Þ

where v ¼ �v þ v 0, and r ¼ ð�q�v ;t � qv � rv þ 2bgr � eðvÞ � rpþr � rp þ qf Þ.

4.5.1. Use of bubble functions to extract the fine scale solutionWe expand the fine scale weighting and test functions via bubble functions be(n).

w0 ¼ bec; dv 0 ¼ bedf; ð35-a;bÞ

where c and df are the coefficients for the weighting and the test functions, respectively. We consider the first three terms onthe left hand side of (34), and expand each term via bubble-functions.

First term:

qðw0; dv 0 � rvÞ ¼ qZ

w0 � ðdv 0 � rvÞdX ¼ c � qZðbeÞ2ðrvÞT dX � df: ð36Þ

Second term:

qðw0;v � rðdv 0ÞÞ ¼ qZ

w0 � ðv � rðdv 0ÞÞdX ¼ c � qZ

bev � rbedXI � df; ð37Þ

where I is the identity tensor.Third term:

ðrw0;2bgeðdv 0ÞÞ ¼Zrw0 : bgðrdv 0 þ ðrdv 0ÞTÞdX ¼ c �

Zbgðrbe �rbeÞdXdfþ c �

Zbgjrbej2dXI � df: ð38Þ

We substitute (36)–(38) in Eq. (34) and solve for the coefficients of the fine-scale velocity field df. These fine scale coeffi-cients are then used to reconstruct the fine-scale solution via Eq. (35-b).

dv 0 ¼ be qRðbeÞ2ðrvÞT dXþ q

Rbev � rbedXI þ

Rbgðrbe �rbeÞdXþ

Rbgjrbej2dXI

h i�1Z

XeberdX: ð39Þ

If we assume the residual vector r to be piecewise constant over the element interior (i.e., mean value of the residual r), wecan simplify the representation of the fine-scale velocity field in terms of the stabilization tensor s.

dv 0 ¼ sr: ð40Þ

Explicit definition of the stabilization tensor that emanates from the derivation presented above is as follows.

s ¼ be qRðbeÞ2ðrvÞT dXþ q

Rbev � rbedXI

þR

bgðrbe �rbeÞdXþR

bgjrbej2dXI

" #�1 ZXe

bedX: ð41Þ

Remark 3. In order to numerically evaluate (41), ideas from our earlier works in [34] and [35] can be employed.

4.6. The resulting coarse-scale sub-problem

We now substitute the fine scale velocity field derived in Section 4.5 into the coarse-scale sub-problem. We rearrange Eqs.(30)–(32), and substitute dv 0 from (40) into the rearranged equations. The modified momentum balance, continuity and vis-coelastic constitutive equations are written as follow.

�ððqð�rv � �wþ ðr � vÞ �wþ v � r �wÞ þ bgðrðr � �wÞ þ D �wÞÞ; srÞ ¼ �R1ð �w; �v ;v 0; p;rpÞ; ð42Þ



�ðrq; srÞ ¼ �R2ðq; �v;v 0Þ; ð43Þ

krrp : wþ krp : ðrwT þrwÞ þ kw � ðr � rpÞþkðr � rpÞ � wþ ð1� bÞgðr � wT þr � wÞ

!; sr

!¼ �R3ðw; �v ;v 0;rpÞ: ð44Þ

4.7. The nonlinear stabilized form

The mixed nonlinear VMS stabilized form for the incompressible viscoelastic fluids is derived from Eqs. (42)–(44).

qð �w; �v ;tÞ þ qð �w;v � rvÞ þ ðr �w;2bgeðvÞÞ � ðr � �w;pÞ þ ðr �w;rpÞ þ ðq;r � vÞ

þ w;krp;t þ rp þ kv � rrp � krp � rv�kðrvÞT � rp � 2ð1� bÞgeðvÞ

! !� ðv1 þ v2; srÞ ¼ qð �w; f Þ þ ð �w;hÞCh

; ð45Þ

where the weighting functions for stabilization term are defined as

v1 ¼ qð�rv � �wþ ðr � vÞ �wþ v � r �wÞ þ bgðrðr � �wÞ þ D �wÞ þ rq; ð46-aÞ

v2 ¼ �krrp : wþ krp : ðrwT þrwÞ þ kw � ðr � rpÞþkðr � rpÞ � wþ ð1� bÞgðr � wT þr � wÞ

!: ð46-bÞ

Remark 4. Eq. (46-a) provides stabilization of the momentum and continuity equations as was presented in our earlierworks [34–36].

