linear stability of three-layer poiseuille flow for oldroyd-b fluids

Linear stability of three-layer Poiseuille flow for Oldroyd-B fluids

SteÂphane Scotto, Patrice Laure*

Institut Non-LineÂaire de Nice, UMR 6618 ± CNRS & UNSA, 1361 route des Lucioles, 06560 Valbonne, France

Received 10 February 1998

Abstract

The linear stability of three-layer plane Poiseuille flow is studied in the longwave limit and for moderate wavelengths. The

fluids are assumed to follow Oldroyd-B constitutive equations with constant viscosities and elasticities. We find that the jumps

of the Poiseuille shear rate at both interfaces which give the convexity of the Poiseuille velocity profile, allow us to determine

the longwave stability for Newtonian fluids. On the other hand, the stability of viscoelastic fluids is analyzed by using the

additive character of the longwave eigenvalues with respect to viscous and elastic terms. The stability with respect to moderate

wavelength disturbances has to deal with two different modes called `shortwave' (SW) and `longwave' (LW), according to

their values at zero wavenumber. The SW eigenvalues can become the most dangerous modes for large Weissenberg numbers

and their influences can be studied by means of shortwave analysis. Moreover, we point out that the longwave stability

analysis and convexity of the Poiseuille velocity profile allow us to determine the LW eigenvalues which are stable with

AMS: 76A10; 76E05; 35Q35

Keywords: Interfacial stability; Viscoelastic fluid; Multilayer flow

1. Introduction

In this paper we study the linear stability of plane Poiseuille flow of three superposed fluids. Werestrict our attention to temporal disturbances and assume that in each fluid the Oldroyd-B equationshold. This configuration is related to the industrial co-extrusion process which allows us to put togetherpolymers having different optical, mechanical and barrier properties. At certain operating conditions,wavy interfaces are observed and are classified as an interfacial instability. In the sequel, we study theirconditions of appearance in order to give simple tools allowing to prevent them.

From the theoretical point of view, the two-layer plane or concentric Poiseuille flow has beenextensively studied. The first work was done on Newtonian liquids [1], but the main contribution aboutthe co-extrusion of polymer can be found in [2±5]. The very comprehensive experiments made byWilson and Khomani [6±8] have showed that the theoretical approach based on the Oldroyd-B model

J. Non-Newtonian Fluid Mech. 83 (1999) 71±92

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* Corresponding author. Fax: +33-493-652517.

PII: S 0 3 7 7 - 0 2 5 7 ( 9 8 ) 0 0 1 4 2 - 6

gave rather good qualitative and quantitative results. So, all these studies are of major importance asthey have described the different destabilization mechanisms and have clarified the role of differentparameters. In this process, the Reynolds number is very small due to high viscosities of polymers. Theparameters controlling the stability of multi-layer flow are the viscosity ratios, the thickness of flow-rate and elasticity ratios. The interfacial tensions of individual layers are also small and do not play animportant role (they mainly prevent the appearance of shortwave instabilities).

The previous studies have demonstrated the nature of interfacial instabilities of two-layer polymersystems. However, a complete understanding of interfacial instabilities for three-layer flows has notbeen established. The stability of such flows is of practical interest as engineering applications involvemore than two layers. In particular, even if you want to put together only two different polymers, thereis always a third thin layer in order to stick them. Unfortunately, there are few theoretical andexperimental studies addressed to such a configuration. The experiments performed by Han and Shetty[9,10] for three- and five-symmetrical layers do not give simple results about stable and unstableconfigurations. More recently, Khomami and Ranjbaran [11] have examined stability of bothsymmetric and asymmetric three-layer plane Poiseuille flow of viscoelastic fluids and haveexperimentally shown that a thin, less viscous fluid adjacent to the wall stabilizes at longwave whileshort and intermediate wavelength disturbances are stabilized when the more elastic fluid is themajority component. The theoretical studies made by Anturkar et al for Newtonian fluids [12] andOldroyd-B fluid [13] are not complete. More recently Wilson and Rallison [14] have performedshortwave analysis of three-symmetrical layer, but they have been mainly concerned with the influenceof the concentration of polymer.

So, in this paper our main goal is to extend to three-layer flow, the results found in our previous paper[15] for two and three symmetrical layers and compare our recipes with the experimental observationsof Khomani and Ranjbaran [11]. In particular, under what conditions is the longwave analysis (thetemporal stability with respect to longwave mode) sufficient to forecast stability with respect tomoderate wavelength perturbations. In Section 2 we introduce the notations and the linear stabilityproblem. The longwave analysis is made in Section 3 for both Newtonian and Oldroyd-B fluids. Wecorrect mistakes found in the Anturkar et al. papers and show that the signs of the jumps of shear stressat both interfaces give the asymptotic stability for Newtonian fluids. In Section 4, stability with respectto moderate wavelength disturbances is analyzed for the configurations which are asymptoticallystable.

