ORIGINAL RESEARCH

The free (open) boundary condition (FBC) in viscoelasticflow simulations

Evan Mitsoulis & Nikolaos A. Malamataris

Received: 11 June 2011 /Accepted: 15 August 2011 /Published online: 27 August 2011# Springer-Verlag France 2011

Abstract The Free (or Open) Boundary Condition (FBC,OBC) was proposed by Papanastasiou et al. (A NewOutflow Boundary Condition, Int. J. Numer. Meth. Fluids,1992; 14:587–608) to handle truncated domains withsynthetic boundaries where the outflow conditions areunknown. In the present work, implementation of theFBC has been extended to viscoelastic fluids governed byexplicit differential constitutive equations. As such weconsider here the Criminale-Ericksen-Filbey (CEF) model,which also reduces to the Second-Order Fluid (SOF) forconstant material parameters. The Finite Element Method(FEM) is used to provide numerical results in simplePoiseuille flow where analytical solutions exist for check-ing purposes. Then previous numerical results are checkedagainst Newtonian highly non-isothermal flows in a 4:1contraction. Finally, the FBC is used with the CEF fluidwith data corresponding to a Boger fluid of constantmaterial properties. Particular emphasis is based on a non-zero second normal-stress difference, which seems respon-sible for earlier loss of convergence. The results with theFBC are in excellent agreement with those obtained from

long domains, due to the highly convective nature ofviscoelastic flows, for which the FBC seems most appro-priate. The FBC formulation for fixed-point (Picard-type)iterations is given in some detail, and the differences withthe Newton–Raphson formulation are highlighted regardingsome computational aspects.

Keywords Free (open) boundary condition . Viscoelasticflows . Non-isothermal flows . Newtonian fluid . Second-order fluid . CEF model

Introduction

The Free (Open) Boundary Condition (FBC, OBC) wasproposed by Papanastasiou et al. [1] to handle truncateddomains with synthetic boundaries where the outflowconditions are unknown. This came in response to aconcerted effort some 20 years ago to solving the problemof boundary conditions at outflows, in a symposiumentitled “Minisymposium on Outflow Boundary Conditionsfor Incompressible Flow”, which took place on July 14,1991 at the University of California at Davis [2]. In therésumé of the minisymposium, the OBC proposed byPapanastasiou et al. [1] was given plaudits for impressivelyaccurate results for the benchmark Backward-Facing Step(BFS) problem [3] and Stratified Backward-Facing Step(SBFS) problem [4], and the method was also implementedby others for the same problem with equally good results.

The original idea by Papanastasiou et al. [1] has beenimplemented also for free-surface flows by Malamataris[5], and Malamataris and Papanastasiou [6]. Since then,several papers have appeared in the literature on the subject[7–11], sometimes referring to the FBC as an “openboundary condition” [4] or a “synthetic outflow boundary

E. Mitsoulis (*)School of Mining Engineering & Metallurgy,National Technical University of Athens,Zografou,157 80 Athens, Greecee-mail: mitsouli@metal.ntua.gr

N. A. MalamatarisDepartment of Mechanical Engineering,TEI of Western Macedonia,GR-50100 Kila-Kozani, Greece

N. A. MalamatarisDepartment of Computational and Data Sciences,George Mason University,Fairfax, VA 22030, USA

Int J Mater Form (2013) 6:49–63DOI 10.1007/s12289-011-1071-6

condition” [1] or a “no boundary condition” [7, 8]. Themathematics behind its workings has also been explained[1, 7, 8]. It seems that the method has been implemented inseveral computational codes for Fluid Mechanics, and inmany papers there is a passing mentioning of its applicationas part of the solution, e.g., [12]. However, details about itsimplementation in other linear-system solution schemes(like Picard iterations) and a more careful analysis of theresults, especially regarding individual effects of variousfluid mechanics parameters, as well as the outflowpressures – a sensitive quantity, appears missing from theliterature.

The authors have improved on this in a recent paper [13]by revisiting the original benchmark problems (BFS andSBFS) and showing detailed results for the primaryvariables (velocities-pressures-temperatures). Furthermore,they have also applied it to the benchmark free-surfaceproblem of extrudate-swell [14, 15]. The Finite ElementMethod (FEM) was used to provide and compare numericalresults with previous solutions [1–4]. The formulation wasgiven in some detail for fixed-point iterative schemes(Picard iteration) as opposed to Newton–Raphson schemes,with regard to computational aspects. Some interestingconclusions from the recent work showed that when theoutflow BC is known (as with surface tension effects inextrudate swell), then the FBC is not needed and should notbe used; also for vertical extrusion under gravity, the FBC isnot valid, since the body force due to gravity always adds aforce and the domain cannot be truncated.

