Linear stability of multilayer plane Poiseuille flows of Oldroyd B fluids

journaloC NoD-Newtonian Fluid Mechanics

ELSEVIER

J. Non-Newtonian Fluid Mech., 71 (1997) 1-23

Linear stability of multilayer plane Poiseuille flows of Oldroyd B fluids P. Laure a,*, H. Le Meur b, Y. Demay a, J.e. Saut b, S. Scotto a a

Institut non-lineaire de Nice (UMR CNRS 129), Universite de Nice, Sophia Antipolis, 06560 Valbonne, France Laboratoire d'Analyse Numerique et EDP (URA 760 CNRS), Universite Paris-Sud, Bru. 425, 914050rsay, Cedex, France

b

Received 5 February 1996; revised 17 January 1997

Abstract The linear stability of plane Poiseuille flows of two and three-symmetrical layers is studied by using both longwave and moderate wavelength analysis. The considered fluids follow Oldroyd-B constitutive equations and hence the stability is controlled by the viscous and elastic stratifications and the layer thicknesses. For the three symmetricallayer Poiseuille flow, competition between varicose (symmetrical) and sinuous (antisymmetrical) mode is considered. In both cases (two and three symmetrical layers), the additive character of the longwave formula with respect to viscous and elastic terms is largely used to determine stable arrangements at vanishing Reynolds number. It is found that if the stability of such arrangements is due simultaneously to viscous and elastic stratification (the flow is stable for longwave disturbance and the Poiseuille velocity profile is convex), then the Poiseuille flow is also stable with respect to moderate wavelength disturbances and the critical thickness ratio around which the configurations becomes unstable is given by longwave analysis. Note that a convex velocity profile means a positive jump or'shear rate at the interface. Finally, the destabilization due to a moderate increase in the Reynolds number is considered and two distinct behaviors are pointed according to the convexity of the Poiseuille velocity profile. Moreover, an important influence of the thickness ratio on the critical wavenumber is found for three symmetrical layer case (for two layer case, the critical wave number is of order one and depends weakly on the thickness ratio). © 1997 Elsevier Science B.V. Keywords: Viscoelastic fluids; Interface; Linear stability; Multifluid flow; Arnoldi; Orr-Sommerfeld

1. Introduction The stability of multilayer Poiseuille flow (tube or parallel plates) has been recently studied by numerous authors due to the growing industrial importance of multilayer products in polymer

* Corresponding author. 0377-0257/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S03 77 -02 57(97)000 11-6

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P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

processing. For example, food packaging commonly uses films formed of three, five or more layers in order to put together polymers having different optical, mechanical and barrier (to air or water) properties. In the industrial process, the polymers are molten in screw extruders, and then pushed through a feedblock and a flat die [1]. This process is limited by the onset in some conditions of a wavy instability deteriorating the quality of the film. Coextrusion of two components has been also used in the fiber industry, whereas concentric coextrusion was motivated by problems associated with the transportation of heavy viscous oil through long pipeline [2]. Various parameters such as viscosity, surface tension, viscoelasticity, density and gravity intervene on the destabilization mechanism. However, it is worth noticing that in the processing of multilayer sheets, the ratios of viscosity and viscoelasticity are the more discriminant parameters. This article is concerned with the linear stability of a plane Poiseuille flow of two viscoelastic fluids. Both fluids follow an Oldroyd-B differential constitutive equation. This law introduces the difference of normal stress (J X,x - (J y,y considered as the characteristic feature of viscoelasticity of polymeric fluid. Stability under infinitesimal disturbances is investigated by the means of generalized Orr-Sommerfeld equations, in each layer. These equations are completed with equilibrium (namely the jump on the normal stress and velocity) and kinematic interface equations. The linear stability is classically studied by the computation of the spectrum of the linearized operator about a basic flow. We restrict our computations to disturbances which are periodic in the direction of the flow motion. The sign of the real part of the eigenvalues gives the stability criterion and it depends on the various parameters of the problem, but also on spatial periodicity. In our problem, the two following methods are mainly used. The longwave analysis allows us to separate the kinematic equation from the equations of motion. In this way, one gets under the Oldroyd-B approximation an analytical formula giving the 'interfacial' eigenvalue as a fractional expression of parameters. However this asymptotic analysis has to be completed by a more general stability analysis or moderate wavelength analysis. This step introduces heavy numerical problems in the approximation of the spectrum. The difficulty mainly comes from the introduction of a continuous spectrum due to the viscoelastic behavior of polymeric fluids [3]. So, this analysis requires accurate numerical analysis and several numerical methods can be used. Among them, the pseudospectral Tau method using an expansion of eigenvectors in Chebyschev polynomials is the most precise [4]. Finally, the discrete problem corresponds to a generalized eigenvalue problem which is solved by an Arnoldi method [5]. We now review briefly previous works on the stability of Poiseuille-Couette multilayer flows. 1.1. The two-layer Newtonian and quasi Newtonian Couette-Poiseuille flow

