J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

A study of viscoelastic free surface flows by the finite elementmethod: Hele–Shaw and slot coating flows

Alex G. Leea,1, Eric S.G. Shaqfehb, Bamin Khomamic,∗a Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA

b Departments of Chemical and Mechanical Engineering, Stanford University, Stanford, CA 94305, USAc Department of Chemical Engineering, Washington University, St. Louis, MO 63130, USA

Abstract

A pseudo-solid domain mapping technique coupled with the DEVSS finite element formulation is applied tostudy the effects of viscoelasticity on free surface flows. Two distinct flow types are analyzed: the flow induced by along air bubble steadily displacing a polymeric liquid confined by two parallel plates, i.e. Hele–Shaw flow, and theslot coating of viscoelastic fluids in the low metering rate limit. The Oldroyd-B, FENE-CR, and FENE-P constitutiveequations are used to model the viscoelastic fluid. Our study reveals the formation of an elastic boundary layer inthe capillary-transition region near the bubble front at moderate Weissenberg numbers while the stress field in theparallel flow region remains largely unaffected by the dynamics of the free surface. Our calculations show that theincrease in the hydrodynamic coating thickness due to viscoelasticity often reported in planar injection experiments[Physica A 220 (1995) 60; J. Non-Newtonian Fluid Mech. 71 (1997) 73] is associated with the onset of these elasticboundary layers and is strongly dependent on the physical properties of the coating fluid.© 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Viscoelastic free surface flows have practical applications in coating technology, polymer processingand enhanced oil recovery[2,3]. Free surfaces in coating and injection flows are liquid/gas interfacesand their presence greatly complicate the flow fields because their positions are unknown a priori. ForNewtonian free surface flows the most important dimensionless parameter in analyzing the flow dynamicsis the Capillary number

Ca = ηU

T, (1)

∗ Corresponding author. Fax:+1-314-935-7211.E-mail address:bam@poly1.che.wustl.edu (B. Khomami).

1 Present address: Corporate Strategic Research, ExxonMobil Research & Engineering Co., P.O. Box 998/1545 Rte. 22E.,Annandale, NJ 08801-0998, USA.

0377-0257/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0377-0257(02)00137-4

328 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

Fig. 1. Schematic of planar injection flow.

whereη is the total viscosity of the liquid phase,T the surface tension, andU the displacement velocity.The Capillary number measures the relative importance of viscous to surface tension forces.

There are two prototype problems in free surface displacement flows. The first is the planar or axisym-metric injection coating (Fig. 1), where the fluid filling a narrow gap or tube is displaced by an advancingair bubble, and the residual coating thickness, hence the flow rate, is determined strictly by hydrody-namics. The second prototype is the slot coating problem (Fig. 2), where the flow rate is pre-meteredand the fluid is carried out of a narrow slot by a moving substrate. Extensive literature can be foundfor both problems but most of the studies conducted thus far have been restricted to Newtonian liquids.The problem of an air bubble steadily displacing a Newtonian fluid in a capillary tube was first studiedexperimentally by Fairbrother and Stubbs[4]. Taylor [5], who subsequently conducted experiments inthe range 10−3 < Ca < 1.9, found that the fractional coverage as a function of the Capillary number wasindependent of the tube diameter or the fluid viscosity. The fractional coverage, which is the ratio of thetube cross sectional area occupied by the film of fluid to the total cross sectional area, is defined as

m = R2o − R2

b

R2o

, (2)

whereRo is the tube radius/gap width (axisymmetric/planar injection) andRb is the radius of the pen-etrating bubble. Bretherton[6] conducted the first theoretical analysis of the problem and reached theconclusion that for small Capillary numbers the thickness of the uniform thin film scales asCa2/3. ForCagreater than 10−3, full solutions of the problem can only be achieved numerically. By using the finitedifference/conformal mapping technique, Reinelt and Saffman[7] obtained results which are in good

Fig. 2. Schematic of slot coating flow.

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 329

agreement with experiments for 10−2 ≤ Ca ≤ 2 for both planar and axisymmetric flow geometries. Luand Chang[8] used the boundary integral method to analyze the planar injection problem and obtainedsolutions forCaas small as 10−4. The asymptotic analysis conducted by Ruschak[9] for the planar casewhich provided aU1/2 scaling for film thickness in the 10−2 ≤ Ca ≤ 10−1 range is also worth mention-ing. Giavedoni and Saita[10], using the polar–spine finite element formulation advanced by Saito andScriven[11], obtained results for the range 10−5 ≤ Ca ≤ 10 which are in good agreement with the finitedifference solutions of Reinelt and Saffman[7] for both planar and axisymmetric flow geometries.

Turning to the literature on the displacement of non-Newtonian fluids, early studies on air bubble dis-placement of non-Newtonian liquids were conducted by Bonn et al.[1]. Using solutions of poly(ethyleneoxide) (PEO) and xanthane, they reported a film thickness increase over that found for Newtonian liquidsin the planar flow geometry. They attributed the cause of the film thickening observed for PEO solutionsto the compressive elastic normal stresses near the bubble tip. Using highly elastic, non-shear-thinning,polyisobutylene/polybutene (PIB/PB) based Boger fluids[16], Huzyak and Koelling[2] reported strongfilm thickening effects in the axisymmetric flow geometry. It is worth noting that a close examination oftheir data at lowWi reveals that the thickening effects, depending on one’s estimate of the experimentalaccuracy, is preceeded by a slight thinning of the film. The extensional thickening of the fluid viscositynear the bubble tip was identified as the probable cause for the significant deviation from coating filmthickness observed in the displacement of Newtonian liquids by an air bubble. Ro and Homsy[17] con-ducted the first theoretical study of free boundary creeping flows created by the air displacement of aviscoelastic fluid initially filling a gap confined by two narrowly separated solid walls, i.e. the Hele–Shawcell. The theory was formulated as a double perturbation expansion in powers ofCa1/3 andWi/Ca1/3,whereWi is the Weissenberg number. The Weissenberg number provides a measure of the elasticity ofthe flow by introducing the relaxation time of the polymer as the second time scale in the problem

Wi = λU

b, (3)

whereb is the gap separation between the plates. Ro and Homsy concluded that for small Capillarynumber and Weissenberg number flows, the fluid elasticity induces resistance to stream-wise strainingand therefore, reduces the coating film thickness.

Extensive studies in slot coating of viscous Newtonian fluids have been conducted using polar–spine[11], cartesian–spine[12], and elliptic mesh generation[13] finite element formulations and via asymptoticanalysis as well[14,15]. We restrict our slot coating analysis to the steady, two-dimensional, low flowrate regime, where a recirculation region is always present in the flow. In this paper we show that thepseudo-solid free surface discretization method can be used to solve the coupled viscoelastic free surfaceflow and predict accurately the highly bent, invading menisci in viscoelastic slot coating.

