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J. Non-Newtonian Fluid Mech. 165 (2010) 631–640

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

umerical simulation of the fountain flow instability in injection molding

.G.H.M. Baltussen, M.A. Hulsen ∗, G.W.M. Petersepartment of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

r t i c l e i n f o

rticle history:eceived 16 December 2008eceived in revised form 4 March 2010ccepted 9 March 2010

ACS:7.50.Cd7.50.Gj

a b s t r a c t

For the first time, the viscoelastic flow front instability is studied in the full non-linear regime by numericalsimulation. A two-component viscoelastic numerical model is developed which can predict fountainflow behavior in a two-dimensional cavity. The eXtended Pom-Pom (XPP) viscoelastic model is used.The levelset method is used for modeling the two-component flow of polymer and gas. The difficultiesarising from the three-phase contact point modeling are addressed, and solved by treating the wall as aninterface and the gas as a compressible fluid with a low viscosity. The resulting set of equations is solvedin a decoupled way using a finite element formulation. Since the model for the polymer does not contain asolvent viscosity, the time discretized evolution equation for the conformation tensor is substituted into

7.55.dr7.20.Gv

eywords:njection moldingountain flow

the momentum balance in order to obtain a Stokes like equation for computing the velocity and pressureat the new time level. Weissenberg numbers range from 0.1 to 10. The simulations reveal a symmetricfountain flow for Wi = 0.1–5. For Wi = 10 however, an oscillating motion of the fountain flow is found witha spatial period of three times the channel height, which corresponds to experimental observations.

© 2010 Elsevier B.V. All rights reserved.

iscoelastic instabilityinite element method

. Introduction

One of the most prominent flow effect which occurs in injectionolding is fountain flow [1], already observed by West in 1911 for aercury filled capillary [2]. This effect occurs during filling near the

olymer melt flow front. Much attention was paid to fountain flowinetics [3,4], since it plays an important role in melt solidificationnd molecular orientation [5,6]. Both influence the final productroperties.

The movement of the contact point is a difficult problem. In aagrangian or ALE framework the free surface is a moving boundaryf the domain and the contact is relatively easy handled by assum-ng, for example, sticky conditions (velocity of the fluid becomesqual to the velocity of the wall) if a node on the surface tries to crosshe wall. This is nicely described in a recent paper by Dimakopou-os and Tsamopoulos [7] (see also [8–10]). In a Lagrangian orLE framework the interface between the two phases (liquid andolid) is aligned with element boundaries. A drawback of these

pproaches is that elements can become too distorted and remesh-ng is required if there is a large deformation of the interface. Sincen injection molding very large deformation is achieved we inves-igate in this paper a different approach using an implicit interface

∗ Corresponding author. Tel: +31 40 247 5081; fax: +31 40 244 7355.E-mail addresses: m.g.h.m.baltussen@tue.nl (M.G.H.M. Baltussen),

.a.hulsen@tue.nl (M.A. Hulsen), g.w.m.peters@tue.nl (G.W.M. Peters).

377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2010.03.001

description in an Eulerian framework. For the interface descriptionwe use a levelset function. The approach is similar to [11], wherea discontinuous color function has been used instead of a levelsetfunction. Since the interface position is implicit, the free surface isnot a boundary of the domain and we need to consider both the fluiddomain and the outside domain. The latter is modeled as a fluid witha low viscosity. An immediate drawback of this approach is thatthe contact point is now a three-phase contact point in an Eulerianframework with all it’s numerical difficulties, such as possible gasinclusion due to inaccurate description of the interface. However alarge deformation limit due to excessive element distortions doesnot exist.

During injection molding several defects can occur, one of thembeing alternating rough and dull bands on the surface of a finishedproduct. The reason for this type of defect to occur is the time peri-odic asymmetric motion of the fountain flow [12–14], which hasbeen shown to be a viscoelastic instability [15–17]. Bogaerds et al.[17] studied the linear stability of a viscoelastic fluid at relativelylow Weissenberg numbers and hence only predicted the criticalconditions for which the flow becomes unstable. The non-linearmotion of the flow cannot be described by this analysis. Deforma-tion rates (�̇) in injection molding are very high, 104 s−1, and the

typical time scale related to the viscoelastic material is �1/(2�),which is of the order of 10−2 s. This leads to Weissenberg numbersof O(100). With the development of the log-conformation methodby Fattal and Kupferman [18] and the implementation using a finiteelement method by Hulsen et al. [19] and Yoon and Kwon [20], the