In the present work, we continue with viscoelastic flows,which admittedly are more difficult to solve due to thehighly non-linear nature of the rheological constitutiveequation for the stresses. We first validate our resultsagainst analytical solutions in simple shear flows with thesecond-order fluid model of viscoelasticity [16]. Then, werevisit the work by Park and Lee [10] to establish thevalidity of our formulation for highly convective non-isothermal Newtonian flows. Finally, we embark on solvingthe Criminale-Ericksen-Filbey viscoelastic model [16] inflow through an abrupt contraction, using experimental datafor a polymer solution and paying attention in particular tosecond normal-stress difference effects, which have notbeen addressed before. The power of the FBC for truncateddomains will be shown in comparison with results fromlong dies.

Mathematical modeling

Governing equations

We consider the conservation equations of mass andmomentum for incompressible fluids under non-

isothermal, steady flow conditions. These are written as[13]:

r � u ¼ 0; ð1Þ

ru � ru ¼ r � s ¼ r � �pI þ t� �

; ð2Þ

rCpu � rT ¼ kr2T þ t : ru; ð3Þ

where ρ is the density, u is the velocity vector, p is thepressure, s is the total stress tensor, I is the identity tensor,t is the extra stress tensor, T is the temperature, Cp is theheat capacity, and k is the thermal conductivity.

Viscoelasticity is included in the present work via anexplicit differential constitutive model for the stresses. Thisis the Criminale-Ericksen-Filbey (CEF) equation [16],which has been used before in polymer melt flowsimulations [17]. The CEF model is written as:

t ¼ h �g þ 1

2Ψ1 þ Ψ2

� ��g � �g

n o� 1

2Ψ1

D �gDt

; ð4Þ

where η is the apparent viscosity, Ψ1 and Ψ2 are the firstand second normal-stress coefficients, respectively; they areall functions of the magnitude of the rate of strain tensor �gjjgiven by:

j �gj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2�g : �g

� �r: ð5Þ

The derivative D/Dt is the “corotational time derivative”(also called the “Jaumann derivative”); this derivative tellshow the components of a tensor change with time aswitnessed by an observer translating with the fluid androtating with it. For a second-order tensor, this derivative isdefined as follows [16]:

D �gDt

¼ @ �g@t

þ u � r �gn o

þ 1

2w � �g

n o� �g � wn o� �

; ð6Þ

in which

�g ¼ ruþ ruð ÞT ; ð7Þ

w ¼ ru� ruð ÞT ; ð8Þare the rate-of-strain and vorticity tensors, respectively.

The material functions η, Ψ1 and Ψ2 are defined inviscometric flows [16], which are simple shear flows with asingle non-zero velocity component u ¼ ðu1; 0; 0Þ and asingle non-zero shear rate component �g12 ¼ @u1=@x2 ¼ �g.Their definitions are as follows:

50 Int J Mater Form (2013) 6:49–63

h ¼ t12�g ; Ψ1 ¼ N1

�g2 ¼ t11 � t22�g2 ; Ψ2 ¼ N2

�g2 ¼ t22 � t33�g2 ;

ð9Þwhere N1 and N2 are the first and second normal-stressdifferences, respectively. The difference N2 is not zero forpolymer melts [14, 16]. The ratio N2 / N1 is negative and itsusual range is between −0.2 and −0.3 in accordance withexperimental findings [14, 16]. This ratio gives rise to amaterial constant θ given by:

N2

N1¼ q

1� q: ð10Þ

Then,

q ¼ N2

N1 þ N2: ð11Þ

When the material functions η, Ψ1 and Ψ2 are constants, theCEF model reduces to the second-order fluid (SOF), whichwas one of the first models used for viscoelastic calcu-lations [18]. The SOF and CEF models are explicit in thestress tensor, unlike the Maxwell model and other modelsbased on that, which are implicit [16, 19]. They constitute agood first point of studying viscoelasticity due to theirsimplicity.

The viscoelastic nature of a polymeric liquid is given bythe Weissenberg number (Ws) or Deborah number (De) orStress Ratio (SR), which can be shown to be equivalent [17,19]. For this we write:

Ws ¼ lU

R¼ Ψ1

2h

� �U

R; ð12Þ

De ¼ l �g; ð13Þ

SR ¼ N1

2t12¼ Ψ1

�g22h �g ¼ Ψ1

2h

� ��g; ð14Þ

where λ=Ψ1/2η is the material relaxation time in s, U is theaverage velocity, and R is a characteristic length (the radiusof a tube for flow through a circular die).

The majority of viscoelastic flows are creeping flows inwhich the inertial terms in the momentum equations arenegligible [17, 19]. These terms give rise to the Reynoldsnumber (Re) defined by:

Re ¼ rURh

: ð15Þ

For creeping flows, such as the flows considered here, Re=0. However, for generality we keep the inertia terms in themomentum equation.

