Research Article A New Method to Simulate Free Surface ...downloads.hindawi.com/journals/amse/2015/159831.pdf · Research Article A New Method to Simulate Free Surface Flows for Viscoelastic

Research ArticleA New Method to Simulate Free Surface Flows forViscoelastic Fluid

Yu Cao123 Xiao-Guang Ren12 Xiao-Wei Guo12 Miao Wang12

Qian Wang12 Xin-Hai Xu12 and Xue-Jun Yang12

1College of Computer National University of Defense Technology Changsha Hunan 410073 China2State Key Laboratory of High Performance Computing National University of Defense Technology Changsha Hunan 410073 China3School of Chemical Engineering and Analytical Science Manchester Institute of Biotechnology University of ManchesterManchester M13 9PL UK

Correspondence should be addressed to Xiao-Guang Ren renxiaoguangnudteducn

Received 11 September 2015 Revised 1 December 2015 Accepted 13 December 2015

Academic Editor Olanrewaju Ojo

Copyright copy 2015 Yu Cao et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Free surface flows arise in a variety of engineering applications To predict the dynamic characteristics of such problems specificnumericalmethods are required to accurately capture the shape of free surfaceThis paper proposed a newmethodwhich combinedthe Arbitrary Lagrangian-Eulerian (ALE) technique with the Finite Volume Method (FVM) to simulate the time-dependentviscoelastic free surface flows Based on an open source CFD toolbox called OpenFOAM we designed an ALE-FVM free surfacesimulation platform In the meantime the die-swell flow had been investigated with our proposed platform to make a furtheranalysis of free surface phenomenonThe results validated the correctness and effectiveness of the proposedmethod for free surfacesimulation in both Newtonian fluid and viscoelastic fluid

1 Introduction

In recent years there is a growing interest in free surfacesimulations (eg die-swell and bubble growing) Die-swellas a typical phenomenon seen in viscoelastic fluids is widelyinvestigated Numerical simulations of extrusion flow needto address free surface problems Numerical schemes forfree surface treatment can be classified into the fixed grid(interface capturing) and themoving grid (interface tracking)methods Fixed grid methods capture the interface in a Eule-rian description including Volume of Fluid (VOF) method[1] Level Set method [2] and Marker and Cell (MAC)method [3] in which the interface is reconstructed throughnumerical schemes in these methodsThough these methodscould address sharp interface and large deformation issueswell the position of the free surface is usually not preciseenough On the contrary moving grid methods explicitlytrack the position of interface by computational grids whichguarantees the precision of the interface position but thequality of the mesh might be reduced during mesh movingin large deformation cases Arbitrary Lagrangian-Eulerian

(ALE) Method [4] is one of the moving grid methodsIt combines the advantage of the Lagrangian method intracking interface and the efficiency of Eulerian method insolving governing equations Therefore it has been widelyused in wave-like free surface simulations

ALE method is usually combined with Finite ElementMethod (FEM) [5] in which the traction boundary conditionon the interface could be treated in a neat and exact way byintegrating the force in the momentum equation by partsHowever with the development of Finite Volume Method(FVM) FVM system is more attractive in fluid simulationsfor its good property in local conservation Therefore com-bining ALE method with FVM is capable of representing theposition of free surface by the computational mesh directlywhile benefitting from a good conservation property withFVM Tukovic and Jasak [6] developed a platform to simulateNewtonian fluid free surface problems by using the ALEmethod in FVM system In order to get the accurate velocityon the surface a new surface mesh is coupled with thetraditional mesh but it increases the computational cost andcomplexity Moreover it could not tackle with viscoelastic

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2015 Article ID 159831 10 pageshttpdxdoiorg1011552015159831

2 Advances in Materials Science and Engineering

fluids The greatest challenge of doing this is setting properboundary conditions on the free surface

In this paper a novel numerical method is proposed for2D viscoelastic extrusion flow simulations To the best of theauthorsrsquo knowledge this paper is the first implementationfor combining ALE technique and FVM in studying time-dependent viscoelastic extrusion simulations which extendsthe application fields of ALE method The main contributionof this paper may be summarized as follows

(i) We addressed the challenges in coupling theALEwithFVM method to simulate time-dependent viscoelas-tic extrusion flows The obstacle of setting properfree surface boundary conditions is well addressed byderiving the Dirichlet pressure boundary conditionand introducing a balance force imposed on the freesurface The proposed method is capable of dealingwith both Newtonian fluids and viscoelastic fluids

(ii) We implemented a coupled ALE-FVM simulationplatform based on OpenFOAM (Open Source FieldOperation and Manipulation) [7] The novel numeri-cal method is validated by comprehensively compar-ing with previous literature Compared to Tukovicand Jasakrsquos work [6] our approach significantlyreduced the storage and computation cost by aban-doning the additional surface mesh Moreover basedon OpenFOAM our free surface numerical solvercould be easily scaled to massive processor cores forlarge-scale parallel simulations

The rest of the paper is organized as follows Details of theproposed method are presented in Section 2 The numericalresults and discussion are reported in Section 3 Conclusionsare drawn in Section 4

2 Methodology

In this section the mathematical formulation of Oldroyd-Bmodel and numerical algorithms used in the paper will bepresented

21 Mathematical Model The governing equations for iso-thermal and incompressible viscoelastic fluid flows includethe mass (continuity) conservation equation momentumequation and constitutive equation The forms of thoseequations for the Oldroyd-B model are given by

120588(120597u120597119905

+ (u minus w) sdot nablau) = minusnabla119901 + nabla sdot 120591 + 120578119904Δu

nabla sdot u = 0

120591 + 120582nabla

120591 = 2120578119901D

(1)

where 120588 is the fluid density u is the fluid velocity w is thevelocity of the cell 119901 is the pressure and 120591 is the polymericcontribution for stress tensor 120578

119904and 120578119901are the solvent viscos-

ity and polymer viscosity respectively 120582 is the characteristic

relaxation time and D is the rate of deformation tensorwhich can be expressed as

D =1

2(nablau + (nablau)119879) (2)

The upper-convected time derivative of the polymer stresstensor is defined as

nabla

120591=120597120591

120597119905+ (u minus w) sdot nabla120591 minus (nablau)119879 sdot 120591 minus 120591 sdot nablau (3)