2. Governing equations

We consider the superposed flow of three distinct Oldroyd-B fluids through a parallel plate channelgeometry (Fig. 1) and we have to deal with two interfaces ys � ds � hs�x; t� with s � 1, 2. Each layer k

has the viscosity �k, the relaxation time �k and the polymeric component �k. Moreover, density isassumed to be the same in each layer (the effect of gravity is not studied).

The equations are standard [15] and are written by using the following scales for the space, velocity,time and stress,

d� � total height; U� � total flow rate

total height; t� � d�

U�; �� 2

72 S. Scotto, P. Laure / J. Non-Newtonian Fluid Mech. 83 (1999) 71±92

For each layer k, the momentum conservation equation, the incompressibility and the Oldroyd-B modelfor the stress tensor are

�kRe@uk

@t� �uk � r�uk

� ��rpk ÿ mk�1ÿ �k��uk � r � � k

r � uk � 0; (2)

�k�Wek

@t� uk � r�k

ÿruk � �kÿ �

k� ruT

� �� mk�k�ruk �ruT

k �; (3)

where the Reynolds number Re � �U�d�=�2, the three Weissenberg numbers Wek � �kU�=d�, thethree viscosity ratios mk � �k=�2 are scaled with respect to the second layer (or the inner layer). Theimmiscibility conditions can be written at each interface ys

@t� uk

@xÿ �k � 0;

and the boundary conditions

u1�0� � �1�0� � 0;

u3�1� � �3�1� � 0;

��u��ys� 0;

��ÿpI � m�1ÿ ��ru�ruT� � �p��ysn � ÿ2HSn;

where the symbol ��fys�� fs�ys� ÿ fs�1�ys� is the jump of the function f at the interface ys. The relevant

parameters are the two thickness ratios, �1 � d1=�d2 ÿ d1� and �3 � �1ÿ d2�=�d2 ÿ d1�, the twoviscosity ratios m1 and m3 (as �2 and m2 are equal to 1), the three Weissenberg numbers Wei andpolymeric components �i. In the sequel, we mainly consider the cases for which the Reynolds number

Fig. 1. Three-layer geometry.

S. Scotto, P. Laure / J. Non-Newtonian Fluid Mech. 83 (1999) 71±92 73

Re and the surface tension S are very small. The Poiseuille solution (labeled as U0) corresponds to flatinterfaces and to a parabolic velocity profile whose coefficients depend on viscosity and thicknessratios.

The Poiseuille profile is perturbed by two-dimensional periodic perturbations having the followingform,

f �x; y; t� � ~f �y� exp�iqx� �t�;where f represents any variable, q is the wavenumber in the flow direction, � the growth rate. Thegoverning equations and boundary conditions are linearized around the Poiseuille flow and give rise tothe Orr±Sommerfeld equation in each layer [15]. After discretization, we have to deal with ageneralized eigenvalue problem for the growth rate �. The temporal stability is governed by the sign ofthe real part of �; if it is positive, the flow is unstable whereas, it is stable for a negative value.

3. Longwave analysis

An asymptotic analysis is carried out for long wavelength disturbances (q! 0). The disturbances invelocity components are expressed in term of a stream function k. The eigenfunctions and theeigenvalues are expanded in the following form:

k � 0k � q1

k � � � � ; k � 1; 2; 3; (4)

hs � h0s � qh1

s � � � � ; s � 1; 2; (5)

� � ÿiq�c0 � qc1 � � � ��: (6)

These expansions are substituted into the whole system and terms of like order are grouped together[1,16]. The expansion of the eigenvalues is computed by means of the kinematic conditions, whereasthe stream functions k are solutions of `o.d.e'. More precisely, the term c0 is an eigenvalue of a realmatrix coming from the two kinematic conditions and there are two possible solutions. Then, for eachzero-order eigenvalue c0, a first-order eigenvalue c1 is deduced from a compatibility condition (orFredholm alternative) due to order 1 terms coming from the two kinematic conditions. Unlike Anturkaret al.'s assertion [12,13], the zero-order coefficients c0 can be complex, even for Newtonian fluids. Insuch a case, both c0 and �c0 are eigenvalues and the system is unstable due to the imaginary part of c0 or�c0. These events are rather interesting because it means that the configuration is unstable independentlyof the viscoelastic characteristics. Li [17] also found such a behavior for a plane Couette flow of threesuperposed layers of fluids with different viscosities. As explained by Taylor [18], Goldstein [19], Li[17], Cairns [20] and more recently Weinstein et al. [21], this instability occurs if both modes solutionshave the same frequency and wave energies with opposite signs. For three-layer Poiseuille flow, theseconditions mean that the two eigenvalues c0 are close together and the jump of the shear rate hasopposite signs (see Fig. 4) as shown in Fig. 2, that occurs for very specific values of m1, m3, �1 and �3.