632 M.G.H.M. Baltussen et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 631–640

Fig. 1. The flow region near the flow front observed in a fixed reference frame (a)aotf

rdobPt

saoetfbmwS

2

2

ciIt

tp��ofla

Fb

nd moving reference frame (b). Streamlines are only drawn in the polymer regionf the domain. In this figure, U is the piston velocity, x′ and y′ are coordinates inhe moving reference frame, whereas x and y are coordinates in the fixed referencerame.

ange of attainable Weissenberg numbers has grown by one or twoecades, in some benchmark problems. In addition to the devel-pment of numerical methods, advanced constitutive models haveeen developed for polymer melts, amongst which the eXtendedom-Pom model (XPP) [21]. This model gives a good description ofhe rheology in shear as well as in elongation [21,22].

The aforementioned progress in numerical techniques and con-titutive models has opened the road to numerical simulationst flow conditions which are closer to processing conditions. Thebjective in this paper is the simulation of non-linear viscoelasticffects during the filling of a simplified injection molding geome-ry, including viscoelastic instabilities. The paper is structured asollows. First, the problem domain and governing equations wille given in Section 2. Next, the numerical implementation of thisodel will be discussed in Section 3 and the simulation resultsill be presented in Section 4. Finally, conclusions will be given in

ection 5.

. Model

.1. Geometry

The filling of a two-dimensional channel is modeled. Initially thehannel is partly filled with a rectangular block of polymer, whichs moved by a piston into the empty part of the channel, see Fig. 1a.n order to avoid problems with modeling a moving flow domain,he piston velocity is subtracted from the velocity field, see Fig. 1b.

Note that two singularities are introduced in the flow field athe corners of the piston due to the jump in velocity between theiston and the wall. Two subdomains are present: the melt domainm and the gas domain �g, see Fig. 2, with the piston boundary

p, the moving wall of the polymer domain �wp, the moving wallf the gas domain �wg, an inflow boundary of the gas �gin and theow front �f. The position of the flow front is part of the solutionnd not known a priori.

ig. 2. The domain with the polymer and gas subdomains and the appropriateoundaries.

Fig. 3. Schematic plot of the streamlines of both the polymer (left) and gas (right)for an incompressible flow, with stick boundary conditions. The curve from c to d isthe flow front, c and d are the contact points and a and b are stagnation points. Twovortices can be seen near the flow front in the gas domain.

2.2. Modeling of the gas domain

The approach adopted is to model the gas domain as a (very)low viscosity viscous fluid. As was pointed out by Dussan [23], theflow of two incompressible fluids with a different viscosity and stickboundary conditions leads to a streamline pattern shown in Fig. 3.At the flow front two vortices are observed in the low-viscosity (gas)domain. Gas traveling from the right to the contact point (c) cannotleave to the left and therefore has to leave to the right causing thevortices.

The vortices result in large velocity gradients near the interface,which can lead to large stresses. If these large stresses are of thesame order of magnitude as the stresses in the melt, the shape ofthe flow front changes due to the stresses in the gas. Thereforethe polymer/gas viscosity ratio should be taken sufficiently large.Different solutions to the undesired motion of the flow front arepossible and all aim to lower the undesirable stresses. Recall thatthe reason for the vortices are the stick boundary conditions incombination with the incompressibility condition. By allowing slipat the gas part of the wall [11], the vortices disappear, see Fig. 4.

This is probably another reason for the success of the slip bound-ary condition in mold filling simulations. The downside of thismethod is that the exact position of the contact point has to beknown, and that the type of the boundary condition is positiondependent. Both of them are non-trivial problems. Another optionis to model the gas as a compressible fluid, also leading to removal ofthe vortices, see Fig. 4 (b). It should be noted that the gas domain isfictitious and it is merely present to allow the flow front to developand move as a traction-free surface, while not causing undesiredflow front motion by developing too high stresses. In this work thegas domain will be modeled as a compressible (fictitious) viscousfluid.