For the convenience of comparison with othersrsquo worksvariables above are usually converted into their dimensionlessform New dimensionless variable Reynolds number Reand Weissenberg number Wi are introduced by scaling thefollowing variables as

1199091015840

=119909

119871

1199101015840

=119910

119871

u1015840 = u119880

1199011015840

=119901119871

120578119880

1205911015840

=120591119871

120578119880

1199051015840

=119871

119880119905

Re =120588119880119871

120578

Wi = 120582119880

119871

120573 =120578119904

120578

(4)

in which 119871 is the characteristic length (the half width of thedie in this work) 119880 is the characteristic velocity (the averagevelocity in this work) and 120578 is the sum of the 120578

119904and 120578

119901

namely total viscosity 120578 = 120578119904+ 120578119901 120573 is the viscosity ratio

parameter

22 Numerical Algorithm

221 Implementation of Free Surface Boundary ConditionThe most difficult problems in die-swell simulation are theimplementation of free surface boundary condition Bound-ary conditions imposed on the free surface could be dividedinto two categories the kinematic boundary condition andthe dynamic boundary condition The kinematic boundarycondition requires that the normal component of the fluidvelocity equals the normal component of the interfacial gridvelocity as denoted by

u sdot n = w sdot n (5)

Advances in Materials Science and Engineering 3

The equation above means the relative speed between thefluid and the grid perpendicular to the interface is zero InOpenFOAM the velocity of fluid is stored at the cell centerwhichmeans it would be hard to get the velocity of interfacialgrid directly However the flux of the fluid through eachcell face could be obtained easily Therefore the velocityof interfacial face could be calculated according to the netmass flux on the surface A detailed description about theimplementation of the kinematic boundary condition can befound in [6 8] After moving the interfacial nodes at theinterface the internal nodes need to be moved according tothe solution of solving a Laplace equation

nabla sdot (Γnablau) = 0 (6)

where Γ is the diffusion coefficient The solver chosen tomove nodes in this work is Laplace Face Decomposition[9] which could maintain a good quality of the mesh inlarge deformation cases And the diffusion coefficient chosenis inversely proportional to the square of distance fromthe free surface Regenerating the internal grids in thisway could form lines of equal potential and decrease thenonorthogonality which in turn increases the quality of themesh

The dynamic boundary condition is also called thetraction boundary condition Ignoring the surface tension itmeans the traction exerted from the fluid to the atmosphereshould be equal to the atmospheric forces acting on the fluidWithout loss of generality the atmosphere pressure is set as1199010= 0 So the boundary condition is given by

120590 sdot n = (minus119901I + 2120573D + 120591) sdot n = (minus119901I + T) sdot n = 0 (7)

where 120590 is the total stress tensor namely the Cauchy stresstensor across the free surface and T = 2120573D + 120591 is theextra stress tensor When solving equations in OpenFOAMPISO algorithm requires the term minusnabla119901 and the term 120578

119904Δu

exists separately in the equation However the term 120590 isa combination of 119901 120591 and D Therefore this boundarycondition is hard to satisfy directly It has been known thatminusnabla119901 + nabla sdot 120591 + 120578

119904Δ119906 = nabla sdot 120590 and in FVM systems every term

in the equation should be integrated over the cell volume 119881Therefore the right side of (1) is actually solved asint

119881

(minusnabla119901+nablasdot

120591+120578119904Δ119906)119889119881 = int

119881

(nablasdot120590)119889119881 According toGaussrsquos theorem thespatial derivative term could be converted to integrals overthe cell surface 119878 bounding the volume Then we have

int119881

(nabla sdot 120590) 119889119881 = int119878

119889S sdot 120590 = sum

119886

120590119886sdot n119886119878119886

= sum

119886 =119887

120590119886sdot n119886119878119886+ 120590119887sdot n119887119878119887

(8)

where S is the face area vector and n is the unit normal vectorof face area 119886 and 119887 represent faces around a cell and freesurface faces respectively

The dynamic boundary condition requires 120590119887sdot n119887= 0

however it would not be the case if any of the boundaryconditions is improper Noting that the boundary conditionwould only influence the calculation of the cell that owns

the boundary patch we introduce a balance force 120590lowast = minus120590

on the free surface (it is zero elsewhere including the internalfield and other boundary fields) to compensate the influenceof nonzero boundary force By adding the term nabla sdot 120590

lowast intothe original momentum equation the right side of (1) can besolved as

int119881

(minusnabla119901 + nabla sdot 120591 + 120578119904Δ119906 + nabla sdot 120590

lowast

) 119889119881

= int119881

(nabla sdot 120590 + nabla sdot 120590lowast

) 119889119881 = int119878

119889S sdot (120590 + 120590lowast

)

= sum

119886

120590119886sdot n119886119878119886minus 120590119887sdot n119887119878119887= sum

119886 =119887

120590119886sdot n119886119878119886

(9)

It can be seen that the influence of the total stress imposed onthe free surface is offset in this way which means the tractionboundary condition is satisfied equivalentlyThemomentumequation is actually solved in the form

120588(120597u120597119905

+ (u minus w) sdot nablau)

= minusnabla119901 + nabla sdot 120591 + 120578119904Δu + ⟨nabla sdot 120590

lowast

⟩Γ119891

(10)

where Γ119891indicates the term is only applied on the free surface

boundary Since the SIMPLE algorithm is used to iterativelysolve the linear momentum equation and the continuityequation a proper pressure boundary condition is requiredduring the step of solving the Poisson equation Accordingto the force balance in the normal direction on the interfaceDirichlet boundary condition for the pressure is given by

119901 = n sdot T sdot n (11)

It is important to note that the pressure condition here is justa derived quantity which has arisen only because of the needto solve the Poisson equation for 119901 in isolation While thetraction boundary condition is the real physical quantity itmust be satisfied on the fluid Provided that119901 andu are solvedjointly no pressure boundary condition is needed anymore[10]

222 DEVSS Method for Numerical Stability In order toenhance the stability of the linear momentum equationespecially for high Wi number flow the discrete elastic splitstress (DEVSS) numerical strategy proposed byMatallah et al[11] is used The momentum equation can be rewritten as