If both eigenvalues obtained at zero-order are real, each c0 is written into the form,

c0 � f ��1; �3;m1;m3�with f real: (7)

Then each c1 is purely imaginary and has the form,

c1 � i Re J0��1; �3;m1;m3� � i�2We2��M�;1 ÿ 1�J1��1; �3;m1;m3� � J2��1; �3;m1;m3�� M�;3 ÿ 1�J3��1; �3;m1;m3��; (8)

where M�;i � ��iWei�=��2We2�. The stability is given by the sign of the imaginary part of c1. As notedin [15], the eigenvalue c1 is the sum of a Newtonian part (J0) and an elastic part (J1, J2, J3). Thesefunctions satisfy symmetry properties which correspond to interchange in the fluid layers 1 and 3.

Ji��1; �3;m1;m3� � Ji��3; �1;m3;m1� for i � 0 and 2; (9)

J1��1; �3;m1;m3� � J3��3; �1;m3;m1�: (10)

The methodology used in the sequel is the following: we fix the two values m1 and m3, then we look atthe stable areas in the plane (�1, �3). Thanks to symmetry properties, we can restrict our computations to

Fig. 2. Location where c0 are complex: (a) m1 � 0.5 and m3 � 2; (b) m1 � 2 and m3 � 4.

the three cases:

m1 � m3 < 1; m1 < m3 and 1 < m1 � m3

labeled respectively, as cases (i), (ii) and (iii)The particular case for which the two outside layers have the same rheological properties and size�� 1 � �3;m � m1 � m3;M� � M�;1 � M�;3�, has already been studied by Laure et al. [15]. In thiscase, the zero-order eigenvalues c0 are always real and are associated to two different modes; thevaricose mode which preserves the reflection symmetry and sinuous mode. Moreover, it is possible toget the analytical expressions of c0 and c1.

3.1. Longwave analysis for the Newtonian fluids

Assuming the zero-order eigenvalues c0 are both real, the stability with respect to longwavedisturbances is given by the sign of the two functions J0. An interesting property is that one of the twofunctions J0 has the same sign as the jump of shear rate ��U00�� of the Poiseuille profile at one of bothinterfaces. The other function J0 has the same sign as the jump of shear stress at the other interface. Inthis way, it is possible to associate at each asymptotic eigenvalue an interface, ys, so that J0 and ��U00��ys

have the same sign.This property has been already proved by Charru and Fabre [22] for two-layer problem and by Laure

et al. [15] for three-symmetrical-layer case. In the latter case �m � m1 � m2 and � � �1 � �3�, this signis simply given by m ÿ 1 and both functions J0 are positive for m > 1 and negative for m < 1. So, theasymptotic stable configurations correspond to the most viscous fluid sandwiched by a viscous fluid.

For a three-layer flow, it is not possible to derive rigorously such a result and we are only able toprove that if one of the two jumps is null then one of the two functions J0 is also null. This is also truefor Oldroyd-B fluids if the viscosity ratio of adjacent layers is equal to elasticity ratio (M�;1 � m1=m2 orM�;3 � m3=m2). Nevertheless, numerical computations allow us to check this property for0:01 � �1; �3 � 100 and 0:01 � m1;m3 � 100. In this way we get a simple criterion which easilygives the asymptotic stability for Newtonian fluids. The only knowledge of the Poiseuille profile givesthe sign of the jump of shear rate at both interfaces:

s1 � sign��U00��y1� � �m1 ÿ 1��m1m3 ÿ m3�

21 � m1��2

3 � 2�3��; (11)

s1 � sign��U00��y2� � �m3 ÿ 1��m1m3 � m3��2

1 � 2�1� ÿ m1�23�; (12)

and the configuration is stable if s1 and s2 are both negative. This means that the process is stable whenthe basic velocity profile is convex (Fig. 3(a) and (b)) and unstable if it is not convex (Fig. 3(c) and (d)).

Now, we have to determine the vanishing points of the two functions s1 and s2. The easiest methodconsists of fixing the values of m1 and m3 and to look for the solutions of equations s1 � 0 and s2 � 0 inthe plane (�1, �3). First, if m1 or m3 is equal to 1, we recover the two-layer case and so, one interfacialeigenvalue is neutrally stable [23] and the other one gives the condition obtained for a two-layer flow(��m3p � �3=�1ÿ �1� or

��m1p � �1=�1� �3�). Otherwise, the vanishing points of the function s1 belong

to the curve

�1 ��m1 � �m1=m3��2

3 � 2�3�q

; (13)

whereas for s2, one gets

�3 ��m3 � �m3=m1��2

1 � 2�1�q

: (14)

Finally, it is clear that when �1 and �3 go together to infinity, the two curves (Eq. (13) and Eq. (14)) donot intersect but tend to the same curve

m3=m1 � ��3=�1�2 (15)

which is the limit line of the vanishing points of the functions s1 and s2. In the plane (�1, �3) the curves1 � 0 is located below the line defined by (Eq. (15)), whereas the curve s2 is above (see Fig. 4).