The non-isothermal version of a viscoelastic model isderived by noting that the variations of viscosity andrelaxation time with temperature are modelled by using atemperature shifting function, aT.

hðTÞ ¼ hðT0ÞaT ðTÞ; ð16Þ

lðTÞ ¼ lðT0ÞaT ðTÞ; ð17Þ

where η(T0) and λ(T0) are the values of viscosity andrelaxation time at the reference temperature T0, respectively.For the temperature shifting function aT , an Arrhenius-typeequation is used:

aT ðTÞ ¼ expE

Rg

1

T� 1

T0

� �� �; ð18Þ

where E is the activation energy of the material, representingthe temperature dependence of the rheological properties, andRg is the ideal gas constant. The temperature T is given in K.

The various thermal and flow parameters are combinedto give appropriate dimensionless numbers [14, 16]. Therelevant ones here are the Peclet number, Pe, and theNahme-Griffith number, Na. These are defined as:

Pe ¼ rCpUR

k; ð19Þ

Na ¼ hEU 2

kRgT20

: ð20Þ

The Pe number represents the ratio of heat convection toconduction, and the Na number represents the ratio ofviscous dissipation to conduction and indicates the extent ofcoupling between the momentum and energy equations.When Pe>>1, there is strong convection, while when Na~1, there is a moderate coupling between momentum andenergy equations. A value of Na>1 indicates temperaturenon-uniformities generated by viscous dissipation, and astrong coupling between momentum and energy equations.

The above rheological model (Eq. 4) is introduced into theconservation of momentum (Eq. 2) and energy (Eq. 3) andcloses the system of equations. Boundary conditions arenecessary for the solution of the above system of equationsand they depend on the problem at hand (see below).

All lengths are scaled by R, all velocities by U, allpressures and stresses by ηU/R.

Method of solution

The numerical solution is obtained with the Galerkin/FiniteElement Method (GFEM), using two different programs,

Int J Mater Form (2013) 6:49–63 51

which employ as primary variables the two velocities,pressure, and temperature (u-v-p-T formulation). The first(called caves) has been developed and used mainly forviscoelastic problems [20]. The second (called uvpth) hasbeen developed for generalized (pseudoplastic and visco-plastic) multilayer problems with free surfaces and/orinterfaces [21].

GFEM casts the differential equations into integral formaccording to the Galerkin principles [22, 23]. For the u-v-p-Τ formulation, GFEM approximates the field variables forthe velocities u-v, pressure p, and temperature T, as follows:

u ¼ 8 TU ¼Xni¼1

8 iUi v ¼ 8 TV ¼Xni¼1

8 iVi

p ¼ yTP ¼Xmi¼1

y iPi T ¼ 8 TT ¼Xni¼1

8 iTi

ð21Þ

where U ;V ;P; T are arrays-columns of the nodalunknowns for each element, and 8 ;y are arrays-rows ofthe basis (interpolation) functions, and the superscript Τrefers to the transpose of a vector. The pressure basisfunctions ψ are of lower order than the other basis functions8, and interpolation for pressure is carried out over mnodes, while for the other variables is carried over n nodes,with m<n. The two codes use Lagrangian isoparametricquadrilateral elements. But caves uses 8-node serendipity

elements, while uvpth uses 9-node elements. These choicesfix the basis functions to n=8, m=4 for caves and to n=9,m=4 for uvpth. Hence, the basis functions ϕ and ψ arequadratic and linear, respectively.

The governing equations, weighted integrally with thebasis functions, result in the following continuity, Ri

C ,momentum, Ri

M , and energy, RiE, residuals in the domain Ω

[1]:

RiC ¼

ZΩ

r � uy idΩ ¼ 0; ð22Þ

RiM ¼

ZΩ

ru � ruð Þfi �r � �pI þ t� �

fidΩ ¼ 0; ð23Þ

RiE ¼

ZΩ

rCpu � rT � kr2T � t : ru

fidΩ ¼ 0: ð24Þ

By applying the divergence theorem in order todecrease the order of differentiation and to project thenatural boundary conditions for heat flux and stress atthe boundaries Γ of the domain Ω, Eqs. 23 and 24reduce to:

RiM ¼

ZΩ

ru � ruð Þfi þ �pI þ t� �

� rfih i

dΩ �ZΓ

n � �pI þ t� �

fidΓ ¼ 0; ð25Þ

RiE ¼

ZΩ

rCpu � rT

fi þ krTð Þ � rfi � t : ru

fi� �

dΩ �ZΓ

n � krTð ÞfidΓ ¼ 0: ð26Þ

Since essential boundary conditions (BCs) for u, v andT will be applied to all but the outflow boundary of thedomain, Eqs. 25 and 26 will be replaced by these BCs.Consequently, the surface integrals in Eqs. 25 and 26 stillneed to be evaluated or replaced or handled in some wayat the synthetic outflow. This is done here by applyingthe free boundary condition (FBC), which simplyevaluates these integrals at each iteration at the outflowand thus finds the local values of the primitive variablesu-v-p-T.

Our recent work [13] contains detailed derivations of theFEM formulation based on the “stiffness” matrix and “load”vector approach advocated by Huebner and Thornton [22].This approach is better suited for Picard iterative schemes(direct substitution) [20] rather than Newton–Raphson

schemes [1], and will be given in the Appendix with theappropriate modifications to incorporate the FBC fordifferential viscoelastic models.