120588(120597u120597119905


= minusnabla119901 + nabla sdot Σ119901+ 120578119905Δu + ⟨nabla sdot 120590

lowast

⟩Γ119891

(12)

where 120578119905= 120578119904+ 120572 and nabla sdot Σ

119901is the added artificial viscosity

ratio The term nabla sdot Σ119901= nabla sdot 120591 minus 120572Δu is treated as a source

termA typical choice of120572 is 120578119901This strategy greatly increases

numerical stability and can be applied in modelling variousconstitutive models


223 The Simulation Procedure Apart from the treatmentof the free surface boundary conditions the simulationprocedure is similar to [6] as shown below

(1) For a new time step 119905119899 the velocity field ulowast the

pressure field 119901lowast the polymer stress field 120591lowast and the

fluid mass fluxes 119891are obtained from the previous

time step The initial net mass fluxes through theinterface faces 120593

119899are calculated according to the

previous interface position by subtracting the fluidmass fluxes by the volumetric face fluxes

119891

(2) Move the interface mesh points so that the net massflux through the interface could be compensated andthe mass conservation law is satisfied

(3) According to the displacement of the interface meshpoints the position of internal mesh points isobtained by solving the Laplacian equation

(4) Beginning of the outer iteration loop

(a) getting the new estimated polymer stress tensor120591lowastlowast by solving the specified constitutive equa-

tion with previous velocity field ulowast(b) setting up the pressure boundary condition and

estimating the balance force 120590lowast(c) calculating the latest velocity field ulowastlowast and the

pressure filed 119901lowastlowast with SIMPLE algorithm

(d) moving the interfacemesh points similar to step(2) until the prescribed accuracy is reached

(5) If the programme has not reached the final timereturn to step (1)

Step (4)(d) is performed again in order to avoid theoperation of step (3) which is a time-consuming step Suchprocedure is reasonable provided that the displacement ofthe interface in one time step is smaller than the thickness ofthe nearest interface cell And this condition could be easilysatisfied by decreasing the time interval

224 Other General Numerical Schemes Used in the Simula-tion In addition to the approaches described above under-relaxation algorithm is also used to enhance the stability ofthe numerical simulation where the relaxation parametersfor 119901 and 119880 are 03 and 05 respectively

Backward Euler method with second order is used fortime derivatives In terms of the linear system solver PCG(Preconditioned Conjugate Gradient method) with AMG(Algebraic Multi-Grid method) preconditioning is used forpressure while BiCGstab (Bi-Conjugate Gradient Stabilizedmethod) with Diagonal Incomplete Lower-Upper (DILU)preconditioning is used for velocity and stress [12 13]Absolute tolerances are 10 times 10

minus6 for the velocity and 10 times

10minus7 for pressure and stress respectively

3 Results and Discussion

31 Basic Geometric Shape and Boundary Conditions Asketch of the geometry and the boundary conditions for

Free surface

Symmetry

OutletInlet

No-slip u = w = 0uin

120591inL

u middot n = w middot n

120590 middot n = 0 p = 0

L1 L2

Figure 1 Die-swell flow configurations

modelling the Newtonian and Oldroyd-B fluid is shownin Figure 1 The half width of the die 119871 is chosen asthe characteristic length The lengths of the upstream andthe downstream are set as 119871

1and 119871

2 respectively They

should be long enough to ensure the flow before entry ofthe die exit and further downstream at the outlet is fullydeveloped In this paper length settings are 119871 = 1 and1198711= 1198712= 8

As the pressure the velocity and the polymer stress aresolved in a coupled way in our system we usually give eitherall the correct boundary conditions or any one of them on theboundary In order to avoid unnecessary numerical failurenear the inlet the conditions of fully developed flow areimposed on the inlet boundaryThe velocity along119909directionat the inlet is uin = 15(1 minus 119910

2

) while the velocity along119910 direction is zero Therefore the average velocity of theinflow is u = 1 and the shear rate near the wall is 120574

119908=

3 The polymer stress tensor at the inlet for the Oldroyd-B model is set according to the analytical solution No-slip boundary condition is set at the wall The symmetryboundary condition is set at the centreline for all fieldsSymmetry boundary condition is provided by OpenFoamitself It means there is neither convection flux nor diffusionflux across the plane that is the vertical velocity and the shearstress component are zero (u

119909= 0 and 120590

119909119910= 0) We assumed

that the flow at the outlet has been already fully developedtherefore the pressure is set as zero at the outlet Note thatthe pressure here is just a reference pressure which is used tosolve the governing equations In terms of the free surface thepressure is initially set as fixed value zero and it would changedynamically during the simulation (see (11)) Without specialdeclaration zero gradient boundary condition is set for theremaining boundaries

32 Validation of ComputationalModel withNewtonian FluidBy setting Re = 001 Wi = 0 120573 = 1 and 120576 = 0 Oldroyd-B model degrades to Newtonian fluid Under isothermalcondition and neglecting the effects of gravity and surfacetension the swelling ratio defined as the maximum diameterof the extrudate divided by the diameter of the die is 1193from our simulation It is in an excellent agreement with theresults of Mitsoulis et al [14] and Georgiou and Boudouvis[15] which is around 119

33 Validation of ComputationalModel with Viscoelastic FluidUsing Oldroyd-B Model Based on K-BKZ models Tanner


Table 1 Mesh parameters for convergence validation withWi = 01

Name Size Δ119909min119871 Δ119910min119871 119878119877

M1 160 lowast 30 01 002 11677M2 320 lowast 30 005 002 11727M3 160 lowast 60 01 001 11691M4 640 lowast 30 0025 002 11789M5 160 lowast 120 01 0005 11697M6 360 lowast 60 0025 001 11770M7 400 lowast 60 00125 001 11787

[16] derived an analytical formula to account for the viscousand elastic contributions to the swell ratio as

119878119877= 019 + [1 +

1

12(1198731

120590119909119910

)

119908

2

]

14

(13)

where 1198731= 120590119909119909

minus 120590119910119910

is the first normal stress differenceand 120590

119909119910is the shear stress Both of them are evaluated at the

wall Introducing the parameter of recoverable shear 119877119904=

(11987312120590119909119910)119908 the swelling ratio can be estimated as

119878119877= 019 + [1 +

1

3119877119904

2

]

14

(14)

For the Oldroyd-B model the recoverable shear of a fullydeveloped Poiseuille flow is equal to

119877119904= [

2Wi (1 minus 120573) 1205742

2 120574]

119908

= (1 minus 120573)Wi 120574119908 (15)