Remark 5. It is important to note that the stabilization of the additional stress equation provided by (46-b) is facilitated bythe VMS based additive decomposition of the velocity field. This additional stabilization helps in reaching higher Weiss-enberg numbers without the need of introducing the rate of deformation as an additional unknown field, as is typically donein the EVSS and DEVSS methods [9,10,12,13,15,16,25,27].

4.8. SUPG stabilization of the constitutive equation

The viscoelastic constitutive equation can be considered as an advection–reaction system. We can write the steady-statecounterpart of (8) as:

F 1ðvÞ � rp þF 2ðvÞ � rrp ¼ GðvÞ; ð47Þ

where F 1ðvÞ 1� 2keðvÞ;F 2ðvÞ ¼ kv and GðvÞ ¼ 2ð1� bÞgeðvÞ. Note that in terms of the shear-rate _c the Weissenbergnumber is defined as Wi ¼ k _c, while in terms of the characteristic velocity vc it is defined as Wi = kvc/lc, where lc is the char-acteristic length. Therefore, the second term ke(v) in F 1 and the term kv in F 2 can be considered as increasing functions ofWi. Accordingly, at low Weissenberg number flows the reaction term dominates the constitutive Eq. (47). However, as theWeissenberg number is increased, the advection term starts dominating, thereby adversely affecting the numerical stabilityof the formulation. In order to augment the stability of the constitutive equation for high Weissenberg number flows, weemploy ideas from Streamline Upwind/Petrov–Galerkin (SUPG) methods and develop a SUPG type stabilization of the ten-sorial field rp [15,25,27]. As a result, an additional weighting function srv � rw is introduced in the stabilized residual formof the viscoelastic constitutive Eq. (44) as follows. The stabilization parameter sr is defined as sr = he/jv j, where he is thecharacteristic size of an element and jv j is the norm of the element velocity.

krrp : wþ krp : ðrwT þrwÞ þ kw � ðr � rpÞþkðr � rpÞ � wþ ð1� bÞgðr � wT þr � wÞ

!; sr

!¼ �R3ðwþ srv � rw; �v ;v 0;rpÞ: ð48Þ

Remark 6. Due to the zero initial conditions, the norm of the element velocity jv j is invariably zero at time t0. Thereforestabilization parameter sr is set equal to zero whenever jv j approaches zero.

The appropriate spaces for the pressure trial solutions and weighting functions for the stabilized form (45) are:eP ¼ fpð�; tÞjpð�; tÞ 2 L2ðXÞ;rp 2 ðL2ðXÞÞnsdg; ð49-aÞeQ ¼ fqjq 2 L2ðXÞ;rq 2 ðL2ðXÞÞnsdg: ð49-bÞ



Let Z ¼ S � ~P � T and Y ¼ V � eQ � X . Find V ¼ fv ; p;rpg 2 Z such that, for all W ¼ fw; q;wg 2 Y,

BstabðW ;VÞ ¼ LstabðWÞ; ð50Þ

where B(�,�) is linear with respect to the weighting function slot and is nonlinear with respect to the trial solution slot.