Finally, we have to study the sign of the two functions s1 and s2 for three cases, m1 � m3 < 1;m1 < m3 and 1 < m1 � m3. The signs of s1 and s2 are given in Fig. 4 and the main results aresummarized below:

1. If the inner fluid is the most viscous (m1 � m3 < 1), there exists an area delimited by the curvess1 � 0 and s2 � 0 and centered around the line (Eq. (15)) (see Fig. 4(a)) for which the configurationis stable. Then, even if the most viscous fluid is sandwiched by another viscous fluid, the thicknessratio of the two external layers (�3 / �1) has to be of same order as the square root of the viscosityratio (m3 / m1).

Fig. 3. Poiseuille velocity profile for different parameter values: convex profiles (a±b) ; not convex profiles (c±d). (a) m1 �0.25, m3 � 0.5, �1 � 0.4, �3 � 0.8; (b) m1 � 0.5, m3 � 8, �1 � 0.4, �3 � 8; (c) m1 � 0.5, m3 � 8, �1 � 0.4, �3 � 1; (d) m1 � 4,

m3 � 8, �1 � 1, �3 � 1.

2. If the least viscous fluid is in the lower layer and the most viscous fluid is in the upper layer(m1 < 1 < m3), then the functions s1 and s2 are negative together in an area above the curve s2 � 0(Fig. 4(b)). The stable configuration corresponds to a flow with the most viscous fluid in the largerexternal layer.

3. If the inner fluid is the least viscous fluid (1 < m1 � m3), then stable areas never exist since s1 and s2

are never negative together (Fig. 4(c)).

We find that the results obtained in the cases (i) and (iii) extend naturally to those addressed to thesymmetrical three-layer case [15,23]. For the concentric co-extrusion [24], we get similar results as thearrangement is always unstable if the least viscous fluid is in the inner layer. In some ways, we alsorecover the rule mentioned in [11] about the stabilization of the longwave disturbances by placing a

Fig. 4. Signs of both jumps of shear stress s1 and s2: (a) m1 � m3 < 1, (b) m1 < 1 < m3, (c) 1 < m1 � m3. The symbol �i (resp.

ÿi) means that the sign of the jump of shear rate at the interface i is positive (resp. negative).

thin, less viscous layer adjacent to the wall. For example, if m1 < m3 < 1, the area which correspondsto small values of both �1 and �3 is stable (see Fig. 4(a)). In the same way if m1 < 1 < m3, the area whichcorresponds to large values of �3 and small values of �1 is also stable (see Fig. 4(b)).

Thanks to Fig. 4, it is easy to determine under which conditions can the influence of the smallestlayer be neglected and the stability of a three-layer flow comes down to analyze the stability of the twolargest layers. If the thickness of an external layer tends to zero (�1 � 0 or �3 � 0), it is not sufficient tolook at the shear rate jump between the two largest layers in order to determine the transition betweenstable and unstable region. In fact, for some viscous ratios the second jump with the smallest layerremains positive and then the approximation does not work as the configuration is always unstable.Finally, one finds that this approximation works well:

� if the most viscous fluid is in the inner layer (case (i)) and if the thickness of one of outer layers tendsto zero (�1 � 0 or �3 � 0).� if the inner fluid has an intermediate viscosity with respect to the two outer fluids (case (ii)), and if

the thickness of the least viscous layer tends to zero (e.g. m1 < 1 < m3 and �1 � 0).� If the thickness of the inner layer tends to zero (�1 and �3 �1), then the viscosities have to be in

increasing or decreasing order (case (ii)).

The three-layer configuration for Newtonian fluids has been already studied by Anturkar et al. [12].Unfortunately, there are some mistakes in this article as they did not find that there is never a longwavestability when the inner layer is less viscous than the two others layers. For example, Fig. 2(a) and (b)in [12] are false and they are corrected in Fig. 5(a) and (b). In order to make an easier comparison withthe previous results, we have used in these pictures the viscosity �i and the thickness ei of each layer i.In Fig. 5(a), the case for which the two outer layers have the same viscosities (� � �1 � �3) is analyzed.The dashed curve is the neutral curve for the two-layer case (e3 � 0). The plain curves are the neutralcurves of the three-layer case for different values of the thickness ratios (e3 / e1). As noted above, for�2 > �1, the configuration is always unstable. For �2 > �1 and the symmetric case (e3 � e1), theconfiguration is stable whatever the thickness of the inner layer e2 [15]. When the thickness ratio e3 / e1

decreases, there exists a curve delimiting the stable and unstable area. This curve is the solution ofequation s1 � 0. Finally, one recovers the two-layer case �e2=e1 �

��2=�1

pwhen e3 goes to zero.