2.3. Governing equations

The flow instabilities which are studied are known to be of a vis-coelastic nature and are not thermally induced [17]. Therefore theproblem is modeled as an isothermal flow. Due to the high viscos-ity of polymer melts, the viscous forces dominate the problem andinertial and gravitational effects are negligible as well. The melt ismodeled as an incompressible viscoelastic fluid. The gas is modeledas a compressible viscous fluid, as discussed in the previous sub-section. This results in the following momentum and continuity

equations:

−∇ · � = 0 in � (1)

∇ · u = 0 in �m (2)

M.G.H.M. Baltussen et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 631–640 633

F ditionc

w�f

�

�

wiufl

wtdt[[Xauar

�

w

w

f

Ho˛sasot

2

vsaaw

ig. 4. Schematic representation of the streamlines in case of (a) slip boundary conontact points.

here � is the Cauchy stress tensor, u is the velocity vector and � =m ∪ �g. The constitutive equations for the Cauchy stress tensor

or the polymer and gas domain are:

= −pI + � in �m (3)

= −pI + 2�gD, with �g∇ · u + p = 0 in �g (4)

here � is the viscoelastic extra-stress tensor, p is the pressure, �g

s the gas viscosity, D is the rate of deformation tensor and I is thenit tensor. Note, that the gas is modeled as a compressible viscousuid with a bulk viscosity equal to the shear viscosity.

For the polymer, an appropriate constitutive relation is neededhich captures the flow behavior in both shear and elongation. In

he past decade several models have been proposed which can pre-ict the polymer flow behavior quantitatively. A popular model ishe eXtended Pom-Pom (XPP) model [21], adapted by van Meerveld24]. The XPP model has been derived from the Pom-Pom model25] which is developed for branched polymers (such as LDPE). ThePP model solves problems of the original Pom-Pom model, suchs a zero second normal-stress difference in shear and a discontin-ous stress response in elongation. The model can be presented insingle mode form in terms of the conformation tensor c, which is

elated to � as follows:

= G(c − I) (5)

here G is the modulus. The evolution equation for c is given by

∂c

∂t+ u · ∇c − (∇u)T · c − c · ∇u + f rel(c) = 0 (6)

here f rel is a non-linear relaxation tensor given by:

rel = 2exp(�(

√trc/3 − 1))

s

(1 − 3

trc

)c

+ 1b

(˛c · c + 3

trc

[1 − ˛ − ˛

3tr(c · c)

]c + (˛ − 1)I

)(7)

ere, � = 2/q with q representing the number of arms, s is therientation relaxation time, b the backbone relaxation time andis the anisotropic slip parameter. The ratio r = b/s is a mea-

ure for the number of tube segments or entanglements. Since thellowable range for the parameter ˛, which mainly controls theecond-normal stress difference in the XPP model, is a topic ofngoing discussion [26,27], it is set to zero. A detailed analysis ofhe allowable range of ˛ is presented in a recent paper [28].

.4. Scaling

The two dimensionless numbers governing this problem are the

iscosity ratio of melt and gas R = bG/�g = �m/�g and the Weis-enberg number Wi = bU/2H, U is the velocity of the moving wallnd H is the height of the channel. All results will be presented indimensionless form, where velocities are scaled with U, lengthith H and time with H/U.

at �wg and (b) compressible gas. Here a is the stagnation point and c and d are the

2.5. Position of the interface

The interface between polymer and gas is computed with thelevelset method [29,30]. It is a reliable, implicit method used intwo-component flow simulations of Newtonian liquids [31] as wellas non-Newtonian liquids [32,33], for which an additional partialdifferential equation has to be solved in the entire domain:

∂

∂t+ u · ∇ = 0 in � (8)

where is the levelset function, which represents the signed dis-tance to the interface, so = 0 is the interface. The sign of determines the subdomain: > 0 is the polymer domain, < 0is the gas domain. Initially is set to the signed distance. However,due to flow does not remain a signed distance function. There-fore a reinitialization step is often applied [30], which regeneratesthe signed distance function. Since the governing equations of ourproblem only depend on the sign of , the reinitialization step willnot be performed in this work.