BstabðW ;VÞ ¼ BðW ;VÞ þ ðv1 þ v2; sðq�v ;t þ qv � rv � 2bgr � eðvÞ þ rpÞÞ

þ srv � rw;krp;t þ rp þ kv � rrp � krp � rv�kðrvÞT � rp � 2ð1� bÞgeðvÞ

! !; ð51Þ

LstabðWÞ ¼ LðWÞ þ ðv1 þ v2; sðqf ÞÞ ð52Þ

and

BðW;VÞ ¼ qð �w; �v ;tÞ þ qð �w;v � rvÞ þ ðr �w;2bgeðvÞÞ � ðr � �w; pÞ þ ðq;r � vÞ þ ðr �w;rpÞ þ kðw;rp;tÞ

þ w;rp þ kv � rrp � krp � rv � kðrvÞT � rp � 2ð1� bÞgeðvÞ� �

; ð53Þ

LðWÞ ¼ qð �w; f Þ þ ð �w;hÞCh: ð54Þ

In (51)–(54) Bstab(W,V) and Lstab (W) are the operators for the nonlinear stabilized form, and B(W,V) and L(W) are the oper-ators for the underlying Galerkin form as presented in (16)–(18).

5. Numerical results

This section presents a series of numerical tests with the proposed formulation. Fig. 1 shows a family of elements withequal-order velocity–pressure–stress fields. For numerical evaluation of the stability parameter s given in (41), quadraticbubbles are employed for the linear triangles and bilinear quadrilaterals, while quartic bubbles are used for the higher-orderelements. Standard Gauss quadrature rules are employed for numerical integration in all the cases. In the numerical imple-mentation of the Newton–Raphson method, nonlinear iterations are carried out on the coarse-scales while the fine-scales aretreated as linear during the iterations for the coarse-scales.

5.1. Rate of convergence study for the Newtonian fluid with additional Newtonian stress

In this section we consider a reduced order model for the present formulation wherein the total viscous stress is split intotwo parts: viscous stress in the momentum balance equation and viscous stress in the additional constitutive equation. Theadditional viscous stress is restricted to be Newtonian by setting k = 0. This annihilates the convective part of the viscousstress in the additional stress equation. Accordingly, we set sr equal to zero in the stabilized form (51). Although the modelis Newtonian, however due to the presence of viscous stress in the additional constitutive equation, the weak form is not thatof the standard Newtonain fluids which is given by the standard Navier–Stokes equations. Numerical convergence estimatesare available for the pressure, velocity and additional stress fields in terms of the order of the interpolation polynomials em-ployed. We perform rate of convergence study to numerically check the mathematical consistency and stability of the pro-posed three-field formulation. In these tests b = 0.01 and therefore a significant portion of viscous stress is modeled by theadditional constitutive equation.

Fig. 1. Family of equal-order linear and quadratic elements.



The exact velocity, pressure and stress fields are:

v ¼ ½vx vy �T ¼ ye5ðx2þy2Þ �xe5ðx2þy2Þ� T

; ð55-aÞ

p ¼ e5=4 sinð2pxÞ sinð2pyÞ; ð55-bÞ

rp ¼ 2ð1� bÞge5ðx2þy2Þ 10xy 5ð�x2 þ y2Þ5ð�x2 þ y2Þ �10xy

" #: ð55-cÞ

It is important to note that the assumed velocity field satisfies the incompressibility condition given by Eq. (7). Substituting(55-a)–(55-b) for the velocity and pressure fields in the governing Eq. (6) yields the body force vector which is then em-ployed to run the discrete problem (see Masud and Kwack [36] for details).

Fig. 2. Structured meshes for convergence rate test.

Fig. 3-a. Equal-order linear triangles.



We are dealing with a nonlinear formulation wherein nonlinearity is engendered by advection terms in momentum bal-ance as well as in the additional stress equation. Therefore we use an iterative solution procedure to get a converged solutionon a given mesh. Acceptable tolerance in the Newton–Raphson convergence check is set equal to 1.0E�16. Once the con-verged solution is attained, the error norms of the computed solution with respect to the exact solution are computed.We report the convergence rates in terms of the L2-norm of the velocity field and H1-norm of the pressure field. Also reportedare the rates for L2div�v , L2�p and L2�rp norms.