In Fig. 5(b) the thickness ratio e3 / e1 between the third and the first layer is fixed and one determinesthe stable area for various viscosity ratios �3 / �1. The same kind of discrepancy with the Anturkar etal.'s results is found. First, the arrangement is never stable for �2 / �1 < min(1, �3 / �1) (case (iii)). Formin(1, �3 / �1) < �2 / �1 < max(1, �3 / �1) and max (1, �3 / �1) < �2 / �1, the configurations are,respectively, related to cases (ii) and (i). Then, the stability depends on the two thickness ratios e3 / e1

and e2 / e1. In the example plotted in Fig. 5(b), one finds that the configuration is always unstable for�2 / �1 < �3 / �1. Otherwise, the curve splitting unstable and stable areas is also given by the equations1 � 0.

3.2. Longwave analysis for Oldroyd-B fluids at zero Reynolds number

The coextrusion of polymers (Re � 0) for three symmetrical layers leads to a rather simple result[15]. The condition m < 1 is not sufficient enough and we have to study the function J2 � �M� ÿ 1�J1.One has found that if the outer fluid is the least elastic �M� < 1�, only the varicose mode candestabilize the flow and in such a situation, it gives the critical thicknesses ratio above which the flow

motion becomes unstable. On the contrary, if the outer fluid is the most elastic �M� > 1�, the sinuousmode gives the thickness ratio under which the Poiseuille flow is no more stable. Finally, if the outerfluid is more viscous (m > 1) and more elastic �M� > 1� than the inner fluid, the configuration is neverstable due to the sinuous mode.

In the section, we try to extend these results to the general three-layer case. As in the in previoussection, we have performed the stability analysis for the three cases mentioned above which depend onthe viscosity of each layer. Note that the case (ii) is not an extension of the three-symmetrical layercase. At present, we have to deal with the sign of the sum,

J� � �M�;1 ÿ 1�J1 � J2 � �M�;3 ÿ 1�J3 (16)

The sign of functions, J1, J2 and J3 is given in Figs. 6±8 for typical parameter values corresponding tothe three cases mentioned above. Without elastic stratification, the sign of functions J2 given inFig. 6(a), Fig. 7(a) and Fig. 8(a) shows that the stable areas found for Newtonian fluids are also stable.

Fig. 5. Neutral stability curves in the (e2 / e1 ÿ �2 / �1) plane: (a) almost symmetric case for various aspect ratios; (b) non-

symmetric case for various viscosity ratios m3 at aspect ratio e3 / e1 � 0.3. ei and �i are respectively, the thickness and the

viscosity of the layer i. With our notations it means that: e2=e1 � 1=�1, �2=�1 � 1=m1, �3=�1 � m3=m1 and e3=e1 � �3=�1.

Fig. 6. Neutral curves in the (�1±�3) plane; case for which the outer fluids are the less viscous (m1 < m3 < 1): (a) J2 � J0 (b) J1 (c) J3 for parameters values

m1 � 0.25 and m3 � 0.5. The symbols S or N mean that functions Ji are both negative, U that one (or both of these for J2) of functions Ji is positive and P that

both functions Ji are positive.

Fig. 7. Neutral curves in the (�1±�3) plane; case for which the less viscous fluid is on one wall whereas the most viscous is on the other wall (m1 < 1 < m3): (a)

J2 (b) J1 (c) J3 for parameter values m1 � 0.5 and m3 � 8. The symbols S and N mean that functions Ji are both negative, U that one (or both of these for J2) of

the functions Ji is positive and P that both functions Ji are positive.

Fig. 8. Neutral curves in the (�1±�3) plane; case for which the outer fluids are more viscous than the inner fluid (1 < m1 < m3): (a) J2 (b) Ji (c) J3 for parameters

values m1 � 4 and m3 � 8; The symbols S or N mean that functions Ji are both negative. U that one (or both of these for J2) of the functions Ji is positive and P

that both functions Ji are positive.

However, new stable regions occur due to elasticity. This is particularly remarkable when the inner fluidis the least viscous (Fig. 8(a)), as the configuration is always unstable for Newtonian fluid or in thethree-symmetrical layer case.

The action of elastic stratification is not very easy to explain. The only simple behavior concerns case(i). In fact, for m1 � m3 < 1, the stable domain due to viscous stratification is centered around the line�3 �

��m3=m1

p�1 and bounded by the cures s1 � 0 and s2 � 0. Fig. 6(b) and (c) shows that for �i > 1 a

part of this area remains stable if both M�;i > 1. Then, each outer layer is larger, less viscous and moreelastic than the inner layer.

For the other parameter values, there is no simple rule and we have to check directly the sign of thefunction J� for both interfacial eigenvalues in order to check if the elastic stratification does not destroythe stability due to viscous stratification.

The last case (iii) which is always unstable for Newtonian fluids can be stabilized if the most elasticfluid is in the largest layer. For example, if �1 > 1 and �3 < 1 then M�;1 > 1 and M�;3 < 1 and so on.

Finally, we can try to compare qualitatively these results with Khomami and Ranjbaran'sexperimental observations [11]. Note that these comparisons are rather roughly done as the viscosities andelasticities depend on the shear rate in the experiments. In fact, they have studied three different systems:

� the PP / HDPE / PP symmetric system which corresponds crudely to the situation m1 � m3 > 1;M�1� M�3

> 1 and �1 � �2.� the HDPE / PP / HDPE system is also symmetrical and corresponds to m1 � m3 > 1; M�1

� M�3< 1

and �1 � �2.� the HDP / PP / LLDPE system for which m1 < m3, M�1

< M�3< 1 and �1 � �2.