2.6. Contact point modeling

The three phase contact point where the flow front touchesthe channel walls is difficult to model [34]. If surface tension canbe neglected, which is the case for a melt/air interface [35], thecontact angle is 180 degrees, so the flow front approaches thewall tangentially. The flow behavior at the contact point cannot bedescribed by the macro-scale physics used in this paper [36,37]. Themotion of this point governs the flow field however. Therefore thefront shape, including the contact point, was either assumed semi-circular [3], taken as an extra unknown [34,38] or an ALE schemewas used for moving the entire fluid mesh [8,9,7]. For a semicircu-lar flow front the x-position of points on the flow front is given by

x(y) − xc =√

(H/2)2 − (y − (H/2))2 ≈ √H

√y for y/H � 1, where xc

is the position of contact point. The square root shape near y = 0cannot be described accurately by the levelset function , sinceit will have a bi-linear approximation in a single finite element.This results in non-attaching flow: the contact point will eventu-ally leave the domain and a thin gas layer will remain betweenthe polymer and the wall. This is an unacceptable numerical solu-tion and the problem can be fixed as follows. By setting the levelsetfunction to zero, = 0, at the walls and thus treating the walls as aninterface too, the flow does attach to the wall. Due to the bi-linearshape function and setting two values of the levelset function tozero, the = 0 contour is a line perpendicular to the wall in theelement attached to the wall, see Fig. 5. The real flow front is thedashed curve and the solid curve is the flow front due to setting = 0 at the wall. The sign of the levelset is given in each of the

nodes of the three elements shown.

After setting = 0 on the wall a certain amount of gas is trans-formed into polymer (the shaded region in Fig. 5). In the actualimplementation there is a continuous transformation of a smallamount of gas into polymer even if the flow is steady. This can be

634 M.G.H.M. Baltussen et al. / J. Non-Newtoni

Fti

etttomemtmgamaF

ceii

ctTa

Fw

ig. 5. The levelset approximation (solid) of the real flow front (dashed) after settinghe wall to = 0. The dashed area between both the curves is transformed from gasnto polymer.

xplained as follows. The wall moves to left, hence the interface inhe element attached to the wall will also have moved slightly tohe left at the end of a time step. The value of on the wall is reseto zero after every time step, which means that a small amountf gas is continuously being transformed into polymer. As a result,ass conservation of both individual phases are not fulfilled. This

ffect is mesh dependent, but converges to zero mass loss withesh refinement. In any case, the result is that the flow front starts

o move to the right with a small velocity, depending on the refine-ent of the mesh. To avoid this and keep the amount of melt and

as constant, melt is allowed to flow out of the domain. In this waysteady state flow front position is obtained. The melt leakage isodeled at the point where the piston touches the wall. By takingquadratic outflow profile, the velocity gradients are smooth, see

ig. 6.The width over which outflow occurs is fixed at 0.1H at both

orners, and the velocity at the wall is taken such that the flux isqual to the fluid which changes phase per unit time. The gas thats transformed into polymer at the contact point has no history ands therefore assumed to be stress-free, i.e. c = I.

In addition to the smooth change of the velocity towards theorners of the piston, a similar transition from the wall velocity −U

o the corner velocities is imposed on the wall towards these points.he same width of 0.1H is used. In this way, the singular behaviort these corner points is eliminated.

ig. 6. The quadratic leak flow outflow profile. The magnitude of the velocity at theall depends on the amount of fluid which changes phase.

an Fluid Mech. 165 (2010) 631–640

3. Numerical methods

The resulting set of equations is solved using a finite elementmethod. Since the problem is non-linear and time dependent, lin-earization and integration techniques will be used to solve the set ofequations. In addition the equations are coupled and solving themsimultaneous leads to large systems which require large CPU timesto solve [39]. Therefore an operator splitting scheme is used wheresubsets of equations are solved in a decoupled manner.

The basic procedure is a time-stepping scheme in which thecurrent time tn and next time tn+1 (the one be solved) is defined forn = 0, 1, . . .. The time step �t = tn+1 − tn is taken to be a constant,but this is not a requirement. The quantities in the system at thediscrete times will also be denoted with a superscript, for examplecn+1 = c(tn+1).

3.1. Implicit stress formulation

The constitutive equation for the polymer (melt) domain doesnot have a viscous contribution (solvent). This is a problem forthe momentum balance and continuity equation of the polymerdomain at the new time tn+1 (substitute Eq. (3) into Eq. (1):

−∇ · �n+1 + ∇pn+1 = 0 in �m (9)

∇ · un+1 = 0 in �m (10)

which cannot be decoupled from the constitutive equation to find(un+1, pn+1). Since a fully coupled procedure would be very expen-sive, the route given by Bogaerds et al. [40] for the linearizedequations and by DAvino et al. [41] for the full non-linear equationsis followed here. The idea is that first the constitutive equation Eq.(6) in the strong form is discretized in time:

cn+1 − cn

�t+ un+1 · ∇cn − (∇un+1)