The paucity of full scale numerical simulations involving viscoelastic free surface flows has led tosometimes conflicting views about the role of elasticity on interfacial dynamics. Although a number ofphysical arguments have been advanced by previous researchers to explain some of the elastically inducedfree surface phenomena, in particular the coating film variation effect due to its importance in many indus-trial applications, no satisfactory analysis has been done to correlate the viscoelastic interfacial behaviorto the physical properties of the coating fluids. Our work is motivated by the desire to achieve a betterunderstanding of the physical processes involved in the free surface displacement of polymeric liquidsand thus, be able to develop more suitable theoretical treatment of multiphase flows. In the followingsection, we shall define the governing equations and give a summary of the numerical formulation for

330 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

the planar injection and slot coating flows. We shall present comparisons of our numerical simulation ofNewtonian flows to previous work as a benchmark for our simulations prior to showing the solutions forviscoelastic free surface flows. We will conclude by providing a brief discussion of instabilities encoun-tered in these flows that have been described previously by Grillet et al.[18] and suggest an instabilitymechanism based on our base state calculations.

2. Mathematical formulation

2.1. Governing equations

The two-dimensional, steady elastic free surface problems depicted inFigs. 1 and 2are defined by thegoverning equations which impose conservation of mass and momentum for an incompressible fluid inthese geometries:

∇ · u¯

= 0, (4)

Reu¯· ∇u

¯= −∇P

Ca+ ∇ · τ

¯, (5)

whereReis the Reynolds number and it is defined as

Re= ρUb

η, (6)

whereρ is the density of the fluid,η the total viscosity of the fluid andU the characteristic velocity ofthe problem. In this paper, only the creeping flow regime,Re= 0, is analyzed for the flow geometriesconsidered.u

¯is the velocity vector,P is the pressure andτ

¯is the total stress tensor formed by the sum of

the Newtonian solvent stress,τ¯s

, and the polymer stress,τ¯p

. The basic equations are non-dimensionalizedas

(x, y)[=](b, b), (u, v)[=]U, P [=]T

b, τ

¯[=]

ηU

b.

whereη is the total viscosity of the fluid at zero shear rate.To analyze the physics of polymer stretching in these flows as well as to determine the flow field

modification by the presence of polymers, three different constitutive models,—the Oldroyd-B, FENE-CR[19], and FENE-P[20]—were chosen for our simulations. This choice was made to meet the followingcriteria:

• to have sufficient mathematical simplicity for results to be interpreted physically;• to reflect the major viscometric behaviors that has been analyzed in previous experiments (both constant

viscosity Boger fluids[16,21]and shear thinning PEO solutions[1]);• to exhibit variable levels of maximum extensional stress attainable in flow.

The constitutive models chosen can be separated into two parts: an evolution equation for the elasticdumbbells (upon which all these models are based), and an expression for the polymer stress. For theFENE-CR model, the dumbbells are taken to evolve in steady flow according to:

u¯· ∇ C

¯= C

¯· ∇ u

¯+ ∇ u

¯T · C

¯− f (R)

Wi(C

¯− I

¯), (7)

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 331

in whichC¯, the polymer conformation tensor, represents an ensemble average of the dyadic productR

¯R¯of the dumbbell end-to-end vectorR

¯, andf (R) is the spring force law of the individual dumbbells. The

FENE-P model, the closed form of the FENE dumbbell evolution equation derived by using Peterlin’sapproximation[22], can be obtained by replacing the term−f (R)(C

¯−I

¯)/Wi in (7)by−(f (R)C

¯−I

¯)/Wi.

In both slow flows(Wi → 0) and for strong flows(C¯

>> 1), the evolution equations for both dumbbellmodels approach the same asymptotic limit. However, in a steady shear flow the latter model exhibitsshear thinning properties as the FENE model while the former remains non-shear-thinning. The springforce law,f (R), is the same for both FENE models and is taken to be

f (R) = 1

1 − (R2/L2), (8)

whereR2 is the trace of the polymer conformation tensorC¯

andL represents the ratio of the length of a fullyextended polymer dumbbell to its equilibrium length. As the value of the finite extensibility parameter,L, decreases, the polymer dumbbell becomes less extendible and as a consequence, the maximum levelof stress attainable in an purely extensional flow is reduced. The polymeric contribution to the total stressfor the Oldroyd-B and the FENE-CR models can be written as,

τ¯

= 2Sγ¯︸︷︷︸

τ¯s

+ (1 − S)

Wif (R)(C

¯− I

¯)︸ ︷︷ ︸

τ¯p

, (9)

where the parameterS is the ratio of the solvent viscosity to the total viscosity and it provides a measureof the polymer concentration in the fluid

S = ηs

ηs + ηp. (10)

The polymer stress for the FENE-P model can again be obtained by substituting the termf (R)(C¯−I

¯)/Wi

in (9) with (f (R)C¯

− I¯)/Wi.

In the limit L → ∞, the evolution equations for both pre-averaged FENE models reduce to that forlinear dumbbells(f (R) → 1), resulting in the third constitutive equation used in our simulations—theOldroyd-B model.

Since one of the major goals of this analysis is to correlate viscoelastic interfacial behavior to thephysical properties of the coating fluid, a direct comparison of the viscometric properties among thechosen constitutive models is provided below to note their adequacy for the current study[23].

• Steady shear flow: In a viscometric shear flow,u = y, v = w = 0, and the FENE-CR equation yields

2Cxy = f (R)

Wi(Cxx − 1), Cyy = f (R)

WiCxy,

f (R)

Wi(Cyy − 1) = 0. (11)

The shear viscosity, scaled by the total viscosity is then 1, independent of theWi. The same conclusionis true for the Oldroyd-B model. The FENE-P model, on the other hand, yields a nonlinear shearviscosity as a function ofWi.

2Cxy = f (R)Cxx − 1

Wi, Cyy = Czz = f (R)

WiCxy,

f (R)Cyy − 1

Wi= 0,

ηshear= S + 1 − S

Wif (R)Cxy. (12)

332 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

• Normal stress difference in shear flow: For the FENE-CR model the stress equation yields (cf.[19])

τxx − τyy = (1 − S)L2

2Wi

(−1 +

√1 + 8Wi2(L2 − 3)

L4

), (13)

likewise for the FENE-P model

τxx − τyy = (1 − S)2Wi

f (R)2, f 3 − 3 + L2

L2f 2 − 2Wi2

L2= 0. (14)

and the normal stress is a linear function ofWi for the Oldroyd-B model

τxx − τyy = 2Wi(1 − S). (15)

• Extensional flow: In planar extension flow, the velocities are given byu = x, v = −y and the dumbbelevolution equation for the FENE-CR model reduces to following (cf.[19])

2Cxx = f (R)

Wi(Cxx − 1), 2Cyy = −f (R)

Wi(Cyy − 1), Czz = 1, Cxy = Cxz = Cyz = 0.