Fig. 2 shows structured triangular and quadrilateral meshes. In each case subsequent meshes are designed such that thecoarser discretization is fully embedded in the refined discretization. Triangular meshes are generated by bisecting the quad-rilateral meshes such that the number of degrees of freedom is same between the two mesh types. Figs. 3-a, 3-b, 3-c, 3-dshow the rates for linear triangles, bilinear quadrilaterals, quadratic triangles and biquadratic quadrilaterals, respectively.In all the cases optimal convergence rates are attained in the norms considered.

Fig. 3-b. Equal-order bilinear quadrilaterals.

Fig. 3-c. Equal-order quadratic triangles.

Fig. 3-d. Equal-order biquadratic quadrilaterals.



5.2. Validation for the additional Newtonian viscous stress

Another check of the consistency and stability of the three-field formulation is provided by the driven cavity problem. Tosuppress viscoelastic effects the relaxation parameter k = 0 and therefore sr is set equal to zero in the stabilized form (51).

Fig. 4. Velocity contours and streamlines at Re = 10,000.

Fig. 5-a. Horizontal velocity along a vertical line passing through the center. (Re = 1000).

Fig. 5-b. Horizontal velocity along a vertical line passing through the center. (Re = 5000).



Though the fluid is Newtonian, the underlying formulation contains the additional stress equation with the viscosity ratiob = 0.5. Computational grid is composed of 80 � 80 bilinear quadrilateral elements.

Fig. 4 shows the resultant velocity contours and streamlines for Re = 10,000. Figs. 5-a, 5-b, 5-c shows the horizontal veloc-ity for Re = 1000, 5000 and 10000, respectively, along a vertical line passing through the center the cavity. For all the testcases a good correlation with published results of Ghia et al. [37] is attained.

Fig. 5-c. Horizontal velocity along a vertical line passing through the center. (Re = 10,000).

Fig. 6. Schematic diagram of the problem.

Table 1The characteristics of the meshes for the narrow channel.

Mesh w Lu Ld Polar mesh size, hp Number of elements Number of unknowns

M1 2 �15 15 p/80 2995 19314M2 2 �15 15 p/160 7500 47232M3 2 �15 15 p/320 18505 114912

Fig. 7. Configuration of the generic mesh M1.



5.3. Flow past a cylinder in a channel

Flow past a circular cylinder in a channel is a popular and standard benchmark problem to validate computational methodsfor viscoelastic fluids [10,13,15,16]. Fig. 6 shows the schematic diagram of the problem description. A fully developed parabolicvelocity profile is applied at the inflow, while traction free boundary conditions are imposed at the outflow. No-slip boundaryconditions are imposed on the surface of the cylinder and the channel walls and symmetry boundary conditions are appliedalong y = 0 edge. The inflow and the symmetric boundary conditions for the extra stress are prescribed as follows.

vx ¼ 32 Vm 1� y2

w2

� �; vy ¼ 0 at x ¼ Lu;

rpxx ¼ 2ð1� bÞ gk3Vmky

w2

�2 rpxy ¼ �3ð1� bÞgVmy

w2 ; rpyy ¼ 0 at x ¼ Lu;

rpxy ¼ 0 at y ¼ 0;

Fig. 8. Drag force on the cylinder versus Wi in the narrow channel (w/Rc = 2.0).

Fig. 9. Viscoelastic stress contours at Wi = 0.85 in the narrow channel, (a)rpxx, (b)rpyy, (c)rpxy.



where Vm is the mean value of the inflow velocity, and w is the width of the channel. The dimensionless Weissenberg numberis defined as Wi = kVm/Rc where Rc is the radius of the cylinder. The drag force Fd on the cylinder is defined as:

Fd ¼Z

Ccylinder

2ðrp þ 2bgeðvÞ � pIÞ : nex ds;

where ex is the unit vector in x-direction.

Fig. 10. Line plots of the viscoelastic stress components at x = 2.0 for Wi = 0.5–0.8 in the narrow channel (a)rpxx, (b)rpyy, (c)rpxy.