Some other features of the above formulation are asfollows. For the temperature T in the program uvpth, asubdivision of the quadratic parent element into 4 bilinearquadrilateral elements is also used for implementing theStreamline-Upwind / Petrov-Galerkin (SU/PG) scheme [24,25], which stabilizes the T-solution in flows dominated byconvection (high Peclet number flows). In this case, thebasis functions ϕi that multiply the convection and viscousdissipation terms of the energy equation (1st and 3rd termsof Eq. 26) are modified according to:

fm ¼ fþ~ku � rf; ð27Þ

52 Int J Mater Form (2013) 6:49–63

where ϕm is the modified basis function according to SUPG

and~k is an artificial thermal conductivity (or diffusivity),

which depends on the velocity and element size as given inthe original paper by Brooks and Hughes [24].

In the program caves, no element subdivision wasperformed for the temperature T, but instead the methodproposed by Payré et al. [26] was used, as implemented byBarakos and Mitsoulis [27] for non-isothermal viscoelasticsimulations. This method consists basically of adjusting theGaussian points for the integration of the convection andviscous dissipation terms according to an upwinding

technique along a streamline crossing an element. It wasinteresting to see that for the problems solved here, cavesgave the same results for the non-isothermal simulationseither with upwinding or without, because of the densegrids used.

In the course of the present work, uvpth was enhanced toinclude explicit differential viscoelastic models, such as theCEF model. The main difference between the two programsis that caves uses a Newtonian (reference) “stiffness matrix”and puts all elastic stresses (or any non-linear viscousstresses) on the RHS “load vector”; uvpth puts the non-

uz=ur=0

z0

r

C

BA

uz=1-r2

ur=0 or FBC

P=0 or free

D

rz=0, ur=0

z=Lz=0

r=0

r =R

uz=1-r2

ur=0

Fig. 1 Poiseuille flow in a tube.Boundary conditions and finiteelement grid (5×5). FBC standsfor the free boundary condition

uz P

Fig. 2 Contours of field variables in Poiseuille flow of a second-order fluid (SOF): De=1 and θ=−0.25

Int J Mater Form (2013) 6:49–63 53

Newtonian viscosity function in the “stiffness matrix” (the1st term of Eq. 4), while the “load vector” takes the extraterms of the viscoelastic stresses (the 2nd and 3rd terms ofEq. 4) and any external load forces, such as gravity,traction, etc.

Both programs apply the Adaptive Viscoelastic StressSplitting (AVSS) scheme for handling viscoelastic stresses[28]. This basically means that the viscosity entering the“stiffness matrix” of element e is a reference viscosity ηref,which depends on the element size he and the maximumcomponent of the elastic stress tensor tel and the velocityvector u, i.e.,

href

e¼ he tel

max

uj jmax

: ð28Þ

The AVSS is capable of providing accurate results for testproblems (e.g. Poiseuille flow of a SOF or CEF fluid) forvery high De numbers (see below), which are impossible toobtain without it.

In the present work, both programs were modified forimplementing the FBC. Some more details of the FEM

formulation relevant to the FBC are given in the Appendix.Both programs use the Picard (direct substitution) scheme. Thecriteria for termination of the iterative process were for boththe norm-of-the-error and the norm-of-the-residuals <10–4.

Results and discussion

Poiseuille flow of a CEF fluid

We first tested the implementation of the FBC with the CEFfluid in simple pressure-driven (Poiseuille) flow in a tubefor code validation. Figure 1 shows the solution domainand boundary conditions, together with a 5×5 finiteelement grid in the understanding that if the method iscorrect it should work even with the sparsest of grids.Because of symmetry only one half of the flow domain isconsidered. The boundary conditions in this type ofproblem are: along the inlet DA a fully-developed velocityprofile is given, uz=1−r2 and ur=0; along the symmetryline AB, τrz=0, ur=0; along the die wall no slip conditionsare imposed, uz=ur=0; along the outlet BC either a fully-

Fig. 3 Axial pressure distribution of a second-order fluid (SOF) at De=1: a θ=0, b θ=−0.25

: =ur=0

uz=1-r2

ur=0 T=T0

uz=1-r2

ur=0 or FBC

uz=ur=0, T=T0Fig. 4 Schematic diagram offlow through a 4:1 contractiongeometry and boundary condi-tions for the numerical simula-tion. In the isothermal case,either fully developed velocityprofile or free boundary condi-tions are used at the outflowboundary

54 Int J Mater Form (2013) 6:49–63

developed profile is given, uz=1−r2 and ur=0 (and this is anecessity for viscoelastic fluids due to the presence of non-zero normal stresses), or the FBC is prescribed. Thepressure P is set to 0 at point C or is left free when usingthe FBC [13].