T · cn − cn · ∇un+1 + f rel(cn) = 0

(11)

The time-discretization scheme is a first-order semi-implicit Eulerscheme where the velocity vector is taken at the new time andthe conformation tensor at the current time. From this equation anexpression for cn+1 can be obtained and, with Eq. (5), an expres-sion for �n+1. The latter is substituted into Eq. (9), the momentumbalance for tn+1, and the following system is found:

−∇ · (G�t[−un+1 · ∇cn + (∇un+1)T · cn + cn · ∇un+1]) + ∇pn+1

= ∇ · (G[cn − �tf rel(cn) − I]) in �m (12)

∇ · un+1 = 0 in �m (13)

This is a Stokes-like system, linear in (un+1, pn+1), from which(un+1, pn+1) can be computed. A weak form of this system can easilybe obtained: find (un+1, pn+1) such that

T n+1 n n+1 T n n n+1

((∇v) , G�t[−u · ∇c + (∇u ) · c + c · ∇u ])

− (∇ · v, pn+1)

= −((∇v)T , G[cn − �tf rel(cn) − I]) (14)

(q, ∇ · un+1) = 0 (15)

for all test functions v and q.

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mtl

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fTdpm

abTt

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is(

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M.G.H.M. Baltussen et al. / J. Non-Ne

For the gas domain a compressible Stokes system is solved: findun+1, pn+1) such that

(∇v)T , �g[(∇un+1)T + ∇un+1]) − (∇ · v, pn+1) = 0 (16)

q, �g∇ · un+1) + (q, pn+1) = 0 (17)

or all test functions v and q.For the trial and test functions of the velocity/pressure, Taylor-

ood Q2/Q1 interpolation is used on quadrilateral elements.Note, that the systems for the polymer and gas domain are build

ogether and during the integration it is decided which equation issed based on the sign of the levelset function . Also note, that noEVSS stabilization [42] has been applied.

.2. Stabilization of the constitutive equation

Once a solution for (un+1, pn+1) has been obtained, the confor-ation tensor needs to be updated to the new time level tn+1. For

his the constitutive equation needs to be solved, however stabi-ization is required here to obtain a smooth solution.

First the velocity gradient (∇u)T is not used directly in the con-titutive equation as written in Eq. (6). It is replaced by a “smoothed’radient G that is found by projection [43]: find G such that

H, ∇uT − G) = 0 (18)

or all test functions H. Once un+1 is known, Gn+1 can be obtained.he replacement of (∇u)T by G gives stabilization in time-ependent shear flows. For the trial and test functions of therojected gradient, Q1 interpolation is used on quadrilateral ele-ents.Exponential profiles near points where the velocity is zero, such

s stagnation points, can generate numerical instabilities. This cane resolved by using the log-conformation formulation [18–20,44].his formulation basically means rewriting the constitutive equa-ion Eq. (6) into a form using s = log c:

∂s

∂t+ u · ∇s = g(G, s) (19)

nce, s has been computed, the conformation tensor can be com-uted by c = exp s.

Finally SUPG [45] has been used to stabilize the convection termn the constitutive equation Eq. (19). The weak form reads: find such that

d + �u · ∇d,∂s

∂t+ u · ∇s − g(G, s)

)= 0 (20)

or any test function d. Here, � = h/2U, where h is a typical size ofhe element in the direction of the velocity and U is a characteris-ic velocity magnitude. A semi-implicit first-order time-integrationill be used and the final time-discretized weak form becomes: find

n+1 such that

d + �un+1 · ∇d,sn+1 − sn

�t+ un+1 · ∇sn+1 − g(Gn+1, sn)

)= 0

(21)

or any test function d. For the trial and test functions of the con-ormation tensor, Q1 continuous interpolation has been used on

uadrilateral elements.

The conformation tensor equation is only valid in the polymeromain. In practice the equation for the conformation tensor isolved in the complete domain. However after each time step, theonformation tensor in the gas domain is set to c = I (or s = 0).

an Fluid Mech. 165 (2010) 631–640 635

3.3. The levelset equation

As a final step in the time-stepping scheme the level set function is updated to the new time level tn+1. The levelset Eq. (8) has beendiscretized in time using a Crank–Nicolson scheme:

n+1 − n

�t+ 1

2(un+1 · ∇n+1 + un · ∇n) = 0 in � (22)

For spatial discretization an SUPG stabilization is employed. Finallythe weak form can be written as: find such that(

r + �un+1 · ∇r,n+1 − n

�t+ 1

2(un+1 · ∇n+1 + un · ∇n)

)= 0

(23)

for all test functions r, where � is the same as for the constitutiveequation. For trial and test functions of the levelset, Q1 interpolationhas been used on quadrilateral elements.