(16)

Similar expressions can be derived for the FENE-P model

2Cxx = f (R)Cxx − 1

Wi, 2Cyy = −f (R)Cyy − 1

Wi, Czz = 1

f (R), Cxy = Cxz = Cyz = 0.

(17)

and for the Oldroyd-B model

ηextension= 1

(1 − 2Wi)+ 1

(1 + 2Wi). (18)

As can be seen elsewhere, both FENE models exclude the stress singularities typically associated withthe linear dumbbell model by bounding the extensional viscosity in the limit asWi approaches infinity[23].

An important observation from the analyses above is the coupling of the out-of-plane axial stress within-plane deformation even in purely two-dimensional flows for the FENE-P model.

2.1.1. Boundary conditionsThe boundary conditions applied on the free surface are

• kinematic condition

u¯· n

¯= 0; (19)

• normal stress balance

n¯· (τ

¯− P I

¯) · n

¯= ∇s · n

¯Ca

; (20)

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 333

• vanishing shear stress

n¯· τ

¯· t¯= 0; (21)

wheren¯

andt¯

are the unit vectors normal and tangent to the free surface, respectively and∇s denotes thesurface divergence operator.

Additional boundary conditions pertinent to each of the two flow types considered are stated below.

(i) Planar injection flow: For this flow, the governing equations are solved by considering a coordinatesystem that moves at the same speed as the bubble tip as shown inFig. 1. In this reference frame, thecorresponding boundary conditions are:

• On the solid wall, non-slip condition applies

u = 1, v = 0; y = 0, 0 ≤ x ≤ ∞. (22)

• At the centerline, symmetry conditions apply

τsy = 0, v = 0, y = 12, 0 ≤ x ≤ xc atxc, u

¯= 0. (23)

• At the outflow, rectilinear uniform flow is assumed (no diffusive outflow of momentum)

n¯· σ

¯· n

¯= 0, v = u

¯· t¯= 0. (24)

• Far from the air–fluid interface, a parabolic velocity profile is an exact solution to the governingequations for the Oldroyd-B and FENE-CR models and is given by

u = 6(1 − 2h∞)y(y − 1) + 1, v = 0. (25)

The quantityh∞ in Eq. (25)is the dimensionless thickness of the hydrodynamic coating left behindby the advancing gas phase, and its value is determined as a part of the solution. The velocityprofile for the FENE-P model is dictated by a highly nonlinear differential equation which requiresnumerical solution. A look-up table has to be assembled to determine the sensitivity of the inflowvelocity to the variation in the thickness of the hydrodynamic coating. Note that for the rangeWiexamined in this paper the velocity profile remains approximately parabolic.

• At the inflow, the conformation tensor components for the Oldroyd-B and FENE-CR models aregiven by

Cxx = 2Wi2

f (R)2

(∂u

∂y

)2

+ 1, Cxy = Wi

f (R)

∂u

∂y, Cyy = Czz = 1, Cxz = Cyz = 0,

(26)

wheref (R) = 1 for the Oldroyd-B model. The inlet stress condition for the FENE-P model is moreinvolved due to the coupling of the out-of-plane axial stress to the in-plane deformation even in apurely two-dimensional flow as shown inSection 2.1

Γ (y) =√

2Wi

L

du

dy, f (y) =

√3Γ (y)

2 sinh(φ/3), Cxx = 1

f (y)

[1 + 2Wi2

f 2(y)

(du

dy

)2]

,

Cxy = Wi

f 2(y)

du

dy, Cyy = Czz = 1

f (y), Cxz = Cyz = 0. (27)

334 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

whereφ = sinh−1(3√

3Ω/2). For the range ofWi investigated in this paper the stress coupling effectare small[24] and was not considered in our simulations.

All of the boundary conditions specified above are essential boundary conditions, i.e. they replacethe respective governing equations at the boundary nodes, with the exception ofn

¯· σ

¯· n

¯which is

imposed naturally.(ii) Sloat coating: As indicated inFig. 2, the distinguishing feature of this flow is the presence of a static

contact/separation line. For the low metering rates considered in this paper, a recirculation region isalways present near the upper stationary wall. Hence, the downstream solution does not get influencedby the stress singularity at the static contact line. The contact angle is therefore strictly dictated bythe hydrodynamics of the flow[11]. Only the Oldroyd-B and FENE-CR models are considered forslot coating and the velocity distribution at the inlet is a fully developed Couette–Poiseuille flow.

• On the moving substrate and the stationary wall, no-slip conditions apply

u = 1, v = 0, 0 ≤ x ≤ ∞; (28)

u = 0, v = 0, 0 ≤ x ≤ xc. (29)

• At the outflow, no diffusive outflow of momentum is assumed

n¯· σ

¯· n

¯= 0, v = u

¯· t¯= 0. (30)

• At the inflow, Couette–Poiseuille velocity profile is again derived using the no-slip boundary con-ditions on the solid walls

u = (12− 48Q)y2 + (24Q − 8)y + 1, v = 0. (31)

Unlike the planar injection flow, the net flow,Q, in slot coating is pre-metered[11]. The inlet stressconditions are shown inEq. (26). Again, except for then

¯·σ

¯·n¯

= 0 condition at the outflow, all boundaryconditions specified above are essential boundary conditions.

2.2. Finite element formulation of the governing equations for flow

The solution method implemented is the DEVSS formulation proposed by Guenette and Fortin[25].Both the free surface parametrization and field equation dependence on mesh deformation were incorpo-rated into a version of the code which has been used previously to solve fixed boundary viscoelastic flowproblems[26].

Hierarchic shape functions[27], Nk, formed by linear combination of Legendre polynomials andparticularly suitable for hp-FEM computations[28,29] were used to approximate the variables of theproblem:

u¯

=9∑

i=1

u¯

iNi, γ¯

=4∑

i=1

γ¯

iNi, P =4∑

i=1

P iNi, τ¯p

=4∑

i=1

τ¯

i

pNi

x¯

=Mx∑i=1

(X¯i + d

¯i)Ni, (32)

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 335

whereX¯i are the initial nodal coordinates and they remain invariant in our numerical solutions(Xi = 0

for i > 4). d¯

is the unknown displacement vector which describes the deformation of the flow domain.This equation constitutes the very essence of the pseudo-solid domain mapping technique which will beintroduced shortly.

To satisfy the Brezzi–Babüska and inf-sup condition, the stresses and the pressure are approximatedusing bilinear (four-noded elements) shape functions while the velocity is discretized using biquadratic(nine-noded elements) shape functions.