The CEF fluid with constant η, Ψ1 and Ψ2 reduces to theSOF. For this case, there is an analytical solution available,which is also the same for the implicit upper-convectedMaxwell (UCM) model. In cylindrical coordinates thesolution is [28]:

uz ¼ 1� r2; �g ¼ �4Ur; ΔP=Lð Þr¼0 ¼ 8U ;

ΔP=Lð Þr¼1 ¼ 8U � 2q � q1� q

� �De �g2;

trz ¼ �g; tzz ¼ 1þ q1� q

� �De �g2;

trr ¼ q1� q

De �g2; tqq ¼ 0:

ð29Þ

Thus, for an average velocity U=0.5, it follows thattrz ¼ �g ¼ 2; furthermore, if De=1 and θ=−0.25, itfollows that the pressure drop at the centerline ΔPcl=4,

at the wall ΔPw=5.2; and the stresses τzz=3.2, τrr=−0.8,

tj j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 t : t q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 t2zz þ t2rr þ t2qq þ 2t2rz q

¼ 3:07 .

The results for De=1 and θ=−0.25 are given in Fig. 2 forthe contours of various flow variables, which are the streamfunction ψ, the axial velocity uz, the pressure P, the normalstresses τzz and τrr, and the shear stress, τrz. Eleven contoursare drawn, equidistant between the maximum and minimumvalues. We observe that all contours (except the pressure) areperfectly parallel to the flow, as they should, since this is afully-developed shear flow. The pressure contours (isobars)show the distinct curvature associated with a non-zero N2.When N2=0, the isobars are perfectly vertical.

Figure 3 shows the axial pressure distributions for De=1and θ=0 or θ=−0.25. In both cases, the axial pressuredistribution is linear. However, when θ=0, the axialpressure distribution along the wall and the centerlinecoincide. When θ≠0, there is a radial distribution ofpressure, which is quadratic in r, and the results betweenthe wall and the centerline are different. In both cases, thenumerical results agree exactly with the analytical solutionsof Eq. 29 either by using the fully-developed profiles at exitor the FBC.

It is interesting to note that the upper limits for convergencewere different for θ=0 and θ≠0. In the former case, no upperlimit was found even for De=10+5. In the latter case, anupper limit was found for De=2 with θ=−0.25.

Newtonian non-isothermal flow

We continue our simulations for the Newtonian non-isothermal flow conditions used by Park and Lee [10] forcode validation. Figure 4 shows the solution domain andboundary conditions for the abrupt 4:1 contraction geom-etry of capillary flow. For the long domain (L/R=20) bothfully-developed boundary conditions and the FBC havebeen tested. Because of symmetry only one half of the flowdomain is considered, as was done in the previous work[10]. Reference results are given for the highest Peclet

Table 1 Material parameters used in the non-isothermal simulationsfor the flow of the IUPAC-LDPE (sample A) melt at 150°C [27, 29]

Parameter (symbol) Value (units)

Density (ρ) 918 kg/m3

Specific heat (cp) 2.302 kJ/kg K

Thermal conductivity (k) 0.26 W/m K

Activation energy (E) 66,514 J/mol [10]

Ideal gas constant (Rg) 8.3143 J/mol K

Reference temperature (T0) 423 K (150°C)

Viscosity (η) 55,000 Pa s

Relaxation time (λ) 0.1 s

Die radius (R) 1 cm

Fig. 5 Meshes used in thisstudy: a mesh1; b mesh2

Int J Mater Form (2013) 6:49–63 55

number Pe=12,200 and die lengths L/R=20 and 10. For thedata of Table 1 given for a benchmark LDPE melt (theIUPAC-LDPE melt A) [27, 29, 30], this value of Pecorresponds to U/R=15 and Na=213, signifying a highlyconvective flow with strong viscous heating effects.

The simulations for the benchmark problem have beenrun with the FEM meshes shown in Fig. 5. These aresimilar to the meshes used by Park and Lee [10], having thesame number of elements and nodes. Park and Lee [10]have used the Newton–Raphson (N-R) scheme in theirformulation together with Newtonian and differentialconstitutive equations for the UCM fluid. The resultsshould be the same for the Newtonian fluid with the Picardscheme used here. As shown, many elements are concen-trated near the die entry due to the singularity there, and atthe outlet to make sure that the FBC is accuratelycalculated.

The results for the axial velocity along the centreline areshown in Fig. 6 together with the results by Park and Lee[10]. Excellent agreement is obtained for all cases, and theFBC provides identical solutions whether the domain istruncated at 10R or at 5R.

The corresponding results for the axial pressure aregiven in Fig. 7. These results are important to show that thepressures values are also correct in the truncated domain.The trick here is not to set the pressure equal to 0 at anynode in the domain (usually the pressure is set to 0 at thecorner node at the outlet). This way the pressure iscalculated at the exit and takes its correct value.

The radial profiles are given in Figs. 8 and 9 for the axialvelocity and the temperature, respectively. They arecompared with the corresponding results given by Parkand Lee [10], with the latter profiles having been read bydata-digitizing software. Therefore, only a qualitativecomparison can be made by visual inspection of the results.