3.4. Spatial integration

The constitutive equation for the fluid stress differs for the poly-mer and gas domain. The interface between the two domains issharp, so large jumps in the conformation tensor and velocity gra-dients will occur. Since the weak form has to be integrated over theentire domain, a proper numerical integration scheme is needed.Standard Gauss-quadrature is too inaccurate for functions havingjumps. Therefore a 10x10 point composite midpoint integrationrule is used in the elements which are intersected by the interface.

4. Results

Two types of simulations are carried out. In one set the poly-mer melt is modeled as a Newtonian liquid with a viscosity ratioR. This first set is used to find appropriate viscosity ratios and tocheck for mesh convergence for simulating fountain flow kinetics.In the second set, the melt is modeled as a viscoelastic fluid and thefountain flow dynamics are studied. In both cases the length L ofthe domain is six times the height H and the initial interface is flatand located at x = 3.5H. The time step �t has been set to 10−2H/Ufor all simulations.

4.1. Newtonian flow

The ratio of melt and gas viscosities R is important, sinceit determines the influence of the lower viscous fluid on theinterface shape. From experiments it is known that the interfaceshape should be about semi-circular when stationary [46]. ForR = 10, 100, 1000, 10000 the flow front shapes and particle pathsat t = 2H/U are given in Fig. 7. The particle paths are calculatedat the final time, by inserting 20 particles at x = 2H (distributedbetween y = 0.25H and y = 0.75H) and convecting them passivelywith the velocity field at that time step. Therefore, they can beinterpreted as streamlines. Note, that R also determines the timeuntil the front becomes stationary, since for low R streamlines crossthe interface indicating interface movement. For large R > 100 thefront becomes stationary at about t = H/U and the flow field showsthe characteristic fountain flow. For R ≥ 1000 the flow front itselfis a streamline.

Is is observed that the flow front profile becomes semi-circularfor R ≥ 1000. Therefore a viscosity ratio of 1000 will be used, from

now on. Note, that the flow front seems to be not perpendicular tothe wall at the contact point, as was stated in Section 2.6. This isa result of the post-processing. In the simulation the flow front isperpendicular to the wall in the element at the wall. In Fig. 8 theV-shaped typical folding of contour lines of the levelset scalar are

636 M.G.H.M. Baltussen et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 631–640

le paths at t = 2H/U for different viscosity ratios.

sa

mtti

aa

Fig. 9. The coarsest mesh M1, the initial interface is located at the center of the

Fig. 7. The shape of the flow front and partic

hown, they compare well with results from, for example, Coyle etl. [4].

Next the mesh convergence of the problem is considered. Threeeshes (M1,M2,M3) are used, see Fig. 9. In the flow front region

he mesh is refined. The amount of elements in this region, theotal number of elements and the total number of nodes are given

n Table 1.

The shape of the upper half of the flow front is given in Fig. 10 forll three meshes at t = 5H/U. At that time the shape is stationary,nd similar for all three meshes. The largest difference can be seen

Fig. 8. Typical deformation pattern of the levelset scalar , t = 2H/U.

square central region of the mesh. The domain size (x, y) = (6H, H).

Table 1Mesh parameters. Nx , Ny are the number of elements in the central region in the x-and y-direction, respectively.

Mesh (Nx, Ny) Total number of elements Number of nodal points

M1 (26, 26) 2252 9257M2 (50, 50) 5276 21413M3 (76, 76) 9828 39677

near the contact point, where a finer mesh gives a better approx-imation of the semi-circular shape. M2 and M3 are converged forthe rest of the flow front. In Fig. 11 the time evolution of the veloc-

ity in x-direction is given at (x, y) = (3.5H, 0.5H). This point is atthe interface at t = 0 and becomes part of the polymer domain fort > 0. From Fig. 11 it can be observed that the x-velocity is station-ary and equal for the two finest meshes, indicating that the solution

Fig. 10. The shape of the upper half of the flow front for meshes M1, M2 and M3 att = 5H/U.