Since in our case 120574119908

= 3 and 120573 = 01 the recoverableshear is 119877

119904= 27Wi

Mesh convergence was tested at Wi = 01 with sevenmeshes which are listed in Table 1 Δ119909min119871 and Δ119910min119871are the relative minimum length along 119909-axis and 119910-axisrespectively These cells appear near the exit of the dieThrough the comparison of all these seven meshes it isfound that the value of Δ119909min119871 plays a dominant role whendetermining the welling ratio Furthermore if Δ119910min119871 le

001 and Δ119909min119871 le 0025 the relative error of the swellingratio is below 02 The solution convergence is reached byusing M6 The time interval is Δ119905 = 10

minus6 for M7 and Δ119905 =

10minus5 for other cases (except for M1)The value of pressure and polymer stress near the sin-

gularity is illustrated in Figure 2 where 119909 is in the rangeof [minus04 04] along with different mesh densities (M4 M6)at Wi = 01 It is shown that the two curves are veryclose to each other which indicates that the given meshdensity is sufficient In terms of the tendency the pressuregradually declines at first and then experiences a steep fallfollowed by a mild elevation before the exit Subsequently itdramatically drops from positive to negative and approacheszero afterwards As shown in Figures 2(b) and 2(c) thereis a highest point for 120591

119909119909and lowest point for 120591

119910119910 where

both points are around the exit Furthermore the curve of120591119910119910

is sharper than the one of 120591119909119909 As 119909 increases the value

of 120591119909119910

decreases at first reaches the bottom and sharply

grows to an apogee before it declines to zero gradually Interms of the trend of the profiles as well as the maximumand the minimum values near the singularity the resultsof the proposed method are close to those of the spectralelement method with 3rd order polynomial adopted in Clauswork [17] which again demonstrates that the current meshis capable of providing reliable results The small deviationin terms of the positions of the maximum value and theminimum value are attributed to the fact that the velocity isstored on the cell center in this paper and on the cell vertexin [17] respectively

In general with the increase of Wi the profile of com-ponents of the polymer stress should be similar but with agreater value However the figure for 120591

119910119910is much different

from the intuition Figure 3 shows that the nadir forWi = 01

and Wi = 02 which is negative becomes the peak for Wi =05which is positiveThe force along119910direction caused by thepolymer stress components 120591

119910119910is 119899119910120591119910119910 where 119899

119910is vertical

component of the standard interfacial normalThus themorelikely 119899

119910is approaching to one (corresponding to a small

swelling ratio with a gentle curve) the greater the influence of120591119910119910

would be As the force increases the extrusion is growinglarger

The swelling ratio is given in Figure 4 with a comparisonwith othersrsquo works Note that in their works the definitionof relative parameters could be different For example Russoand Phillips [18] and Tome et al [19] used the full width as thecharacteristic lengthWe just altered them to be the same onein our workMoreover Crochet and Keunings [20] and Tomeet al [19] took the Reynolds number as 0 and 05 respectivelyRusso and Phillips [18] tested two situations with Re = 0001

and Re = 05 and the results shed light on the fact thatno significant change was achieved by varying the Reynoldsnumber when Re is small enough All of these three paperstook 120573 = 19 in their similar simulations Our simulationis kind of similar to Crochet and Keunings [20] and Tomeet al [19] who varied the relaxation time 120582 in different casestherefore the trend of the curves is the same However Russoand Phillips [18] changed the input velocity that is both shearrate near the wall 120574

119908and the Weissenberg number Wi are

changed In his case the recoverable shear rate is far differentfrom the one in our simulation even with the same Wi Thatis the reason the trend of his curve seems different fromothers The swelling ratio of all these curves shown increaseswith the increase of theWeissenberg number From Figure 4our results are above Crochet and Keunings [20] and Tomeet al [19] but below Russo and Phillips [18]With the increaseof Wi number the elastic effect plays a dominant role indetermining the die-swell In addition slight drop of swellingratio at low Wi number illustrated in Figure 4 is due to thenegative value of 120591

119910119910and was also observed in [20]

The steady state profiles of the free surface at various Winumbers are shown in Figure 5 It can be seen that a largerWi value would result in an increased slope of the curveThisis because the first normal stress at the exit of the channelfor bigger Wi number is larger than the smaller ones Suchphenomenon also indicates the main factor of the swelling isthe value of the first normal stress near the exit of the dieUnder higher Wi number the hexahedron cell is deformed


0 01 02 03 04minus02minus03 minus01minus04

x

minus4

minus3

minus2

minus1p

0

1

2

3

M4

M6Claus (P = 3)

(a) 119901


x

0

5

10

120591x

x 15

20

25

30

M4

M6Claus (P = 3)

(b) 120591119909119909

0 02 04minus02minus04

x

minus4

minus3

minus2

minus1

0

120591y

y

M4

M6Claus (P = 3)

(c) 120591119910119910

0 0402minus02minus04

x

minus8

minus6

minus4

minus2

0

2

4

120591x

y

M4

M6Claus (P = 3)

(d) 120591119909119910

Figure 2 Pressure and components of polymer stress near the singularity at Wi = 01

120591y

y

minus4

minus2

0

2

4

6

8

10

0minus02minus04 0402x

Wi = 01

Wi = 02

Wi = 05

Figure 3 120591119910119910

near the singularity for various Wi

Sr

04 06 08 1 12 14 1602Wi

1

12

14

16

18

2

22

Our resultTannerrsquos theory

Crochet and Keunings 1982Russo and Phillips 2011

Tomeacute et al 2002

Figure 4 Swelling ratio with the increase of Wi


18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1

Figure 5 Free surface profile for various Wi for 120573 = 01

greatly which increases the mesh nonorthogonality Themaximum mesh nonorthogonality has reached as high as 40at Wi = 10 and the threshold in OpenFOAM is usually setas 70 Though the nonorthogonality value could be reducedby mesh refinement the computational time would increaseThemaximum swelling ratio obtained from our simulation isabout 226 at Wi = 25 using Mesh M1

Contour plots of velocity components 119880119909for Wi = 01

05 08 10 and119880119910for Wi = 05 10 are displayed in Figure 6

By increasing Wi number the horizontal components of thevelocity 119880

119909increase along 119910 directions The contour line

of 119880119909

= 14 expands upward and forward The verticalcomponents of the velocity 119880

119910also increase due to larger

elastic effects which results in a larger swelling ratioThe profile of the first normal stress difference 119873