The stability of these systems with respect to longwave disturbances is given by looking at theirbehaviors for small wavenumbers. Experimental data are collected respectively, in Figs. 24, 29 and 30of Ref. [11]. For the first system, encapsulation prevents from getting experimental data for smallwavenumbers and comparison is not really possible (in this case theoretical results forecast that thissystem is unstable for longwave disturbance). Nevertheless, for all thickness ratios considered in theexperiments, the two last systems are stable for small wavenumbers. Therefore, the agreement betweenexperiments and our computations is rather good. For the HDPE / PP / HDPE system, the stability ismainly due to viscous stratification (the Poiseuille velocity profile is convex) whereas, for the HDPE /PP / LLDPE system it comes from elasticity (it is the stable area between the two curves s1 � 0 ands2 � 0 in Fig. 7(a)). However, this kind of comparison is not very easy to make because the growth rateand its first derivative tend to zero as the wavenumber decreases and therefore, a positive growth ratefor small wavenumbers would be difficult to detect experimentally.

4. Stability analysis for moderate wavelength

In the latter section, we have found the parameter values for which the three-layer Poiseuille flow isstable with respect to longwave disturbances. Nevertheless, longwave analysis is not sufficient to ensurestability with respect to some wavenumbers. Therefore, it is necessary to look at the evolution of thereal part of the growth rate � as a function of the wavenumber q. The numerical procedure is standard:each variable is discretized along the y-direction by using Chebyschev Polynomial; the boundaryconditions are taken into account by applying the Tau method [25]; the eigenvalue � is the solution of a

generalized eigenvalue problem solved by means of an Arnoldi decomposition coupled with an inverseiterative method [26,27].

As we are mainly concerned by the co-extrusion process, we assume in sequel that the Reynoldsnumber and surface tension are null. In fact, we have found in our previous paper [15] that stabledomains found for null Reynolds number remain stable for small Reynolds number. For example, in theexperiments made by the CEMEF and Elf ATOCHem [28], they are, respectively, about 10ÿ6 and 10ÿ3.The computations are made for two Weissenberg numbers We2 � .1 and 1 and for polymericcomponents �i � 0.9.

The previous studies performed for two-layer or three-symmetrical layer flow [14,29], have shownthat there are two kinds of interfacial eigenvalues: the first ones, labeled as LW eigenvalues, verify�(q � 0) � 0, and are obtained by longwave analysis; the second ones, called SW eigenvalues, verify�(q � 0) � ÿ1 / Wei, and can cross the imaginary axes for moderate or large wavenumber if theWeissenberg number is large enough. Examples of variation of all these interfacial eigenvalues with thewavenumber q are plotted in Fig. 9(a) for the three-symmetrical layers. As shown in this figure we haveto deal with six interfacial eigenvalues as for each mode ��q � 0� � 0;ÿ1=We1;ÿ1=We2�, we haveto deal with the varicose and the sinuous mode modes. In the general case, there are eight eigenvalues.The destabilization coming from the SW eigenvalues can be found by using shortwave analysis [30]. InFig. 9(a), the real part of SW modes starting from ÿ1, become positive for large values of q. If thecomputations are made for small Weissenberg numbers, the destabilization due to these SW eigenvaluesis not observed [2].

As we have seen that the influence of SW eigenvalue can be evaluated by means of shortwaveanalysis, we focus our further discussion on the two LW eigenvalues. In our previous paper [15] aboutthe two-layer or three symmetrical layer Poiseuille flow, we have pointed that in some circumstances,longwave analysis allows finding of stable configurations with respect to all wavenumbers. This occurswhen the elasticity does not remove stability due to viscous stratification. We want to check if thisresult can be extended to three-layer flow. With the notation introduced in the previous section, thiscondition means that for each LW eigenvalue the functions J0 and J� � �M�;1 ÿ 1�J1 � J2��M�;3 ÿ 1�J3 are both negative. Under this condition, we expect that the real part of the LWeigenvalues remains negative for larger values of the wavenumber q.

In Fig. 9, all possible behaviors are illustrated according to the sign of the functions J0 and J�:

� In Fig. 9(a), J0 and J� are negative for both eigenvalues and the real part of the LW eigenvalues isalways negative for larger values of q.� In Fig. 9(b) the LW eigenvalue with negative J0 and J� has a negative real part for larger q whereas

this, with positive J0 and J�, has always a positive real part.� In Fig. 9(c), the two eigenvalues are asymptotically stable and have negative functions J0. The real

part of one eigenvalue remains negative whereas, the other one has a positive real part for larger q.The destabilization due to the latter eigenvalue can be detected by shortwave analysis.� In Fig. 9(d), both LW eigenvalues are asymptotically stable, but the functions J0 are positive. One

eigenvalue has a positive real part for wavenumber around 1. This behavior cannot be detected byshortwave analysis

The next figures in which the stable and unstable areas are plotted in the plane (�1, q), show in a morecomprehensive way the relation between longwave analysis and stability with respect to moderatewavelength disturbances.