M.G.H.M. Baltussen et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 631–640 637

F

hltpnefd

4

etousFesfbira

F(F

ig. 11. Velocity in x-direction for meshes M1, M2 and M3 at (x, y) = (3.5H, 0.5H).

as converged, so mesh M2 will be used from now on. Small oscil-ations are present in all three solutions. These are explained byhe varying leak flow of gas into the polymer domain at the contactoint. As the amount of leak flow is determined by the mesh sizeear the contact point, the amplitude of these oscillations is alsoxpected to be lower for a finer mesh, which is the case, exceptor 1.5H/U < t < 2H/U. The reason for the relatively large jumpsuring this period is unknown.

.2. Viscoelastic flow

Viscoelastic flow simulations have been carried out using differ-nt Weissenberg numbers. In order to perform these simulationshe non-linear parameters have to be chosen. Some combinationsf relaxation times and non-linear viscoelastic parameters result innphysical behavior in shear and/or elongation, such as too muchhear thinning and maxima in the steady state shear stress. Fromig. 12 the steady state shear response of the XPP model for differ-nt parameter sets is shown. It can be seen that the steady statehear stress shows a maximum and minimum in the shear stressor r = (4, 6) and q = (5, 9, 13). This is considered as unphysical

ehavior, since polymer melts in general show a shear stress that

s continuously increasing or at least constant for increasing shearates. Shear flow with (r, q) = (2, 5) does not show this behaviornd therefore this parameter set will be used.

ig. 12. Steady state shear viscosities for single mode XPP, for r = 2 (top), r = 4middle, shifted down one decade), and r = 6 (bottom, shifted down two decades).or each relaxation time ratio four different q are given.

Fig. 13. The flow front shape for Wi= 0.1, 1.0, 2.5, 5.0, 10.0, for XPP. The greyunsymmetrical shape in the right figure is for Wi= 10.

With the material properties fixed, first the shape of the flowfront is investigated for different Wi, since from [17] it is knownthat for a Weissenberg number that is lower or higher than unity theflow front shape will be flattened. In Fig. 13 the flow front shape isshown for Wi = 0.1, 1.0, 2.5, 5.0, 10.0 at t = 50H/U. In agreementwith Bogaerds et al. [17], both for lower and higher Wi than unitythe flow front is flattened. For Wi= 10, the flow becomes unstable,hence an asymmetric flow front is observed.

4.3. Unstable fountain flow

The unstable motion of the flow front is further investigatedby computing the behavior of uy at the center line of the channel.For stable flow, this velocity component is zero everywhere on thecenter line. Any deviation from zero, is an indication that the flowis asymmetric. In order to initialize the possible slow growth of theasymmetric flow, a random disturbance of order 10−3 is added tothe initial fully relaxed conformation tensor, similar to [47]. Thevelocity at (x, y) = (3.5H, 0.5H) is shown. The results are given forWi= 0.1, 1.0, 2.5, 5.0, 10.0, see Fig. 14.

The flow of XPP is stable for Wi= 0.1, 1.0, 2.5, since uy onlyshows numerical noise due to the varying leak flow. For Wi= 5.0,the flow is stable too, but the appearance of two entrapped “fic-titious gas” bubbles causes a asymmetrical motion at t = 33H/U,which disappears again afterwards. For Wi= 10.0, however, theflow is unstable, resulting in a oscillating velocity in y-direction.Notice that the mean velocity in y-direction is negative, indicatingthat the center line fluctuation is not symmetrical around the initialposition. At t = 26H/U, two spurious entrapped bubbles of the “fic-titious gas” appear in the melt, causing large jumps in the velocity iny-direction. After the “bubbles” are convected away from the flowfront region, the oscillatory behavior is regained. Since the oscillat-ing motion is present before the occurrence of the gas “bubbles”, thespurious gas is not the cause of the unstable motion. The growth ofthe instability is shown in more detail in Fig. 15, where the norm ofthe velocity in both the x- and y-direction are given. Due to the ran-dom disturbance to the initial conformation tensor, a disturbanceof 10−3U of the y-velocity at the centerline is present after the firsttime step. First a linear growth of uy is observed until t = 7H/U,after which the oscillating behavior shows up. It should be notedhere, that based on the center line velocity, the time needed forparticles to flow from the piston to the flow front is at least 10H/U.This is beyond the time the actual flow front dynamics sets in (seeFig. 15). Therefore it can be assumed that the piston is positionedfar enough to have negligible influence on the local flow frontdynamics.