A Galerkin formulation is used for the elliptic saddle point problem resulting from the equations ofmotion and continuity. The Petrov–Galerkin streamline upwinding method (SUPG) proposed by Brooksand Hughes[30] is used to integrate the constitutive equations which are hyperbolic in nature. Followingthis procedure, the weak form of the governing equations is obtained:

〈∇ · u¯

h Np

i 〉 = 0, (33)

〈(σ¯

h + (∇u¯

h + ∇uh

¯T) − 2γ

¯h), ∇Nu

i 〉 =∫

Γn

Nui σ

¯h · n

¯h dΓn, (34)

FENE-CR:⟨τ¯

h

p+ Wi

τ¯

h

p

∇

f− 2(1 − S)γ

¯h

,

(Nτ

i + h

|u¯

h|u¯h · ∇Nτ

i

)⟩= 0, f =

L2 + (Wi/(1 − S))tr(τ¯

h

p)

L2 − 3.

(35)

FENE-P:⟨τ¯

h

p+ Wi

τ¯

h

p

∇

f+ 2(1 − S)

I¯∇

f

,

(Nτ

i + h

|u¯

h|u¯h · ∇Nτ

i

)⟩= 0,

f =L2 + 3 + (Wi/(1 − S))tr(τ

¯h

p)

L2, (36)

where the symbol∇ represents the upper convective derivative. Its operation on an arbitrary tensor,S¯, is

defined as

S∇

= S − ∇ v¯

T · S¯

− S¯

· ∇ v¯. (37)

The Oldroyd -B formulation can be obtained by settingf = 1 in Eq. (35). In the above formalism,σ¯

h isthe total stress tensor and it is defined asσ

¯h = −phI

¯+ τ

¯h. n

¯is the outward, unit vector normal to the

boundary,Γn, and〈a, b〉 is the standard inner product defined onΩ [31]. Nui , Nτ

i andNp

i are the weightingfunctions for the velocity, stress, and pressure variables, respectively. In addition, we approximateγ

¯as

follows:

〈 (γ¯

− 12(∇ u

¯h + ∇ uh

¯T), N

γ

i 〉) = 0. (38)

It should be noted that this approximation is essential in providing the discrete stabilization introducedin the momentum equation. The addition of this term does not alter the final solution while providing

336 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

stabilization commonly observed in EVSS based schemes[25] without the necessity of explicitly splittingthe viscous and elastic stresses.

The integral along the boundary,Γn, which results from applying the divergence theorem to the weakform of the momentum equation is split into contributions from each segment of the boundary (Γ 1

n , Γ 2n ,

Γ 3n , Γ 4

n ). The contributions along the solid boundary and the inlet are not included because the velocitycomponents along these boundaries are substituted by the essential conditions of the problem, as will beshown in the next section. The remaining contribution to the boundary integral is calculated along the freesurface, where the normal stress balance and vanishing tangential stress conditions apply. The quantityn¯

h · σ¯

h contained in the boundary integral inEq. (34)can be decomposed into its normal and tangentialcomponents to facilitate the application of the stress boundary conditions

n¯

h · σ¯

h = (n¯

hn¯

h : σ¯

h)n¯

h + (t¯hn¯

h : σ¯)t¯h = − 1

Ca

dt¯h

ds, (39)

where dt¯h/ds is the arclength derivative of the local tangent vector and it provides a measure of the local

mean curvature.Substituting(39) into (34)and integrating the term containing dt

¯h/ds by parts[32] gives∫

Γn

Nui (n

¯h · σ

¯h) ds =

∫Γn

(1

Ca

dNui

dst¯h

)ds − 1

Ca(Nu

i t¯h|s2 + Nu

i t¯h|s1) = 0, (40)

wheres1 ands2 are the ends of the free boundary curve. The formulation of the problem is complete oncethe free boundary is parametrized.

2.3. Free surface parametrization

Free surface problems are inherently nonlinear even for Newtonian fluids under creeping flowconditions because of the nonlinearities introduced by the boundary conditions (Eqs. (20) and (21)).The prediction of the steady flow free surface profile requires therefore a convergent iteration scheme.The flexibility and simplicity of the free boundary parametrization and tessellation of the flow domainplay a crucial role in the efficient and accurate readjustment of the free surface at eachiteration.

The free boundary tracking is accomplished by means of an implicit, pseudo-solid domain mappingtechnique proposed by Sackinger et al.[33]. The most attractive feature of this method is that it providesa simple way to relate the free boundary position to the internal mesh discretization which permits thesimultaneous solution of all the variables pertinent to the problem via a full Newton–Raphson iterationframework. Coupled to the solution of the governing equations of the problem and the distinguishingcondition, i.e. the condition which is used to effectively constrain the unknown location of the free ormoving boundary, is a displacement field that maps the initial domainΩinitial to the final domainΩfinal,and a node in the computational mesh (material point in the pseudo-solid) will be displaced to a newposition according to

x¯final = x

¯initial + d¯, (41)

so the components of the mapping vector,d¯, are introduced as new variables into the problem. The

vector of unknowns, ordered such that the variables associated with the governing equations appear first,

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 337

followed by the discretized mesh mapping variables, can be partitioned as

u =

uf

D

uf

Γ

udD

udΓ

. (42)

In the equation above, the superscriptsf and d represent variables associated with the flow domainequations and nodal displacement equations, respectively. The subscriptD denotes unknowns corre-sponding to the interior values of the conserved field variables while the variables on the flow boundariesor interfaces are denoted byΓ .

In order to compute the flow variables and the free surface position simultaneously, additional equationshave to be introduced into the computational matrix to describe the motion of the boundary nodes aswell as the dependence of the internal mesh points on the boundary displacements. The central idea inthe pseudo-solid technique is to introduce an independent set of field equations, i.e. the pseudo-soliddescription, to govern the deformation of the internal mesh points in the flow domain which are thencoupled to the flow field equations through the boundary conditions at the free surface.

Applying the assumptions of zero body force and zero density for the pseudo-solid and using a simpleisotropic linear elastic constitutive equation for the displacement of the nodal points in the computationalmesh, the following equations are embedded into the computational matrix and solved simultaneouslyalong with the governing equations of the problem:

0 = ∇ · S¯

· e¯k, (43)

wheree¯k are the unit vectors in the Cartesian coordinate.S

¯is defined as

S¯

= tr(E¯) + 2E

¯, (44)

E¯

= 12(∇ d

¯+ ∇ d

¯T). (45)

2.4. Finite element formulation of the free boundary problem

2.4.1. Boundary conditions for the pseudo-solidThe use of a separate field equation to describe the deformation of the mesh points requires the intro-

duction of a separate set of boundary conditions for the pseudo-solid except at the free surface, wherethe distinguishing condition would be enforced so that the displacement of the boundary nodes of thepseudo-solid would conform to the free surface problem under study. In this study, every boundary of thepseudo-solid is treated as a moving or free boundary, constrained by either the prescribed geometry orthe kinematic condition at the free surface. The boundaries which are constrained to stay at a fixed spatiallocation (the edges adjacent to the solid walls or the symmetry line, and both in- and out-flow planes) areformulated with vanishing tangential traction so that the mesh points on the boundaries may slide freelyalong them.