Newtonian Fluid

Distance, z / R

-20 -10 0 10 20

Axi

al V

eloc

ity, u

z / U

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Park and Lee [10](caves) F-D(uvpth) F-D(caves,FBC) L=20(caves,FBC) L=10(caves,FBC) L=5

Fig. 6 Axial velocity profiles along the symmetry line in the non-isothermal Newtonian flow simulation at Pe=12,200 and Na=213. F-D stands for fully-developed outlet boundary condition and FBCstands for the free boundary condition

Newtonian Fluid

Distance, z / R-20 -10 0 10 20

Pre

ssur

e, P

*=P

/(U

/R)

0

10

20

30

40

50

60

(caves) F-D(uvpth) F-D(caves, FBC) L=20(caves, FBC) L=10(caves, FBC) L=5

Fig. 7 Axial pressure profiles along the wall in the non-isothermalNewtonian flow simulation at Pe=12,200 and Na=213. Solutionsobtained from different programs and different boundary conditions

Newtonian Fluid

Axial Velocity, uz / U

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

r / R

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Park and Lee [10]This work (caves) F-DThis work (uvpth) FBCThis work (caves) FBC

Fig. 8 Radial velocity profiles at the exit (z=20) in the non-isothermal Newtonian flow simulation at Pe=12,200 and Na=213.Results obtained from different programs and different boundaryconditions

56 Int J Mater Form (2013) 6:49–63

In the present work, the profiles have been obtained withboth programs, caves and uvpth. The results virtuallycoincide from all three sources.

CEF isothermal flow

We continue with the isothermal CEF flow of a Boger fluid[31, 32], which is a fluid exhibiting a constant-viscosity ηand a quadratic first-normal stress coefficient Ψ1. Thus, as afirst approximation, it can be said to obey the SOF, i.e., theCEF fluid with constant material parameters. Theseparameters are given in Table 2, where different levels ofthe θ-ratio have been assumed (0, −0.25, −0.45) in theabsence of experimental data. The novelty here is theinclusion of a non-zero second normal-stress difference. Wepursue calculations at the highest elasticity level of De=1.14 for which convergence was possible. We present

results for both a long domain with a L/R=20 and a shorterone with L/R=10 by applying the FBC.

Figure 10 shows axial distributions of the velocity andpressure for different domain lengths obtained with De=1.14 and θ=−0.45. The axial velocity profile is given atthe centreline while the axial pressure profile is givenalong the walls, hence the peaks at z=0 where asingularity is present. We observe an excellent agreement

Table 2 Material parameter values used in Eq. 4 for fitting data for aBoger fluid at 21°C [31, 32]

Parameter (symbol) Value (units)

Reference temperature (T0) 294 K (21°C)

Viscosity (η) 18.8 Pa s

First normal-stress coefficient (Ψ1) 1.67 Pa s2

Normal-stress ratio (θ) 0, −0.25, −0.45Die radius (R) 1 cm

(a)

CEF Fluid

Distance, z / R

Axi

al V

eloc

ity, u

z / U

0.0

0.5

1.0

1.5

2.0

FBC, L/R=20FBC, L/R=10

(b)

CEF Fluid

Distance, z / R

-20 -10 0 10 20

-20 -10 0 10 20

Pre

ssur

e, P

* =

P/(

U/R

)

0

20

40

60

80

100

120

140

160

180

200

FBC, L/R=20FBC, L/R=10

Fig. 10 a Axial velocity profiles along the symmetry line and b axialpressure profiles along the walls in the isothermal CEF flowsimulation of a Boger fluid (De=1.14 and θ=−0.45). Results obtainedwith different die lengths

Newtonian Fluid

Temperature, T (oC) 150 160 170 180 190 200 210 220 230 240

r / R

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Park and Lee [10]This work (caves)This work (caves) FBC

Fig. 9 Radial temperature profiles at the exit (z=20) in the non-isothermal Newtonian flow simulation at Pe=12,200 and Na=213.Results obtained from different programs

Int J Mater Form (2013) 6:49–63 57

from the two domains, meaning that the FBC can be safelyused in truncated domains with this type of viscoelasticfluids. Thus, we can confidently say that the FBC has beencorrectly implemented for viscoelastic flows, albeit forisothermal ones.

The streamlines and isobars from the simulations aregiven in Fig. 11a and b for De=1.14 and θ=−0.45. Thesalient features of this flow are the existence of a vortex inthe reservoir and the parabolic contours of the pressure inthe die due to a non-zero θ-value (cf. Fig. 2). The values θ=0 and θ=−0.25 did not change in any visible way thestreamline patterns, while the pressures were affected more,as shown in Fig. 11c, where for θ=0 the isobars are nowvertical lines in the die.