The model shows high frequency oscillations in u for both

y

stable and unstable flow conditions. The amplitude of these oscil-lations decreases with increasing Wi. This is explained by therelatively large momentum gain from the artificial creation of meltdue to leak at the interface for low Wi. For higher Wi this momen-

638 M.G.H.M. Baltussen et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 631–640

the po

tm

w(tafrwwc

tdh

of

Fig. 14. The y-velocity at

um gain is relatively less important, since the polymer stresses areuch higher.The main frequency of the unstable motion is about 0.3U/H,

hich can be found from the power spectrum of uy in (x, y) =3.5H, 0.5H), see Fig. 16. The power spectrum gives the energy dis-ribution for all frequencies present in uy. Most energy is presentt low frequencies, with a maximum at 0.33U/H and at higherrequencies which are multiples of the lower frequencies. Thisesults in a typical length scale of the repeating motion of 3H,hich is in good agreement with experiments by Bulters et al. [16],here the repeating distance is three to ten times the height of the

hannel.In addition to the frequency, the direction of the motion dis-

urbance is important. In order to visualize the relatively small

isturbance, the difference in velocity between the upper and loweralf is shown in Fig. 17, at t = 21.4H/U.

From Fig. 17, the swirling motion can be observed in a large partf the melt region, also observed by Bogaerds et al. [17]. Near theront however, the flow is reversed. This can partly be explained

int (x, y) = (3.5H, 0.5H).

by the fact that the flow front is not symmetrical at this time.Therefore opposing points near the interface, can be of differentmaterial, which results in a comparison between the velocity in thegas with the velocity in the melt. Bogaerds et al. [17] performed alinear stability analysis, thus predicting the critical Wi and the mostunstable eigenmode of the flow. In this work all non-linear termsare taken into account and in addition to the linear growth of theinstability, the non-linear oscillating motion is observed. In [17]the critical Wi for (r, q) = (2, 5) is 2.5, the growth rate is very smallhowever. Although simulations of Wi > 2.5 have not been per-formed by Bogaerds for this particular parameter set, it is expectedthat the flow at higher Wi will be unstable too. In our study onlyWi= 10 shows the unstable motion. A reason that the critical Wifor the non-linear computations is higher than for the linear insta-

bility simulations might be the different modeling of the interface,including the contact point, using a level set function compared tothe height function used in [17]. Further investigation of the com-parison between the linear and non-linear instability will be partof future work.

M.G.H.M. Baltussen et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 631–640 639

Fig. 15. The evolution of both ux (top) and uy (bottom) at (x, y) = (3.5H, 0.5H),Wi= 10.

Fig. 16. Power spectrum of uy at (x, y) = (3.5H, 0.5H) for 10H/U < t < 50H/U.

Fig. 17. The difference between velocity in the upper and lower half of the channelat t = 21.4H/U.

Fig. 18. Particle paths of 20 particles at t = 21.4H/U.The flow front is the right curve.

In order to illustrate the asymmetrical fountain flow, the pathsof 20 particles are shown in Fig. 18. The asymmetrical motion ofthe fountain flow is clearly visible.

Due to the large deformations during one simulation (materialmakes approximately 3–4 cycles from piston to flow-front), thegradients in the levelset function become large and cannot be cap-tured well anymore. This leads to interface generation, especiallynear the contact point and near the piston corners. A likely solutionto this problem is the reinitialization of the levelset function to asigned distance function at regular intervals.

5. Conclusions

The viscoelastic flow front instability has been studied bynumerical simulation. A two-phase viscoelastic model in twodimensions has been developed which predicts a fountain flowinstability and is able to monitor this instability in the full non-linear regime. The modeling of the three phase contact point hasbeen addressed and solved by modeling the gas as a compressibleviscous fluid and treating both flow front and wall as an interface.An operator splitting scheme is applied in such a way that a sta-ble set of equations is obtained without the need for additionalviscous solvents. The final set of equations is solved with a finiteelement method. The discontinuous properties over the interfaceare captured well, by applying local spatial refinement of the inte-gration points. At Wi = 10 a periodic motion of the flow front isobserved with a spatial period of three times the channel height,which is in agreement with experiments. This is an extension of themodel developed by Bogaerds et al. [17], where only the onset ofthis instability was studied.

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