If we apply the Galerkin approximation and divergence theorem to the constitutive equation for thepseudo-solid(43), we obtain

Ndi (∇ · S

¯) · e

¯k = ∇ · (Ndi e

¯k · S¯) − ∇(Nd

i e¯k) : S

¯, (46)

338 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

(Ndi , e

¯k · (∇ · S¯)) = 0 = −

∫Ω

∇(Ndi e

¯k) : S¯

dV +∫

Γn

Ndi (n

¯e¯k : S

¯) ds. (47)

The boundary traction term is then rotated into its normal and tangential components with respect to thefree surface. Then, a Neumann condition (vanishing shear stress) is applied to the tangential componentof the traction to allow unconstrained sliding of the mesh in those directions to reduce element distortions[33]. The normal component of the traction equation is subsequently replaced by the distinguishingcondition for the free surface, which for our problem is the kinematic condition. As explained above,this last step is crucial since it provides the only coupling between the physics of the problem and theequations defining the deformation of the pseudo-solid.

The number of unknown nodal coordinates or displacement mapping variables (dx anddy) is signif-icantly reduced by adopting a subparametric mapping[34] in the interior of the mesh. This is done bymaking all element sides, which are not part of the free surface, straight lines while the free surface isapproximated by biquadratic functions. This choice is found to minimize the distortion of the elements[34], which is a major source of discretization error.

2.4.2. Solution of the problem via Newton iterationsExpressed in terms of a vector of finite element coefficientsu

¯(42), which contains all the variables in

the problem, and a vector of weighted residualsR¯(u¯), Newton’s method is an iterative solution procedure

for the linearized system

J¯(u¯

n) δ¯

n = −R¯(u¯

n), (48)

u¯

n+1 = u¯

n + δ¯

n, (49)

where the values of the variablesu¯

n+1 are updated from the last computed setu¯

n. Key to this iterativesolution procedure is the structure of the Jacobian matrix,J

¯. The assembly of the Jacobian matrix

containing the sensitivity (derivatives) of all equations with respect to all unknowns for the implementationof the Newton–Raphson iteration method is straightforward. The basic structure is shown below

J =

∂Rf

D

∂uf

D

∂Rf

D

∂uf

Γ

∂Rf

D

∂udD

∂Rf

D

∂udΓ

∂Rf

Γ

∂uf

D

∂Rf

D

∂uf

Γ

∂RdD

∂udD

∂RdΓ

∂udΓ

∂RdD

∂uf

D

∂Rf

D

∂uf

Γ

∂Rf

D

∂udD

∂Rf

D

∂udΓ

∂RdΓ

∂uf

D

∂RdΓ

∂uf

Γ

∂RdΓ

∂udD

∂RdΓ

∂udΓ

. (50)

The upper left 2×2 block describes the sensitivity of the flow equations with respect to the flow variables.The 2× 2 submatrix on the lower right corner describes the sensitivity of the chosen field equation forthe pseudo-solid, i.e. mesh-position equations, with respect to the coordinates of the nodal points.

The upper right corner submatrix contains the sensitivity of the governing equations for flow to themesh point displacements. The differentiation of spatial gradients in the discretized residual equations

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 339

Fig. 3. Computational domain.

340 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

with respect to the mesh displacement variables are shown inEqs. (53) and (54). The remaining submatrixdescribes the dependence of the boundary residual equations (i.e. the kinematic condition in our case)on the flow variables. The derivatives contained in this last submatrix provide a way to couple the meshdisplacement with the kinematic conditions the latter of which determines the profile of the air–fluidinterface.

The sensitivity of the discretized governing equations with respect to mesh deformation can be derivedin a straightforward manner by differentiating the integrands in the weak form of the governing equationswhich are functions of position with respect to the mesh displacement vector. Thus, the global Jacobianmatrix would receive additional contributions from two sources:

• Terms arising from the domain transformation (physical to computational) of the unit element∫. . . dA =

∫. . . |Je|dξdη, (51)

where|Je| = ∂x¯/∂ξ

¯is the determinant of the elemental Jacobian matrix for domain transformation.

From here, contributions of the form

∂|Je|∂dij

, (52)

would appear.

Fig. 4. Convergence study:τxx component of the polymeric stress.

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Fig. 5. Convergence study: trace (τii ) of the polymeric stress.

• Implicit dependence of the spatial gradient operators upon the mesh coordinates. Terms arising formthis source are derived using the transformation

∂∇∂dij

= ∂ (J¯

−1e

· ∇ξ )

∂dij, (53)

where∇ξ represents spatial gradients in the computational domain. Using the result fromEq. (32), thesensitivity of the mesh displacement gradient to mesh displacements can be derived

∂∇dk

∂dij= ∂J −1

e · ∇ξ dk

∂dij= ∂J −1

e

∂dij· ∇ξ dk + δikJ

−1e · ∂Nj

∂ξ. (54)

Table 1Convergence of calculations for Oldroyd-B model forCa = 0.2

Mesh (element) Degrees of freedom (NDF) Oldroyd-BWimax FENE-CRWimax FENE-PWimax

9,662 166,582 0.4 0.45 0.45512,730 219,617 0.5 0.575 0.619,515 336,212 0.525 0.61 0.61

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3. Finite element solutions

The resulting system of equations for the unknown coefficientsu¯

i , γ¯

i , P i , τ¯

i , and d¯

i are solvedsimultaneously via Newton’s method with first order continuation in both Capillary and Weissenbergnumbers.

Fig. 6. Convergence study: maximum stress value vs. Wi–Oldroyd-B model.

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3.1. Convergence study

The numerical solutions for both geometries were tested for convergence by varying the density of themesh and the location of the inlet and outlet boundaries. The convergence criterion adopted is that the totalsummation of the square of the difference between all variables, i.e. the difference of the norm, between

Fig. 7. Convergence study: maximum stress value vs. Wi–FENE-CR model.

344 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

two consecutive approximations is equal to or smaller than 10−5. Three different structured meshes areused in our studies ranging from 9662 to 19515 elements; the number of degrees of freedom rangesfrom 166,582 to 336,212. The geometric characteristics of these meshes are displayed inFig. 3a–c. Themost important distinction between the three mesh sizes lies in the density of element distribution in thecapillary-transition/thin film region of the air/fluid interface. The mesh with the lowest element density(Fig. 3a) has nine elements stacked across the fluid domain while the mesh with the highest elementdensity (Fig. 3c) has 31 elements in the same computational space.

The development of the stress field in the planar injection flow at a Capillary number of 0.2 andWeissenberg number of 0.4 for the Oldroyd-B constitutive model for all three meshes is shown inFigs. 4and 5. The large oscillations in the stress field contours for the 9662 element mesh clearly shows thatconverged results are obtained only for highly refined discretizations.

In all of our viscoelastic simulations at higher Weissenberg numbers, steep stress boundary layers areformed in both the capillary transition region and at the bubble vertex. Above a certain Weissenbergnumber, these gradients cannot be captured for the given mesh resolution with our choice of polynomialspaces. As we increase mesh refinement, we are able to proceed to higher values of the Weissenbergnumber (Wimax in Table 1) for all three constitutive model.