5.3.1. Cylinder in a narrow channel, w/Rc = 2.0The parameters of the three finite element meshes used in the narrow channel problem are presented in Table 1. Various

constants are: Vm = 1.0, Rc = 1.0 and b = 0.59. Fig. 7 shows spatial discretization of one of the meshes M1 with polar mesh sizehp = p/80. In order to capture the high gradients in rpxx, downstream mesh size at the tail of the cylinder is reduced to 1

4 hp.Fig. 8 shows the drag force on the cylinder and comparison is made with published results from the literature [10,13]. The

horizontal axis represents the Weissenberg number and the vertical axis represents the drag force. Coronado et al. [10] havepresented results for a range in Wi up to 0.7 with their four-field GLS method, and Hulsen et al. [13] have reported stablesolution up to Wi = 2.0 with a logarithmic conformation martix. Our results for meshes M2 and M3 match very well withHulsen et al. [13]. It is important to note that in all the meshes we can reach up to Wi = 0.85 without introducing the addi-tional rate of deformation tensor as an unknown field. Fig. 9 shows the viscoelastic stress contours near the cylinder atWi = 0.85. As reported, rpxx contour plot shows steep gradients at the perimeter of the cylinder as well as downstream fromthe cylinder. In contours for rpyy and rpxy, high gradients in viscoelastic stresses are observed along the circumference of thecylinder.

Fig. 10 presents line plots for viscoelastic stress at x = 2.0 for Wi = 0.5–0.8. The x-axis represents the viscoelastic stressesand the y-axis represents the y-coordinates through the depth of the channel. Also presented are the results from Coronadoet al. [10] obtained via their conformation tensor defined as M = k rp/(1�b)g + I. Once again a good comparison with pub-lished results is attained. Fig. 11 presents the rpxx plot along the symmetry line for Wi = 0.6–0.8. The horizontal axis repre-sents s which is defined as the perimeter of the cylinder on 0 < s < pRc and as points on the symmetry line (s = x + pRc � Rc)when s > pRc. The vertical axis represent rpxx. Computed results show good correlation with Hulsen et al. [13].

Fig. 11. Viscoelastic stress component rpxx along the symmetry line for Wi = 0.6–0.8.

Table 2Number of iterations at various steps for mesh M3.

Step 1 Step 2–16 Step 17

Wi 0.05 0.10–0.80 0.85No. of Iter. 5 4 5

Table 3Residual norms at various steps for mesh M3.

Iter. No Step 1 Step 2 Step 17

1 1.92E+03 2.17E+03 9.45E+032 3.52E�02 3.62E�02 1.70E�013 4.21E�08 1.78E�08 7.21E�064 1.07E�11 2.39E�14 1.05E�075 2.75E�15 7.54E�09



Table 2 presents the number of nonlinear iterations for the various steps for mesh M3. Step 1 corresponds to Wi = 0.05,and thereafter Wi increases by 0.05 at every quasi-Newton step. As can be seen in Table 2, four to five iterations are sufficientto attain convergence in any given step. Table 3 presents reduction in the residual norm at various steps for mesh M3. Thetangent tensor presented in the Appendix A yields quadratic convergence as shown in Table 3. Convergence tolerance for theresidual norm is 1.0E�16.

5.3.2. Cylinder in a wide channel, w/Rc = 8.0This test case investigates flow around a circular cylinder in a wide channel. Characteristics of the various meshes used

are presented in Table 4. Various constants are: Vm = 1.0, Rc = 1.0 and b = 0.59. Fig. 12 shows the coarsest mesh M4 with polarmesh size hp = p/40. Mesh downstream of the cylinder is refined such that element characteristic length is 1

4 hp.Fig. 13 represents the drag force on the cylinder for the wide channel case. Comparison is made with the four-field GLS

method presented in Coronado et al. [10], with results obtained with their finest mesh for Wi up to 2.4. Present study obtainsresults up to Wi = 2.45 with mesh M6 employing the three-field VMS stabilization formulation. Fig. 14 shows the viscoelasticstress contours around the cylinder at Wi = 2.45. The rpxx contour shows steep gradients at the perimeter of the cylinder aswell as downstream of the cylinder. In contours for rpyy and rpxy, high gradients in the viscoelastic stresses can be seen alongthe circumference of the cylinder.