Fig. 9. Real part of the eigenvalues vs. wavenumber. Parameters: m � 0.5 Re � 0, �i � 0.9, S � 0, r � 1; (a) �3 � �1 � 0:2, M�;1 � M�;3 � 0:5, W2 � 1.0; (b)

e1 � 3, e3 � 0.2, M�;1 � M�;3 � 0:5, W2 � 1.0; (c) �1 � 1.3, �3 � 2, M�;1 � M�;3 � 2, W2 � 1.0, (d) m1 � 0.5, m3 � 8, �1 � 1.8, �3 � 0.4, M�;1 � 2,

M�;3 � 0:5, W2 � 1.0. (±) LW eigenvalues, (Ð Ð) SW eigenvalues.

In Figs. 10±12 the parameters mi, M�i and �3 are chosen so that there exist values of �1 for which thetwo functions J0 and J� are negative for both LW eigenvalues. Figs. 10 and 11 correspond to case (i) forwhich the outer layers are less viscous than the inner layer. In particular, the two outer layers have thesame viscosity in Fig. 10, and then, one recovers the three symmetrical results [13] along the line�1 � �3. The Fig. 12 corresponds to case (ii) for which the viscosities are in increasing order.

For small values of the Weissenberg number �We2 � 0:1�, Figs. 10±12 shows that stability withrespect to longwave disturbance implies stability to moderate wavelength disturbances. If theWeissenberg numbers increase �We2 � 1:0�, a SW eigenvalue can destabilize the Poiseuille flow (seethe horizontal dashed lines in Fig. 10(a), Fig. 11(a) and Fig. 12(a)) or the interval of �1, for which theconfiguration is stable, decreases as the wavenumber q increases (see Fig. 10(b) and Fig. 11(b)). In thislatter case, the final width of the stable domain can also be determined by means of shortwave analysis.

Fig. 10. Neutral curves in the (�1±q) plane. The viscosities of the outer fluids are equal and less viscous than the inner fluid

(m1 � m � m3 < 1) and the Weissenberg number W e2 of the inner fluid is 0.1 (±) or 1 (Ð Ð). Parameters: m � 0.5, Re � 0,

�i � 0.9, S � 0, r � 1; (a) �3 � 0:2, M�;1 � M�;3 � 0:5; (b) �3 � 2, M�;1 � M�;3 � 2.

The Fig. 13 illustrates the case (iii) (both J0 functions are positive) and the parameters are chosen sothat there exist values of the thickness ratios �1 for which the functions J� are negative for both LWeigenvalues. This figure clearly demonstrates that there can exist a perturbation damped at small andlarge wave numbers and amplified at moderate wavenumbers. This situation occurs for configurationswhere the Poiseuille velocity profile is not convex.

However, in the experiment made by Elf ATOChem and the CEMEF [28] for which the thickness ofthe die is around 1±2 mm, the spatial periodicity of the interfacial perturbation is larger than 5 mm.Therefore, it seems that the most dangerous disturbances are those with wavenumber around 1. Thosewith a large wavenumber are not sufficiently amplified to be shown during the experiments. It followsthat the most dangerous unstable modes should correspond to asymptotic unstable configuration(J� < 0) or asymptotic stable configurations (J� < 0) with a not convex Poiseuille velocity profile

Fig. 11. Neutral curves in the (�1±q) plane. The viscosities of the outer fluids are less viscous than the inner fluid

(m1 < m3 < 1) and the Weissenberg number W e2 of the inner fluid is 0.1 (±) or 1 (Ð Ð). Parameters: m � 0.25, m3 � 0.5,

Re � 0, �i � 0.9 S � 0, r � 1: (a) �3 � 0.2, M�;1 � M�;3 � 0:5; (b) �3 � 1, M�;1 � M�;3 � 2.

(J0 > 0). On the other hand, interfacial perturbations are not expected, if the configuration isasymptotically stable and corresponds to a convex Poiseuille profile.

Moreover, as mentioned in Section 3.1, each LW eigenvalue can be associated to an interface (so thatfunctions J0 and (si; i � 1 or 2) have the same sign). Therefore, by following separately the evolutionwith the wavenumber of each LW mode (see Fig. 9), it is possible to point out the interface which ismainly responsible of the loss of stability.

This property of LW eigenvalues can be tested on Khomami and Ranibaran's experimentalobservation [11]. In fact, there are only two experimental situations which have a convex Poiseuilleprofile and are stable with respect to longwave disturbances:

� the PP / HDPE system which is composed of two layers (see Fig. 10 in [11]).� the HDPE / PP / HPDE system which corresponds to a three-symmetrical layer flow (see Fig. 24 in

[11]).