Also, with increasing mesh refinement the maximum values (in magnitude) of the three componentsof polymeric stress (τxx, τxy, τyy) converge to a finite value, indicating that the resolution of the stressboundary layer is improved.Figs. 6 and 7show the maximum values of the stress components as afunction of the flow Weissenberg number for the Oldroyd-B and FENE-CR models, respectively. As

Fig. 8. Newtonian flow field and second invariant of the rate of strain tensor evolution as a function ofCa.

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Fig. 9. Newtonian flow field (top) and second invariant of the rate of strain tensor (bottom) atCa = 0.75.

Fig. 10. Newtonian (left) and viscoelastic (right) flow field comparison.

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Fig. 11. Interfacial shape variation due to elasticity.

shown in the figures, convergence is obtained for all three meshes up toWi = 0.2. As the value ofWiincreases, more refined meshes are needed to resolve the stress gradients in the flow. AtWi = 0.4, thedifference in the maximum stress values between the two most refined meshes is about 6%. In whatfollows, the numerical results for each flow type will first be compared to Newtonian results previouslyreported in the literature. Then we will examine the effect of elasticity on the flow dynamics.

3.2. Planar injection flow

3.2.1. Flow field comparisonIn Fig. 8a–c we show the evolution of Newtonian flow field (left column) and the value of the second

invariant of the rate of strain tensor (right column) as a function ofCa; the shading reflects the magnitude

Fig. 12. Newtonian fluid film thickness as a function ofCa.

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of the quantities in flow. Note the disappearance of the recirculation region and the reduction in themagnitude of the rate of strain as the thickness of hydrodynamic coating increases. As will be shown inthe next section, the higher rate of strain at small values ofCa will lead to significantly higher polymerstresses in viscoelastic flows. For small values ofCa, the bubble front adopts almost a perfect semicircledue to dominant surface tension forces as predicted by perturbation analysis[35]. As the value of theCa is increased, the enhancement of viscous stresses results in an increase in the thickness of fluid filmentrained on the confining walls. In addition, the stagnation lines move away from the uniform filmregion into the capillary-transition region. WhenCa approaches unity, the interfacial flow undergoes arapid transition from a triple stagnation line flow field to that of a single stagnation line at the vertexof the bubble, which results in the complete disappearance of the recirculation region. An intermediateand rather peculiar recirculation pattern in the transition(0.72 ≤ Ca ≤ 0.85) has been observed inthe simulations of Giavedoni and Saita[10] and is reproduced in our numerical results (Fig. 9). As therecirculation region reduces in size, the lateral stagnation lines move toward the vertex of the bubble.But before the recirculation completely vanishes, two stagnation lines are detected on the symmetry linewhich gives rise to a recirculation pattern within the fluid domain. Although our simulations suggest thatthis new recirculation region would persist when the Capillary number is in the range 0.73 ≤ Ca ≤ 0.86,in good agreement with the results of Giavedoni and Saita, it is worth noting that this structure has neverbeen detected under experimental conditions[5].

Fig. 10 shows the side by side comparison between the Newtonian (Ca = 0.2) and viscoelastic(FENE-CR model,Ca = 0.2, Wi = 0.6, S = 0.85, L = 10) flow fields. The numerical results showthat the large increase in the thickness of the hydrodynamic coating in both FENE-CR and Oldroyd-Bfluids significantly weaken the recirculation flow and draws the lateral stagnation lines toward the vertexof the bubble. A reduction in the magnitude of the rate of strain due to elasticity is also captured inour computational results (Fig. 10bottom).Fig. 11shows the superposed interfacial profiles between

Fig. 13. Pressure drop across the vertex of the air bubble as a function ofCa.

348 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

the FENE-CR and Newtonian fluids. The compression of the bubble cap due to viscoelastic effects isin good qualitative agreement with our experimental observations[39] and results reported by other re-searchers[1,2]. Finally, the property of chain extensibility on the interfacial shape is also studied. Thehigher stress attainable in simulations with higher extensional parameter value(L = 10) leads to a morecompressed bubble cap while little change is observed for simulations with smallerL value (L = 5), ingood qualitative agreement with the experimental observation of Bonn et al.[1] using highly flexible PEOpolymers versus rigid Xanthan gum additives. The changes in coating film thickness due to elasticity willbe examined in detail in the next section.

3.2.2. Film thickness and pressure jump conditionsFig. 12illustrates the computed values ofh∞, the fluid film thickness far downstream from the interface

for Newtonian flow, versusCa. Experimental and simulation results from previous authors have beenadded to the same graph for comparison purposes. The figure shows that the film coverage for Newtonian

Fig. 14. Polymer stress evolution as a function ofWi (FENE-CR fluid,Ca = 0.2, S = 0.85,L = 10).

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fluids asymptotes toCa2/3 in the small Capillary number regime, in agreement with the predictionsof Bretherton, and reaches a maximum plateau value ofh∞ ≈ 0.4. The deviation from Bretherton’sprediction at higherCa is expected since the lubrication approximation used in his analysis is only validat very low displacement velocities. Excellent agreement is observed between our numerical predictionsand those obtained by Giadoveni and Saita[10] for all Capillary number values investigated.

The pressure drop between the vertex of the bubble and the thin film region is plotted inFig. 13asa function ofCa. Again, excellent agreement is obtained between our numerical solutions and thoseobtained by previous researchers.

Fig. 15. Polymer concentration effect on coating film thickness (Oldroyd-B fluid).

350 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

We begin our discussion of viscoelastic properties on planar injection flows by recalling our resultsin Fig. 8 where the computed second invariant of the rate of strain tensor is plotted forCa = 0.01,Ca = 0.2 andCa = 1. High strain rate regions can be observed close to the wall in front of the bubbleand in the region past the vertex and adjacent to the interface. Although the region near the confining wallis characterized by a higher strain rate value, the region adjacent to the gas/fluid interface is of greaterinterest since it directly affects the viscous stresses being exerted on the advancing bubble. It is also

Fig. 16. Normal stress gradient vs.x-coordinate (FENE-CR fluid,Ca = 0.2, S = 0.85,L = 10).

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important to note that shear flow dominates near the walls while straining flow is more important nearthe bubble cap.

The evolution of the polymeric stresses for the FENE-CR constitutive model with increasing Weis-senberg number is displayed inFig. 14a–c. For all Capillary numbers studied, a stress boundary layer isfound to form on the free surface at higher values ofWi (Fig. 14c). Although steep stress gradients areobserved for all three components of the polymeric stress (τxx, τxy, τyy), the plot of the trace of the stress

Fig. 17. Normal stress gradient vs.x-coordinate (Oldroyd-B fluid,Ca = 0.05,S = 0.85,L = 10).