1just

before the exit (at 119909 = minus004 and the exit is at 119909 = 0) of the diefor a large range of Wi is shown in Figure 7 These lines aresampled along 119910-axis at fixed 119909 Apparently the maximumvalue of each line increases as Wi becomes larger which isconsistent with (15) All of these four lines begin with zeroat the middle of the die (119910 = 0) As for Wi = 10 it goes upgradually until it approaches thewall where it grows instantlyto the peak Subsequently it drops dramatically The profilesof otherWi values smaller than 10 are something like shiftingthe curve of Wi = 10 downwards to the right In detail thereis no peak in the case ofWi = 01 Generally speaking a largerWi results in a higher peak which indicates the first normalstress dominates the extrusion process

Decreasing the viscosity ratio 120573 from 1 to 0 meansgradually increasing the contribution of elastic stress anddecreasing the effects of purely viscous stressWhen120573 reacheszero the model degrades to the upper-convected Maxwell(UCM) model In order to investigate the role of 120573 in die-swell simulations simulations under 120573 = 05 and 120573 = 09

are carried out Figure 8 illustrates the trend that the swellingratio would increase as Wi number increases regardless of

the value of 120573 Moreover the result departs fromTanners the-ory obviously Considering an extreme case as 120573 approachesone the swelling ratio almost remains a constant even for alarge value of Wi according to Tanners theory However thisdoes not match the experimental result For the case 120573 = 09Wi = 1 and the case 120573 = 01 Wi = 1 the recoverable shear119877119904are 3 and 27 respectively According to (15) the swelling

ratio of the former one should be larger than the later onewhile the fact is contrary Russo and Phillips [18] attribute thediscrepancy to the fact that Tanners formula derived froma KBK-Z integral constitutive equation only considered asingle relaxation time however the Oldroyd-Bmodel owns aretardation time in addition to the relaxation timeThis resultmight be related to the fact that the elongational responsefor 120573 = 01 is much stronger than the one for 120573 = 09

as illustrated in Figure 9 Thus the extensional responsealso plays an important role in swelling process and shouldbe taken into account when predicting the final swellingratio The simulation results in [21] demonstrate that flowswith a sufficiently high elongational viscosity are able toswell to an apparently large state even at a relatively lowflow rate ignoring their performance in simple shear flowsNevertheless decreasing the viscosity ratio would lead to anincrease in the recoverable shear and in turn increases thevalue of swelling ration according to (15) By comparing thevalue in Figures 4 and 8 it is reasonable to say that Tannerstheory still makes some sense Apart from that with a carefulcomparison it is found that when the viscosity ratio 120573 isvery large (for the case of 120573 = 09) which represents a verydilute fluid the swelling ratio suffers a significant drop Incomparison when 120573 is relatively small (for the case of 120573 =

05) the swelling ratio ismuch less sensitiveThis phenomenawas also observed in [22] Note that because of differentdefinitions of the viscosity ratio 120573 in this paper is equivalentto 1 minus 120573 in their works

As can be seen from (13) the value of the shearstress would not change once the shear rate is determinedConsequently the first normal stress difference is the onlyvariable which determines the swelling ratio Figures 7 and10 demonstrate that a high proportion of polymeric stresscontributionwould result in a larger value of119873

1 whichwould

provide greater power to swell According to the analyticalsolution of the Oldroyd-B model smaller 120573 would result ina higher value of the first normal stress Therefore the firstnormal stress plays a vital role in determining the behaviourof extrusion flows and could be used to predict the finalresults of swelling ratio

4 Conclusion

In this work a new method to track the free surface forviscoelastic fluids was proposed The obstacle of setting aproper free surface boundary condition was well resolved byderiving a Dirichlet pressure boundary condition and intro-ducing a balance force The proposed numerical approachwas implemented opon OpenFOAM which is widely usedin CFD simulations [23 24] Comparing with previousliterature the results showed that our solutions were closeto those with other methods (eg FEM Spectral Element


0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1

Figure 6 Contour plots of velocity components 119880119909(a)ndash(d) and 119880

119910(e)-(f) for a range of Wi

N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

Figure 7 First normal stress difference at 119909 = minus004 for various Wi with 120573 = 01


04 06 08 102Wi

Our result 120573 = 05


Tannerrsquos theory 120573 = 05


1

12

14

16

18

2

22

Sr

Figure 8 Swelling ratio with the increase of Wi for 120573 = 05 and120573 = 09

100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576

Figure 9 Planar extensional viscosity for Oldroyd-B model withvarious 120573

Method and MAC) which validated the correctness andeffectiveness of the proposed numerical method Moreoverthe results showed that the swelling ratio would declinegradually with the increase of the viscosity ratio In themeantime brief analysis of the influence of the viscosity ratiowas performed Moreover providing appropriate constitu-tive equations investigation of die-swell behaviour for newpolymeric materials can be performed based on our newplatform In addition since the FVM systems are suitable forparallelization the free surface simulations in our platformwould be easily scaled to massive processors for parallelcomputing

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09

Figure 10 First normal stress difference at 119909 = minus004 for variousWiwith 120573 = 05 and 120573 = 09

Acknowledgments

Thanks are due to the computational resources providedby the University of Manchester partial funding from theNational Natural Science Foundation of China under Grantsno 61221491 no 61303071 and no 61120106005 and Openfund (no 201503-01 and no 201503-02) from State KeyLaboratory of High Performance Computing

References

[1] J L Favero A R Secchi N S M Cardozo and H JasakldquoViscoelastic fluid analysis in internal and in free surface flowsusing the software OpenFOAMrdquo Computers and ChemicalEngineering vol 34 no 12 pp 1984ndash1993 2010

[2] S Osher and J A Sethian ldquoFronts propagating with curvature-dependent speed algorithms based onHamilton-Jacobi formu-lationsrdquo Journal of Computational Physics vol 79 no 1 pp 12ndash49 1988


[3] S McKee M F Tome V G Ferreira et al ldquoTheMACmethodrdquoComputers amp Fluids vol 37 no 8 pp 907ndash930 2008