Remark 7. The present method can produce same engineering accuracy as in reference [10], however with an order ofmagnitude reduction in the cost of computation.

Fig. 15 shows the viscoelastic stress plots at x = 4.0 for a range in Wi from 1.0 to 2.0. For Wi = 1.0, 1.5 and 2.0, our resultscompare very well with those reported in Coronado et al. [10]. Fig. 16 presents the rpxx component along the symmetry linefor Wi = 1.5, 2.0 and 2.45. As described in the previous case, the horizontal line represents the perimeter of the cylinder andthe downstream distance along the line of symmetry. Once again a good correlation with published results is attained.

Fig. 12. Configuration of the generic mesh M4.

Table 4The characteristics of the meshes for the wide channel

Mesh w Lu Ld Polar mesh size, hp Number of elements Number of unknowns

M4 8 �40 40 p/40 3954 25560M5 8 �40 40 p/80 6158 39258M6 8 �40 40 p/160 15101 94098



6. Conclusions

We have presented a new mixed finite element method for incompressible viscoelastic fluids. A three-field formulation isdeveloped that is based on the equations of balance of mass and momentum, together with the Oldroyd-B constitutive modelfor viscoelastic fluids. The multiscale stabilized form is derived via the variational multiscale (VMS) method. The notion ofconsistent linearization of the fine scale problem only with respect to the fine scale fields simplifies the sub-grid scale mod-eling of the problem. Embedding the fine scale solution into coarse scale weak form leads to the stabilized three-field veloc-ity–pressure–stress method. Convergence rate study for a Newtonian fluid is presented wherein, due to the presence of

Fig. 14. Viscoelastic stress contours at Wi = 2.45 in the wide channel. (a)rpxx, (b)rpyy, (c)rpxy.

Fig. 13. Drag force on the cylinder versus Wi in the wide channel (w/Rc = 8.0).



viscous stress in the reduced form of the additional constitutive equation, the weak form is based on the three-field varia-tional equation. Optimal convergence rates are attained in the norms considered, showing the consistency and stability ofthe underlying numerical method. A detailed study of flow past a circular cylinder in narrow and wide channels is carriedout. Comparisons with published results show high engineering accuracy on relatively cruder spatial discretizations, high-lighting computational economy for this important class of problems.

Fig. 15. Line plots of the viscoelastic stress components at x = 4.0 for Wi = 1.0–2.0 in the wide channel (a)rpxx, (b)rpyy, (c)rpxy.



Acknowledgements

Authors wish to thank Prof. K.R. Rajagopal and Dr. David Gartling for many helpful discussions. Partial support for thiswork was provided by the National Academies Grant NAS 7251-05-005. This support is gratefully acknowledged.

Appendix A. The consistent tangent tensor

The variational multiscale method is based on the notion of an overlapping sum decomposition of the velocity field. Thenumerical solution of the nonlinear stabilized formulation in (50) is obtained via an iterative procedure employing Newton–Raphson or a modified Newton procedure. To simplify the solution of the fine-scale problem in the Newton–Raphson iter-ations, we assume a linear approximation of the fine-scale velocity dv 0. This simplifying approximation leads to a definitionof the stabilization tensor s, and the need to update the fine-scale velocity field is suppressed. Only the coarse-scale solutioncomputed from the stabilized formulation is iterated upon in the Newton–Raphson scheme. Accordingly, the coarse-scalesolution represents the total solution where the fine-scale field is mathematically embedded in via the stabilization terms.To derive the consistent tangent we consider the nonlinear stabilized form (50) and rewrite it in terms of the coarse-scalefields.