CEF non-isothermal flow

The Boger fluid does not have any thermal fluid propertieswhich can give appreciable thermal effects in the range ofsimulations, since it is a corn syrup-solution of lowviscosity that does not generate any viscous heating effects.For this reason and to continue with studying a non-isothermal CEF flow of a Boger fluid, we have assumedhere thermal data borrowed from the LDPE melt (Table 1),which of course are artificial and help only see that the FBCis correctly implemented for non-isothermal viscoelasticflows. Furthermore we have taken its viscosity η and firstnormal stress coefficient Ψ1 to be 100 times higher togenerate enough viscous heating, otherwise the flow was

Fig. 11 a Streamlines and b isobars in flow through a 4:1 axisymmetric contraction of a CEF fluid (SOF): De=1.14 and θ=−0.45; c isobars for θ=0.Results obtained in a truncated domain with L/R=10 and FBC at the outlet

58 Int J Mater Form (2013) 6:49–63

isothermal with the real properties of Table 2. Under theseconditions and for De=1.14, the Peclet number was Pe=5202 and the Nahme number was Na=2.74. The maximumtemperature rise ΔTmax even under these conditions hardlyreached 3°C.

The boundary condition at the outlet is only theFBC either for L/R=10 or 20. A denser grid wasnecessary in the non-isothermal viscoelastic simulations,

having more cuts near the outflow for good results in thelonger domain (L/R=20). All non-isothermal results areobtained with θ=−0.45.

We present results first for the axial velocity andpressure distributions in Fig. 12. The velocity distribu-tions are along the centerline while those for the pressureare along the wall. In both cases the truncated domain atL/R=10 gave the same results as those from the longdomain with L/R=20 up to 10R. Temperature results are

(a)

CEF FluidNon-isothermal

Distance, z / R

Axi

al V

eloc

ity, u

z / U

0.0

0.5

1.0

1.5

2.0

FBC, L/R=20FBC, L/R=10

(b)

CEF FluidNon-isothermal

Distance, z / R

-20 -10 0 10 20

-20 -10 0 10 20

Pre

ssur

e, P

* =

P/(

U/R

)

0

20

40

60

80

100

120

140

160

180

200

FBC, L/R=20FBC, L/R=10

Fig. 12 a Axial velocity profiles along the symmetry line and b axialpressure profiles along the walls in the non-isothermal CEF flowsimulation of a Boger fluid (De=1.14, θ=−0.45, Pe=5202, Na=2.74).Results obtained with different die lengths

(a)

CEF FluidNon-isothermal

Distance, z / R

Tem

pera

ture

, T (

o C)

20.98

20.99

21.00

21.01

21.02

21.03

21.04

FBC, L/R=20FBC, L/R=10

(b)

CEF FluidNon-isothermal

Temperature, T (oC)

-20 -10 0 10 20

21.0 21.5 22.0 22.5 23.0 23.5

r / R

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

FBC, L/R=20 at 10RFBC, L/R=10 at 10RFBC, L/R=20 at 20R

Fig. 13 a Axial temperature profiles along the symmetry line and bradial temperature profiles in the non-isothermal CEF flow simulationof a Boger fluid (De=1.14, θ=−0.45, Pe=5202, Na=2.74). Resultsobtained with different die lengths

Int J Mater Form (2013) 6:49–63 59

given in Fig. 13. In particular, Fig. 13a shows thecenterline temperature rise, which as explained above isminimal, hardly rising a fraction of a degree. Bothdomains give identical results in the common length of10R. Figure 13b shows the radial temperature profile at z=10R either using a truncated domain with L/R=10 or 20.The results coincide. The maximum temperature rise isclose to the walls and is about 1.5°C higher than the walltemperature of 21°C. When using a longer domain with L/R=20, the maximum temperature rise is higher (about 2.2°C) asexpected.

Figure 14 shows streamlines, isobars and isothermsobtained from the CEF non-isothermal flow simulations.

The vortex results are not affected in a visible way bythe non-isothermal simulations. The same can also besaid for the pressures, where the isobars have beennormalized between 0 and 1. The curved isobars reflectthe presence of a non-zero second normal stressdifference N2. Finally, the isotherms show that thetemperature rises towards the exit and near the walls,where the thermal boundary layers developed due toviscous dissipation effects that are convected downstreamby the flow. All results at the outflow are well behaved, sowe can confidently say that the FBC has been imple-mented correctly and works well for non-isothermalviscoelastic flows.