[4] C W Hirt A A Amsden and J L Cook ldquoAn arbitrarylagrangianeulerian computing method for all flow speeds(reprinted from the Journal of Computational Physics vol 14pg 227ndash253 1974)rdquo Journal of Computational Physics vol 135no 2 pp 203ndash216 1997

[5] M Fortin and D Esselaoui ldquoA finite-element procedure forviscoelastic flowsrdquo International Journal for Numerical Methodsin Fluids vol 7 no 10 pp 1035ndash1052 1987

[6] Z Tukovic andH Jasak ldquoAmovingmesh finite volume interfacetracking method for surface tension dominated interfacial fluidflowrdquo Computers amp Fluids vol 55 pp 70ndash84 2012

[7] H Jasak ldquoOpenFOAM open source CFD in research andindustryrdquo International Journal of Naval Architecture and OceanEngineering vol 1 no 2 pp 89ndash94 2009

[8] S Muzaferija and M Peric ldquoComputation of free-surface flowsusing the finite-volume method and moving gridsrdquo NumericalHeat Transfer Part B Fundamentals vol 32 no 4 pp 369ndash3841997

[9] H Jasak and Z Tukovic ldquoAutomatic mesh motion for theunstructured finite volume methodrdquo Transactions of Famenavol 30 no 2 pp 1ndash20 2004

[10] N Phanthien and R Tanner ldquoViscoelastic finite volumemethodrdquo Rheology Series vol 8 pp 331ndash359 1999

[11] H Matallah P Townsend and M F Webster ldquoRecovery andstress-splitting schemes for viscoelastic flowsrdquo Journal of Non-Newtonian Fluid Mechanics vol 75 no 2-3 pp 139ndash166 1998

[12] M A Ajiz and A Jennings ldquoA robust incomplete Choleski-conjugate gradient algorithmrdquo International Journal for Numer-ical Methods in Engineering vol 20 no 5 pp 949ndash966 1984

[13] J Lee J Zhang and C-C Lu ldquoIncomplete LU preconditioningfor large scale dense complex linear systems from electro-magnetic wave scattering problemsrdquo Journal of ComputationalPhysics vol 185 no 1 pp 158ndash175 2003

[14] E Mitsoulis G C Georgiou and Z Kountouriotis ldquoA study ofvarious factors affectingNewtonian extrudate swellrdquoComputersamp Fluids vol 57 pp 195ndash207 2012

[15] G C Georgiou and A G Boudouvis ldquoConverged solutions ofthe Newtonian extrudate-swell problemrdquo International Journalfor NumericalMethods in Fluids vol 29 no 3 pp 363ndash371 1999

[16] R I Tanner Engineering Rheology Oxford University PressOxford UK 2002

[17] S Claus Numerical simulation of complex viscoelastic flowsusing discontinuous galerkin spectralhp element methods [PhDthesis] 2013

[18] G Russo andTN Phillips ldquoSpectral element predictions of die-swell for Oldroyd-B fluidsrdquo Computers amp Fluids vol 43 no 1pp 107ndash118 2011

[19] M F Tome N Mangiavacchi J A Cuminato A Castelo andSMcKee ldquoA finite difference technique for simulating unsteadyviscoelastic free surface flowsrdquo Journal of Non-Newtonian FluidMechanics vol 106 no 2-3 pp 61ndash106 2002

[20] M J Crochet and R Keunings ldquoFinite element analysis of dieswell of a highly elastic fluidrdquo Journal of Non-Newtonian FluidMechanics vol 10 no 3-4 pp 339ndash356 1982

[21] A Al-Muslimawi H R Tamaddon-Jahromi and M F Web-ster ldquoSimulation of viscoelastic and viscoelastoplastic die-swellflowsrdquo Journal of Non-Newtonian Fluid Mechanics vol 191 pp45ndash56 2013

[22] J-R Clermont and M Normandin ldquoNumerical simulation ofextrudate swell for Oldroyd-B fluids using the stream-tubeanalysis and a streamline approximationrdquo Journal of Non-Newtonian Fluid Mechanics vol 50 no 2-3 pp 193ndash215 1993

[23] X-WGuo S Zou X Yang X-F Yuan andMWang ldquoInterfaceinstabilities and chaotic rheological responses in binary poly-mer mixtures under shear flowrdquo RSC Advances vol 4 no 105pp 61167ndash61177 2014

[24] X-W GuoW-G Yang X-H Xu Y Cao and X-J Yang ldquoNon-equilibrium steady states of entangled polymer mixtures undershear flowrdquoAdvances inMechanical Engineering vol 7 no 6 pp1ndash10 2015

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


fluids The greatest challenge of doing this is setting properboundary conditions on the free surface

In this paper a novel numerical method is proposed for2D viscoelastic extrusion flow simulations To the best of theauthorsrsquo knowledge this paper is the first implementationfor combining ALE technique and FVM in studying time-dependent viscoelastic extrusion simulations which extendsthe application fields of ALE method The main contributionof this paper may be summarized as follows

(i) We addressed the challenges in coupling theALEwithFVM method to simulate time-dependent viscoelas-tic extrusion flows The obstacle of setting properfree surface boundary conditions is well addressed byderiving the Dirichlet pressure boundary conditionand introducing a balance force imposed on the freesurface The proposed method is capable of dealingwith both Newtonian fluids and viscoelastic fluids

(ii) We implemented a coupled ALE-FVM simulationplatform based on OpenFOAM (Open Source FieldOperation and Manipulation) [7] The novel numeri-cal method is validated by comprehensively compar-ing with previous literature Compared to Tukovicand Jasakrsquos work [6] our approach significantlyreduced the storage and computation cost by aban-doning the additional surface mesh Moreover basedon OpenFOAM our free surface numerical solvercould be easily scaled to massive processor cores forlarge-scale parallel simulations

The rest of the paper is organized as follows Details of theproposed method are presented in Section 2 The numericalresults and discussion are reported in Section 3 Conclusionsare drawn in Section 4

2 Methodology

In this section the mathematical formulation of Oldroyd-Bmodel and numerical algorithms used in the paper will bepresented

21 Mathematical Model The governing equations for iso-thermal and incompressible viscoelastic fluid flows includethe mass (continuity) conservation equation momentumequation and constitutive equation The forms of thoseequations for the Oldroyd-B model are given by