Our rule works well for the two-layer system as the change from the stable to the unstable areacorresponds to a change in the convexity of the Poiseuille velocity profile. However, it does not work inthe second case as the system is unstable with respect to O(1) wavenumber disturbances for � > 0.6although the Poiseuille velocity profile remains convex and stable with respect to longwavedisturbances. This partial discrepancy could have two explanations. Firstly, the Oldroyd-B model doesnot take into account the shear thinning of viscosity and elasticity and it is too basic to forecast in aright way the appearance of interfacial instabilities. A possibility of allowing the improvement of thecomparisons would be the computation of the Poiseuille profile for generalized Newtonian fluids (thatmeans that the viscosities follow a Carreau law). In this way, one gets the viscous and elasticity ratios atthe interface which will be used for comparisons. This method is actually in progress and seems to givea better agreement between experiments and theoretical results [28]. Secondly, as mentioned above, it

Fig. 12. Neutral curves in the (�1±q) plane. The viscosities are in increasing order from the bottom layer up to the top layer

(m1 < 1 < m3) and the Weissenberg number W e2 of the inner fluid is 0.1 (±) or 1 (Ð Ð). Parameters m1 � 0.5, m3 � 8,

Re � 0, �i � 0.9, S � 0, r � 1; �3 � 7, M�;1 � M�;3 � 0:5.

is perhaps difficult to find experimentally if a configuration is unstable with respect longwavedisturbances since the growth rate vanishes as the wavenumber tends to zero.

5. Conclusion

The interfacial stability of Oldroyd-B fluids with constant viscosities has been investigated for three-layer Poiseuille flow. The aim is to give practical tools which allow prevention of the appearance ofinterfacial instabilities in the co-extrusion process. As of usual in this context, we have computed thetemporal stability with respect to longwave and moderate wavelength disturbances.

In our longwave analysis, we have obtained new results. First, for specific values of the viscosity andthickness ratios, destabilization can be given by the zero order in the asymptotic expansion of thegrowth rate and then it does not depend on elasticity. Otherwise the term which gives stability can besplit into a Newtonian part (J0) and an elastic part (J�). The sign of the Newtonian part can be deducedfrom the sign of the jump of the Poiseuille shear stress at the interfaces (that is to say the convexity ofthe Poiseuille velocity profile). This last point allows us to correct some mistakes found in a previouspaper [12]. Finally, we have also clarified the conditions which allow neglect of the most thin layer. Inparticular, if the viscosities are in an increasing order, it is possible to neglect the thin layer which sticksthe two other main layers. In addition, some indications are given on the action of elasticity and elasticstratifications. However, it is difficult to point out general guidelines and in most cases the longwavestability can only be determined by computation.

The main interest of longwave analysis is that it consumes low time computer and can be easily usedto determine the stability of a given configuration. Therefore, we have tried to find under whichconditions a stable configuration found by means of longwave analysis remains stable with respect tomoderate disturbances. Such a result is very interesting because it allows getting of a stable region in an

Fig. 13. Neutral curves in the (�1±q) plane. The viscosities of the outer layers are more viscous than the viscosity of the inner

layer (1 < m1 < m3) and the Weissenberg number W e2 of the inner fluid is 0.1 (±) or 1 (Ð Ð). Parameters: m1 � 4, m3 � 8,

Re � 0, �i � 0.9, S � 0, r � 1; �3 � 0.8, M�;1 � 1:2; M�;3 � 0:9.

easy and quick way. The stability analyses are made for zero Reynolds number and we have checked ifthe rule given in our previous paper [15] also works for the three-layer flow: if the configuration isstable with respect to longwave disturbance and the Poiseuille profile is convex then the real part of theinterfacial eigenvalues remains negative (that means J0 < 0 and J� < 0 with our notation). Obviously,the rule concerns only the LW interfacial eigenvalues which are null in the longwave limit (q! 0 ). Ifthe Weissenberg numbers are small, it seems that the above criterion works well. For higher values ofthe Weissenberg number, the width of stable area delimited by the LW eigenvalue can decrease as thewavenumber increases, whereas destabilization due to SW interfacial eigenvalues can occur for a largevalue of q. These behavior occurring for large Weissenberg numbers can be detected by shortwaveanalysis. Nevertheless, it we consider the pattern of interfacial perturbations observed in theexperiments, the criterion based on longwave analysis seems to be relevant to predict the stableconfiguration even for large Weissenberg numbers. However, this recipe has only been testednumerically on Oldroyd-B fluid and would be extended to shear-dependent viscosity and elasticity (e.g.by using the White±Metzner constitutive model) in order to have better agreement with theexperiments.

Acknowledgements

The authors thank Y. Demay for his advice and constant encouragement. P.L. has also been supportedby a grant awarded by Elf-ATOChem (Serquiny, France).

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linear stability of three-layer poiseuille flow for oldroyd-b fluids

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