352 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

tensor (τii ) indicates that the maximum stretch of the polymer molecules occurs downstream from thelateral stagnation line (vertex stagnation line in high Capillary number flows), i.e. theτxx component ofthe stress tensor (flow direction) is the most dominant in the flow.

Although stress boundary layer formation has been identified in purely extensional flows near aninterfacial stagnation point before[36–38], the boundary layer structure obtained in our simulations isvery different from that examined previously. In purely (uniaxial/planar) extensional flows, as shown inthe simulations by Yao and Mckinley[38] and Rasmussen and Hassager[37], the maximum stress isachieved at the stagnation point/line whereas in our numerical results the maximum polymeric stress islocated downstream from the lateral stagnation line. The stress boundary layer obtained in our simulationsappears to occur due to the highly convergent flow field in the capillary-transition region of the interface.A quick examination of the flow streamlines (Fig. 10) reveals that the contraction nature of the flow only

Fig. 18. Normal stress gradient vs.x-coordinate (FENE-CR fluid,Ca = 0.2, S = 0.85,L = 10).

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affects fluid elements in the vicinity of the interface while most of the streamlines remain rectilinear andparallel to the solid walls; this observation further supports the proposed mechanism that extensional flowalong the interface is responsible for the stress boundary layer formation.

Along with the onset of the polymeric stress boundary layer, a rapid increase in coating film thicknesswith Weissenberg number is observed relative to the Newtonian case as shown inFig. 15a and b. Thenumerical results inFig. 15a and b display a weak film thinning regime at lowWi and a rather strongfilm thickening response at moderate values ofWi. To understand the cause of film variation, we examinebriefly the mechanism of film thinning proposed by Ro and Homsy. In their perturbation analysis theyfound two major competing forces due to elasticity in the capillary-transition region of the bubble front:gradients in the shear stress in the gap direction (y-axis) and gradients in the normal stress in the flowdirection (x-axis). Shear stress gradients along they-axis represent momentum transfer between theconfining walls and the fluid domain, i.e. viscous stresses responsible for dragging a fluid element pastthe bubble front thus forming the hydrodynamic coating on the walls, these forces act to form the fluidfilm; gradients of the normal stress along thex-axis, on the other hand, arise because of the restoringforce in polymer molecules to oppose stretch in the flow direction. If the normal stress gradients werenegative, they would act to resist the passage of fluid elements in the direction along the wall velocity,and hence, they would trigger film thinning. In the limit of small Capillary number and weak elasticityRo and Homsy[17] found that negative gradients of normal stress in the flow direction were larger inmagnitude than the gradients in shear stress, so the prediction was that the addition of polymers in flowleads to coating film thickness reduction. These negative normal stress gradients were captured in ourlow Wi numerical simulations (Figs. 16a and 17a). To examine this proposed mechanism, we plottedthe values of the normal stress gradient (∂τxx/∂x) at y = 2/3h∞ (solid line) andy = h∞ (dash line),whereh∞ is the constant coating film thickness, as functions of thex-coordinate inFigs. 16 and 17forCa = 0.2 andCa = 0.05 using the FENE-CR and the Oldroyd-B constitutive models, respectively. Thetwo vertical dash lines indicate the location of the lateral stagnation line (left dash line, L) and the point at

Fig. 19. Effect of the finite extensibility parameter (L) on film thickness (Ca = 0.2).

354 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

which the gradient of the free surface variation (dh/dx) falls below 10−4 (right dash line, R). In essense,they roughly demark the capillary-transition region of the bubble front.

A close examination ofFigs. 16 and 17reveals that the viscoelastic effects can be classified into threeseparate regimes. At low Weissenberg numbers (Figs. 16a and 17a), a negative normal stress gradient isformedupstreamof the capillary-transition region of the bubble front. This stress gradient acts to opposethe flow of fluid elements in the direction of the wall velocity (with the frame of reference situated at thebubble tip), hence film thinning effects ensue.

At moderate values ofWi (Figs. 16b and 17b), the polymer stresses in the capillary-transition regionslowly becomes dominant. Although relatively high level of stress is achieved at this stage (strong positivenormal stress gradient), most of the stress accumulated is quickly relaxed in the capillary-transition regionas well (high negative normal stress gradient). Therefore, the coating film thickness remains fairly constantin this parameter range (plateau inFig. 15).

Fig. 20. Shear thinning effects on film thickness: (a)Ca = 0.2, S = 0.85,L = 10; (b)Ca = 0.2, Wi = 0.55,S = 0.85,L = 10.

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As the value ofWi is further increased (Figs. 16c and 17c), a steep stress boundary layer is formeddownstream from the lateral stagnation line. The appearance of the stress boundary layer then propelsthe accumulation of the normal stress to continue into the thin film parallel flow region, and thus, astrictly positivenormal stress gradient is formed in the capillary-transition region. This gradient, alsodue to the restoring force in polymer molecules to oppose stretch, would act to reduce the strength ofthe recirculation region (move the stagnation line towards the tip of the advancing bubble) by draggingmore fluid into the thin film region, hence, the coating film thickens. As the coating film starts to thickenrapidly, the strain rate near the free surface is reduced and a slight decrease in the magnitude of the elasticstress gradient is observed (Fig. 18).

The effect of variation in the extensibility parameter value,L, on film thickening is shown inFig. 19. Asthe value ofL is decreased, the polymer molecule behaves more and more like a rigid rod so the maximumstress attainable is reduced and film thickening is smaller and delayed to a higher Weissenberg number.This observation is in good agreement with the experimental results obtained by Bonn et al.[1] wherethey observed strong film thickening effects for PEO solutions (flexible polymer, highL value) and littlechange in the coating film thickness for Xanthan gum fluid (rigid polymer, lowL value). Computed resultsusing the FENE-P constitutive model, which exhibits shear thinning, are compared to those for FENE-CRin Fig. 20a. In agreement with our physical arguments describing the variation of film thickness, filmthickening is reduced due to shear thinning which lowers the magnitude of elastic stress gradient in thecapillary-transition region of flow as shown inFig. 20b. The effects of elasticity on the stress conditionsat the bubble vertex have also been examined in our study[39].

Fig. 21. Stress fields obtained using decoupled vs. full solution.

356 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

3.3. Decoupled solutions

An alternative method for obtaining approximate solutions to viscoelastic problems is by decouplingthe constitutive and conservation equations. By assuming that the presence of polymer molecules does notaffect the flow field, the conservation equations can be solved separately from the constitutive equations.Note that this represents the limiting form of zero polymer concentration (i.e.S → 1). This assump-tion allows the system to be divided into the solution of an elliptic saddle-point problem (Newtonianflow), and the integration of hyperbolic equations (constitutive equations) using fixed flow kinemat-ics. This way, the elastic stresses are treated as body forces and are updated once the velocity fieldis known.