120588(120597u120597119905

+ (u minus w) sdot nablau) = minusnabla119901 + nabla sdot 120591 + 120578119904Δu

nabla sdot u = 0

120591 + 120582nabla

120591 = 2120578119901D

(1)

where 120588 is the fluid density u is the fluid velocity w is thevelocity of the cell 119901 is the pressure and 120591 is the polymericcontribution for stress tensor 120578

119904and 120578119901are the solvent viscos-

ity and polymer viscosity respectively 120582 is the characteristic

relaxation time and D is the rate of deformation tensorwhich can be expressed as

D =1

2(nablau + (nablau)119879) (2)

The upper-convected time derivative of the polymer stresstensor is defined as

nabla

120591=120597120591

120597119905+ (u minus w) sdot nabla120591 minus (nablau)119879 sdot 120591 minus 120591 sdot nablau (3)

For the convenience of comparison with othersrsquo worksvariables above are usually converted into their dimensionlessform New dimensionless variable Reynolds number Reand Weissenberg number Wi are introduced by scaling thefollowing variables as

1199091015840

=119909

119871

1199101015840

=119910

119871

u1015840 = u119880

1199011015840

=119901119871

120578119880

1205911015840

=120591119871

120578119880

1199051015840

=119871

119880119905

Re =120588119880119871

120578

Wi = 120582119880

119871

120573 =120578119904

120578

(4)

in which 119871 is the characteristic length (the half width of thedie in this work) 119880 is the characteristic velocity (the averagevelocity in this work) and 120578 is the sum of the 120578

119904and 120578

119901

namely total viscosity 120578 = 120578119904+ 120578119901 120573 is the viscosity ratio

parameter

22 Numerical Algorithm

221 Implementation of Free Surface Boundary ConditionThe most difficult problems in die-swell simulation are theimplementation of free surface boundary condition Bound-ary conditions imposed on the free surface could be dividedinto two categories the kinematic boundary condition andthe dynamic boundary condition The kinematic boundarycondition requires that the normal component of the fluidvelocity equals the normal component of the interfacial gridvelocity as denoted by

u sdot n = w sdot n (5)








119904Δu




119881


120591+120578119904Δ119906)119889119881 = int

119881


int119881

(nabla sdot 120590) 119889119881 = int119878

119889S sdot 120590 = sum

119886

120590119886sdot n119886119878119886

= sum

119886 =119887

120590119886sdot n119886119878119886+ 120590119887sdot n119887119878119887

(8)







int119881


lowast

) 119889119881

= int119881


) 119889119881 = int119878

119889S sdot (120590 + 120590lowast

)

= sum

119886


119886 =119887

120590119886sdot n119886119878119886

(9)


120588(120597u120597119905



lowast

⟩Γ119891

(10)



119901 = n sdot T sdot n (11)



120588(120597u120597119905



lowast

⟩Γ119891

(12)














119891

















Free surface

Symmetry

OutletInlet


120591inL


120590 middot n = 0 p = 0

L1 L2



1and 119871

2 respectively They



2


119908=


119909= 0 and 120590

119909119910= 0) We assumed









119878119877= 019 + [1 +

1

12(1198731

120590119909119910

)

119908

2

]

14

(13)

where 1198731= 120590119909119909

minus 120590119910119910





119878119877= 019 + [1 +

1

3119877119904

2

]

14

(14)


119877119904= [

2Wi (1 minus 120573) 1205742

2 120574]

119908

= (1 minus 120573)Wi 120574119908 (15)



119904= 27Wi







119910119910 where



of 120591119909119910







119910119910is 119899119910120591119910119910 where 119899

119910is vertical













x

minus4

minus3

minus2

minus1p

0

1

2

3

M4

M6Claus (P = 3)

(a) 119901


x

0

5

10

120591x

x 15

20

25

30

M4

M6Claus (P = 3)

(b) 120591119909119909


x

minus4

minus3

minus2

minus1

0

120591y

y

M4

M6Claus (P = 3)

(c) 120591119910119910


x

minus8

minus6

minus4

minus2

0

2

4

120591x

y

M4

M6Claus (P = 3)

(d) 120591119909119910


120591y

y

minus4

minus2

0

2

4

6

8

10


Wi = 01

Wi = 02

Wi = 05

Figure 3 120591119910119910


Sr

04 06 08 1 12 14 1602Wi

1

12

14

16

18

2

22






18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1







of 119880119909




1just









1 whichwould


4 Conclusion



0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










119904Δu




119881


120591+120578119904Δ119906)119889119881 = int

119881


int119881

(nabla sdot 120590) 119889119881 = int119878

119889S sdot 120590 = sum

119886

120590119886sdot n119886119878119886

= sum

119886 =119887

120590119886sdot n119886119878119886+ 120590119887sdot n119887119878119887

(8)







int119881


lowast

) 119889119881

= int119881


) 119889119881 = int119878

119889S sdot (120590 + 120590lowast

)

= sum

119886


119886 =119887

120590119886sdot n119886119878119886

(9)


120588(120597u120597119905



lowast

⟩Γ119891

(10)



119901 = n sdot T sdot n (11)



120588(120597u120597119905



lowast

⟩Γ119891

(12)














119891

















Free surface

Symmetry

OutletInlet


120591inL


120590 middot n = 0 p = 0

L1 L2



1and 119871

2 respectively They



2


119908=


119909= 0 and 120590

119909119910= 0) We assumed









119878119877= 019 + [1 +

1

12(1198731

120590119909119910

)

119908

2

]

14

(13)

where 1198731= 120590119909119909

minus 120590119910119910





119878119877= 019 + [1 +

1

3119877119904

2

]

14

(14)


119877119904= [

2Wi (1 minus 120573) 1205742

2 120574]

119908

= (1 minus 120573)Wi 120574119908 (15)



119904= 27Wi







119910119910 where



of 120591119909119910







119910119910is 119899119910120591119910119910 where 119899

119910is vertical













x

minus4

minus3

minus2

minus1p

0

1

2

3

M4

M6Claus (P = 3)

(a) 119901


x

0

5

10

120591x

x 15

20

25

30

M4

M6Claus (P = 3)

(b) 120591119909119909


x

minus4

minus3

minus2

minus1

0

120591y

y

M4

M6Claus (P = 3)

(c) 120591119910119910


x

minus8

minus6

minus4

minus2

0

2

4

120591x

y

M4

M6Claus (P = 3)