Fig. 14 a Streamlines, b isobars, and c isotherms in non-isothermal flow through a 4:1 axisymmetric contraction of a CEF fluid (SOF): De=1.14,θ=−0.45, Pe=5202, Na=2.74. Results obtained in a truncated domain with L/R=10 and FBC at the outlet

60 Int J Mater Form (2013) 6:49–63

Conclusions

We have implemented the free (open) boundary condition toviscoelastic non-isothermal flows in the context ofexplicit differential constitutive equations of the CEFtype. Code validation was carried out first againstanalytical solutions for the SOF in simple Poiseuilleflow, and then against previous results by Park and Lee[10] in the benchmark problem of a 4:1 axisymmetriccontraction. The Newtonian non-isothermal flows withstrong convection were successfully tested (for Pecletnumber, Pe=12,200) in truncated domains of die length L/R=10 and 5. Then, viscoelastic isothermal and non-isothermal flows were successfully used for the CEF fluidwith constants given for a benchmark Boger fluid atDeborah number, De=1.14. The emphasis was put on anon-zero second normal-stress difference, which is moredifficult to solve and causes an earlier loss of convergence.In all cases studied, the FBC worked very well, givingresults with truncated domains that were the same as withthose from long domains. This bodes well for other morecomplicated viscoelastic models, in particular integralmodels of the K-BKZ type used for fitting rheologicaldata for various polymeric liquids [33–35]. The FBC isparticularly suited for strong viscoelastic flows, due to thehighly convective nature of these flows.

Acknowledgements Financial assistance for one of the authors(EM) from the programme “PEBE 2009–2011” for basic researchfrom NTUA is gratefully acknowledged.

Appendix

Our recent work [13] contains detailed derivations of theFEM formulation based on the “stiffness” matrix and“load” vector approach advocated by Huebner and Thorn-ton [22]. Here we concentrate on the appropriate modifica-tions to incorporate the FBC for differential viscoelasticmodels.

Mass and momentum discrete equations

Combining the discrete forms of the conservation equationsof mass and momentum (including compressibility) intoone matrix equation leads (in two-dimensional axisymmet-ric domains, r-z-θ corresponding to 1-2-3) to the followingsystem of an element (stiffness) matrix [S], a vector ofunknowns {x}, and a RHS (load) vector {F} for eachelement:

S11 S12 S13S21 S22 S23S31 S32 S33

24

35 U

VP

24

35 ¼

F1

F2

0

24

35 ðA:1Þ

The entries for each term in the above system are given indetail in [13].

Contribution from the FBC

With the FBC, the extra terms along the outflow boundaryare:

F ¼Z

ΓFBC

n � �pI þ t� �� �

8dΓ ¼|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

free boundary condition

ZΓFBC

nr �pþtrrð Þþnztrznrtrzþnz �pþtzzð Þ

� �8 idΓ ;

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}free boundary condition

i ¼ 1; 3 ðA:2Þ

Fq ¼Z

ΓFBC

n � krTð Þ8dΓ ¼|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

free boundary condition

ZΓFBC

k nr@T

@rþ nz

@T

@z

� �8 idΓ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}free boundary condition

; i ¼ 1; 3 ðA:3Þ

After the appropriate manipulations, the following matrixsystem is obtained:

SO11 SO12 SO13SO21 SO22 SO23

� � UVP

24

35 ¼ FO1

FO2

� �ðA:4Þ

where the components of the element (stiffness) matrix [SO]of Eq. A.1 are:

SO11 ¼ZΓ

h@8

@z8 idΓ ; i ¼ 1; 3 ðA:5Þ

SO12 ¼ZΓ

h@8

@r8 idΓ ; i ¼ 1; 3 ðA:6Þ

SO13 ¼ 0 ðA:7Þ

Int J Mater Form (2013) 6:49–63 61

SO21 ¼ZΓ

Ψ1 þ Ψ2ð Þ �grz@8

@z8 id�

ZΓ

2h3

@8

@rþ 8

r

� �8 idΓ ; i ¼ 1; 3 2nd term ¼ 0 for incompressible fluids

ðA:8Þ

SO22 ¼ZΓ

2h@8

@z8 idΓ þ

ZΓ

Ψ1 þ Ψ2ð Þ �grz@8

@r8 idΓ �

ZΓ

2h3

@8

@z8 idΓ ; i ¼ 1; 3

3rd term ¼ 0 for incomp: fluids ðA:9Þ

SO23 ¼ �ZΓ

y8 idΓ �ZΓ

@r@p

y8 idΓ ; i ¼ 1; 3

2ndterm ¼ 0 for incomp: fluids ðA:10Þ

FO1 ¼ FO2 ¼ 0 ðA:11ÞThe above contributions of [SO] and [FO] must be added tothe corresponding terms of Eq. A.1 for the elements havingthe FBC on one side.

For the contribution to the energy equation from the

FBC (Eq. A.3), the matrix Eq. A.4 has a 4th unknown, T ,and the stiffness element corresponding to this is simply:

SO44 ¼ZΓ

k nr@8

@rþ nz

@8

@z

� �8 idΓ ; i ¼ 1; 3 ðA:12Þ

It should be noted that when using the N-R iteration, Eqs.

A.2 and A.3, as such, simply constitute the residuals, fRg,from which the Jacobian ½J � ¼ ½@R=@x� is derived, and thesystem is solved for the vector of unknowns fΔxg,according to ½J �fΔxg ¼ �fRg. Thus, it is not necessaryto derive “stiffness” matrices and “load” vectors, as in theabove.

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