In this section, we compare the numerical results obtained by using the decoupled scheme versus thoseobtained by using the fully coupled technique (described inSection 2) for the Hele–Shaw flow problem.

In Fig. 21theτxx components of the polymeric stress tensor for an Oldroyd-B fluid obtained from bothsolution techniques are compared side by side for two different values ofWi. At low values of the Weis-

Fig. 22. Newtonian slot coating flow field.

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senberg number, the decoupled solution yielded a good quantitative description of the stress distributionin flow. For moderate values of theWi, however, the decoupled method overestimated the maximum stressin the flow field, hence the steepness of the stress boundary layer, by over 70%. As shown inSection 3.2.2,due to the onset of the stress boundary layer, significant film thickening can occur at moderate valuesof Wi. The increase in hydrodynamic coating leads to a weakened recirculation region which results inreduced stress gradients because the resistance to flow past the bubble front becomes attenuated. In adecoupled calculation the film thickness remains constant, hence, for viscoelastic injection flows suchanalyses are applicable only to vanishingly small values ofWi.

Fig. 23. Result comparison: (a) the variation of curvature versus inclination, theta, along the free surface for Capillary number,Ca = 0.4, and dimensionless flow rate,Q = 0.25; (b) the variation of pressure,−P , and normal stress, Tnn= −P + τnn,along the free surface for Capillary number,Ca = 0.4 and dimensionless flow rate,Q = 0.25.

358 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

Fig. 24. Stress field evolution in slot coating flow (FENE-CR,Ca = 0.5, S = 0.85).

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3.4. Slot coating flow

As stated before, we limit our slot coating flow simulations to cases where a recirculation region ispresent in the flow domain.

3.5. Newtonian flow field comparison

For our Newtonian calculations of slot coating, the parameter ranges investigated were 0.15 ≤ Q ≤0.25, 0.08 ≤ Ca ≤ 0.7. The major limitation to examining flow parameter regimes outside this range isthe distortion of the element near the static contact line. When the separation angle drops to values below20, the element deformation becomes too great for the computed results to be reliable. The Newtonianflow field as a function of Capillary number is shown inFig. 22for a flow rate of 0.25. Clearly, as theCapillary number increases, the meniscus becomes more invasive. As a consequence, the free surfacecurvature increases and the separation/contact angle diminishes.

As shown inFig. 23, our computed results are in excellent agreement with those obtained by Saito andScriven[11].

3.6. Viscoelastic effects on interfacial profile

The range of Weissenberg numbers investigated for viscoelastic flows was 0≤ Wi ≤ 1.0. The stressfield evolution for slot coating of viscoelastic fluids are shown inFig. 24 for Ca = 0.5, whereτttp isthe stress component tangent to the free surface(τ

¯: t

¯t¯). In a similar manner as in the Hele–Shaw flow,

an elastic stress boundary layer appears at moderateWi. At still higher Weissenberg numbers, the strainhardening liquid in the capillary-transition region of the flow becomes increasingly difficult to stretch,and it becomes relatively easy to draw the unstretched material out of the fluid reservoir near the staticcontact line. This leads to an increase in meniscus invasion into the gap (Fig. 25) and a further reductionin the contact angle value. The stability of the contact line, i.e. its ability to remain attached to the exitof the stationary surface, has often been identified as the main driving force for the onset of viscous

Fig. 25. Increase in meniscus invasion due to elasticity.

360 A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362

Fig. 26. Numerical error due to excessive element distortion at the contact line.

fingering in slot coating operations[40]. Therefore, the increase in meniscus invasion due to elasticity,and the resulting reduction in separation angle relative to Newtonian coating, may explain the reducedstability observed in viscoelastic slot coating flows[41]. Even though values of the Weissenberg numbergreater than unity have been achieved in our simulations for the slot coating of elastic fluids, it was foundfrom the estimation of the free surface curvature (Fig. 26) that the distortion of the corner element wastoo significant for the computed values to be reliable (separation angle at static contact line fell below20).

4. Conclusions

In our computational work, a DEVSS finite element method coupled with pseudo-solid domain mappingtechnique was used to analyze a number of free surface problems. Specifically, we simultaneously solvedthe governing equations for the velocity and stress fields as well as the equations describing the freesurface deformation for Hele–Shaw and slot coating flows for both Newtonian and viscoelastic fluids.The excellent quantitative agreement with existing computational and experimental Newtonian flow dataas well as the good qualitative agreement with experimental observations of interfacial deformations forviscoelastic fluids over a wide range of Capillary and Weissenberg numbers demonstrate the quality androbustness of our numerical scheme.

Our computational results captured for the first time both elastically driven film thinning at small valuesof Wi [2,17] and the strong elastically driven film thickening at moderateWi as witnessed in a variety ofexperimental studies[1,2,39]. In our simulations for the Hele–Shaw flow we have shown that thickeningof the hydrodynamic coating film due to viscoelasticity is triggered by the onset of a stress boundarylayer at the air–fluid interface, which weakens the strength of the recirculating flow by generating astrong positive normal stress gradient in the flow direction. The structure of this boundary layer in thecapillary-transition region suggests that its formation is caused by the extensional nature of the contractionflow near the free surface downstream from the planar extensional stagnation line. By using a variety ofdifferent constitutive models, we have also shown how the physical properties of the polymer moleculescan affect the thickness of the coating film.

A.G. Lee et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 327–362 361

We have also computed the slot coating flows of viscoelastic fluids and a similar elastic stress boundarylayer structure was observed. Viscoelasticity was found to increase the meniscus invasion in slot coatingflows, and thus, reduce the separation angle at the static contact point. This observation has significantimplications in the stability of viscoelastic slot coating flows since fluid detachment from the stationarywall has often been identified as the triggering mechanism for the onset of fingering instabilities.

The accumulation of polymer stress near the free surface in the capillary-transition region presents aformidable challenge when it comes to conducting stability analysis of viscoelastic Hele–Shaw flows.These stress boundary layers imply that the stress jump condition at the interface cannot be derived simplyby using the stress profile in the bulk flow as attempted by Wilson[42]. In fact, since our simulationsrevealed no significant changes in the stress field in the parallel flow region due to the presence of polymermolecules, the onset of fingering phenomena in viscoelastic interfacial flows should be closely linked tothe response of the interfacial stress boundary layers to free surface perturbations.

Our work provides an overview on viscoelastic interfacial phenomena, covering different coating flowtypes. In capturing both the stress and flow field evolutions in viscoelastic planar injection flows our workmay provide stronger physical grounds for performing future theoretical and computational analysis onthis subject.

Acknowledgements

ESGS would like to thank the 3M Corporation and NSF for supporting this work through Grants1DCH652 and 0090428, respectively. BK would like to thank NSF for supporting this work throughGrant CTS-0089502. AGL would also like to thank the National Science Foundation for support throughtheir graduate research fellowship program. The authors are particularly grateful to Prof. G.M. Homsyfor his helpful comments and suggestions.

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