(d) 120591119909119910


120591y

y

minus4

minus2

0

2

4

6

8

10


Wi = 01

Wi = 02

Wi = 05

Figure 3 120591119910119910


Sr

04 06 08 1 12 14 1602Wi

1

12

14

16

18

2

22






18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1







of 119880119909




1just









1 whichwould


4 Conclusion



0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials











119891

















Free surface

Symmetry

OutletInlet


120591inL


120590 middot n = 0 p = 0

L1 L2



1and 119871

2 respectively They



2


119908=


119909= 0 and 120590

119909119910= 0) We assumed









119878119877= 019 + [1 +

1

12(1198731

120590119909119910

)

119908

2

]

14

(13)

where 1198731= 120590119909119909

minus 120590119910119910





119878119877= 019 + [1 +

1

3119877119904

2

]

14

(14)


119877119904= [

2Wi (1 minus 120573) 1205742

2 120574]

119908

= (1 minus 120573)Wi 120574119908 (15)



119904= 27Wi







119910119910 where



of 120591119909119910







119910119910is 119899119910120591119910119910 where 119899

119910is vertical













x

minus4

minus3

minus2

minus1p

0

1

2

3

M4

M6Claus (P = 3)

(a) 119901


x

0

5

10

120591x

x 15

20

25

30

M4

M6Claus (P = 3)

(b) 120591119909119909


x

minus4

minus3

minus2

minus1

0

120591y

y

M4

M6Claus (P = 3)

(c) 120591119910119910


x

minus8

minus6

minus4

minus2

0

2

4

120591x

y

M4

M6Claus (P = 3)

(d) 120591119909119910


120591y

y

minus4

minus2

0

2

4

6

8

10


Wi = 01

Wi = 02

Wi = 05

Figure 3 120591119910119910


Sr

04 06 08 1 12 14 1602Wi

1

12

14

16

18

2

22






18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1







of 119880119909




1just









1 whichwould


4 Conclusion



0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials








119878119877= 019 + [1 +

1

12(1198731

120590119909119910

)

119908

2

]

14

(13)

where 1198731= 120590119909119909

minus 120590119910119910





119878119877= 019 + [1 +

1

3119877119904

2

]

14

(14)


119877119904= [

2Wi (1 minus 120573) 1205742

2 120574]

119908

= (1 minus 120573)Wi 120574119908 (15)



119904= 27Wi







119910119910 where



of 120591119909119910







119910119910is 119899119910120591119910119910 where 119899

119910is vertical













x

minus4

minus3

minus2

minus1p

0

1

2

3

M4

M6Claus (P = 3)

(a) 119901


x

0

5

10

120591x

x 15

20

25

30

M4

M6Claus (P = 3)

(b) 120591119909119909


x

minus4

minus3

minus2

minus1

0

120591y

y

M4

M6Claus (P = 3)

(c) 120591119910119910


x

minus8

minus6

minus4

minus2

0

2

4

120591x

y

M4

M6Claus (P = 3)

(d) 120591119909119910


120591y

y

minus4

minus2

0

2

4

6

8

10


Wi = 01

Wi = 02

Wi = 05

Figure 3 120591119910119910


Sr

04 06 08 1 12 14 1602Wi

1

12

14

16

18

2

22






18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1







of 119880119909




1just









1 whichwould


4 Conclusion



0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





x

minus4

minus3

minus2

minus1p

0

1

2

3

M4

M6Claus (P = 3)

(a) 119901


x

0

5

10

120591x

x 15

20

25

30

M4

M6Claus (P = 3)

(b) 120591119909119909


x

minus4

minus3

minus2

minus1

0

120591y

y

M4

M6Claus (P = 3)

(c) 120591119910119910


x

minus8

minus6

minus4

minus2

0

2

4

120591x

y

M4

M6Claus (P = 3)

(d) 120591119909119910


120591y

y

minus4

minus2

0

2

4

6

8

10


Wi = 01

Wi = 02

Wi = 05

Figure 3 120591119910119910


Sr

04 06 08 1 12 14 1602Wi

1

12

14

16

18

2

22






18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1







of 119880119909




1just









1 whichwould


4 Conclusion



0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




18

17

16

15

14

13

12

11

1

Y

1 2 3 4 5 6 7 80X

Wi = 01

Wi = 05

Wi = 08

Wi = 1







of 119880119909




1just









1 whichwould


4 Conclusion



0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0

04 02

1214

1 1 112

14

04 04 04

02 04 06 08 1 12 14

minus2minus4 2 40X

0

1Y

2

Ux

060606

06

0808

1

1212

(a) Wi = 01

0

02

02 04 06 08 1 12 14

minus2minus4 2 40X

Y

2

0

1

Ux

206

2 12 1214

0404 0408

1214

1 106

0202 02 0206 06

12

1212

(b) Wi = 05

0 02 04 06 08 1 12 14

minus2 2 40X

Y

2

0

1

Ux

06 060202

12 1214 14 14

121 1

02 0406

06

08

08

06

12121

04

0404

(c) Wi = 08

0 02 04 06 08 1 12 14

Y

Ux

minus2 0 2 4minus4

X

0

1

2

0408

12

02 020408 08

1214 14

12

0404

0606 0606

12

1

1

14 14

1 0804

(d) Wi = 1

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0

1Y

2

0 0 0 0 0 02

01

02

02

02

02

0

(e) Wi = 05

Uy

minus2 0 2 4minus4

X

0 01 0302 04

0 0 0 0

02 02 010

0

Y 1

2

02 0

1

03

02 0

30 03

030201

02

(f) Wi = 1



N1

minus10

0

10

20

30

40

50

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10



04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




04 06 08 102Wi





1

12

14

16

18

2

22

Sr


100

101

102

120573 = 01

120573 = 05120573 = 09

120573 = 099

120578E

120578

10minus2 10minus1

120582 120576





N1

02 04 06 08 10Y

minus5

0

5

10

15

20

25

30

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(a) 120573 = 05

N1

minus2

0

2

4

6

8

10

12

14

02 04 06 08 10Y

Wi = 01

Wi = 05

Wi = 08

Wi = 10

(b) 120573 = 09


Acknowledgments


References

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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Research Article A New Method to Simulate Free Surface ...downloads.hindawi.com/journals/amse/2015/159831.pdf · Research Article A New Method to Simulate Free Surface Flows for Viscoelastic