qð �w; �v ;tÞ þ qð �w; �v � r�vÞ þ ðr �w;2bgeð�vÞÞ � ðr � �w;pÞ þ ðr �w;rpÞ þ ðq;r � �vÞ � ðv1; s � rÞþ ðwþ srv � rw; krp;t þ XÞ � ðv2; s � rÞ ¼ qð �w; f Þ þ ð �w;hÞCh

; ðA-1Þ

where

X ¼ rp þ k�v � rrp � krp � r�v � kðr�vÞT � rp � 2ð1� bÞgeð�vÞ; ðA-2Þr ¼ ð�q�v ;t � q�v � r�v þ 2bgr � eð�vÞ � rpþr � rp þ qf Þ ðA-3Þ

and v1 and v2 are defined by (46-a) and (46-b), respectively by setting v ¼ �v .The residual vector for the Newton–Raphson scheme based on the nonlinear stabilized form (A-1), is defined as

Rð �w;wþ srv � rw; �v ;p;rpÞ ¼ qð �w; �v ;tÞ þ qð �w; �v � r�vÞ þ ðr �w;2bgeð�vÞÞ � ðr � �w; pÞ þ ðr �w;rpÞ þ ðq;r � �vÞ� ðv1; srÞ � qð �w; f Þ � ð �w;hÞCh

þ ðwþ srv � rw; krp;t þ XÞ � ðv2; srÞ: ðA-4Þ

We linearize the nonlinear residual with respect to the coarse-scale fields. The linearization operator is defined as

LðRð �w;wþ srv � rw; �v ;p;rpÞÞ ¼def d

deRð �w;wþ srv � rw; �v þ ed�v ;pþ edp;rp þ edrpÞ

��e¼0: ðA-5Þ

Applying (A-5) to (A-4) leads to

qð �w; d�v � r�vÞ þ qð �w; �v � rd�vÞ þ ðr �w;2bgd�eÞ � ðr � �w; dpÞ þ ðr �w; drpÞ þ ðq;r � d�vÞ � ðv1; sdrÞþ ðwþ srv � rw; dXÞ � ðv2; sdrÞ ¼ �Rð �w;wþ srv � rw; �v; p;rpÞðiÞ; ðA-6Þ

where �v ¼ �v ðiÞ; �e ¼ eð�vÞ; d�e ¼ eðd�vÞ and (i) is an index for iteration. The linearized residuals d r and dX in the last term on lefthand side of (A-6) are defined as

Fig. 16. Viscoelastic stress component rpxx along the symmetry line for Wi = 1.5–2.45.



dr ¼ �qd�v � r�v � q�v � rðd�vÞ þ 2bgr � d�e�rdpþr � drp; ðA-7ÞdX ¼ drp þ kd�v � rrp þ k�v � rdrp � kdrp � r�v � krp � rd�v � kðrd�vÞT � rp � kðr�vÞT � drp � 2ð1� bÞgd�e: ðA-8Þ

We can now write the stabilized form with the consistent tangent tensor as

qð �w; d�v � r�vÞ þ qð �w; �v � rd�vÞ þ ðr �w;2bgd�eÞ � ðr � �w; dpÞ þ ðr �w; drpÞ þ ðq;r � d�vÞ � ðv1; sdrÞþ ðwþ srv � rw; dXÞ � ðv2; sdrÞ ¼ �R1ð �w; �v; p;rpÞðiÞ � R2ðq; �vÞðiÞ þ ðv1; srÞ � R3ðwþ srv � rw; �v ;rpÞðiÞ þ ðv2; srÞ;

ðA-9Þ

where the left hand side is the consistent tangent tensor written in terms of the incremental solution fields fd�v; dp; drpg. Theright hand side is the residual vector at (i)th iteration and is composed of three parts where �R1ð �w; �v ; p;rpÞðiÞ;�R2ðq; �vÞðiÞ;�R3ðwþ srv � rw; �v ;rpÞðiÞ; ðv1; srÞ and (v2,sr) are the residuals from the momentum balance Eq. (25), the conti-nuity Eq. (26) the viscoelastic constitutive Eq. (27) and the stabilization terms, respectively.

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