The longwave stability analysis of the two-layer flow of viscous fluids with different viscosities was studied by Yih [6]. It was found that, whatever the Reynolds number is, the stability is governed for both plane Couette and Poiseuille flows by a function of viscosity and thickness ratios m and E. Moreover, he has also pointed out that some regions in the parameter plane (E, m) are stable when the less viscous layer is the thinner ('thin layer effect'). Yiantsious and Higgins [7] have completed for Poiseuille flow the previous longwave analysis and have also taken into account moderate wavelength. Moreover, a shortwave analysis was introduced in

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

3

order to analyze the effect of the surface tension. Charru and Fabre [8] have shown that, for longwaves and moderate surface tension, the nonlinear behavior at small amplitudes is given by a Kuramoto-Sivashinky equation. They have also pointed out that the Poiseuille flow is stable if the basic velocity profile is convex. Khomami [9,10] has studied the longwave stability and the effect of the converging channel for power law fluids. Pinarbasi and Liakopoulos [11] have computed stability of a two-layer Poiseuille flow of Carreau-Yasuda and Bingham-like fluids at moderate wavenumbers (using a Tau method and a QZ algorithm). The three-layer vertical flow was considered by Renardy [12] whereas the three-layer horizontal Poiseuille flow has been studied by Anturkar et al. [13]. 1.2. The multilayer viscoelastic Couette- Poiseuille flow

Generalization of the longwave stability analysis of plane Couette flow has been carried out by Li [14] for Oldroyd-B fluids, and by Waters and Keeley [15] for Oldroyd-B fluids with a shear-dependent Carreau viscosity. Anturkar et al. [16] have also examined the stability of the two and three-layer viscoelastic Poiseuille flow with respect to longwave and moderate wavelength disturbances. Later, Chen [17] has pointed out an error in the shear stress condition at the interface used in these previous works. Moreover, he has shown for Couette flow that the jump of the normal stress arises in the interfacial instability. This author has also studied interfacial instability due to elastic stratification in the concentric coextrusion of two viscoelastic .~uids [18]. The shortwave analysis of a two-layer Poiseuille flow of UCM fluids was considered by Renardy [19]. Finally, the most complete results on the interfacial instability for Oldroyd-B liquids with constant and shear rate dependent viscosities are given by Su and Khomami [20]. The influence of the die geometry has also been considered by these authors and this paper is in a way the starting point of our present study. 1.3. Experiments

Han and Shetty [21,22], Anturkar et al. [23] have published experimental results and considered industrial configurations. In particular, the influence of a convergent geometry was numerically studied by Anturkar et al. [24] using a Galerkin method and experimentally by Khomarni and Wilson [25]. Nevertheless, despite a great interest for this subject in recent past years, comparison with experiment is not clear. This is due to difficulties encountered in industrial or laboratory observations. One of the difficulties is due to encapsulation effect (the less viscous fluid tends to encapsulate the more viscous component and the thickness ratio does not remain constant in the transverse direction [1]). In the same manner, it is difficult to ensure a constant temperature along the die and the viscosity and elastic properties of each layer are not well known and homogeneous. Finally, characterization of irregular interface needs a suitable optical apparatus and it is not always possible to measure precisely the spatial and temporal periods of the perturbed interface (see photographs in [22]). One way to eliminate such a difficulty is to impose external temporally regular disturbances at the interface [25]. The aim of this paper is to perform both longwave and moderate wavelength stability analysis in two different geometries. In the first one, both fluids are in contact with the wall. In the second one, one of the fluid is in contact with the upper and lower walls and the second fluid

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

4

is symmetrically encapsulated. In this latter case, the geometry is assumed to be symmetric, the gravity is neglected and the stability with respect to varicose and sinuous perturbations is studied. In Section 2, we give the governing equations and the Poiseuille profile around which linear stability is performed. In Section 3, we perform the longwave stability analysis of the Poiseuille flow (it can be found for the Couette flow in [17,19]), and give new explicit formula. This allows us to provide areas of stable regions in the plane (€", m) for vanishing Reynolds number. In this way, we identify some general asymptotically stable configurations with respect to longwave disturbances. These configurations are destabilized by an increase of the Reynolds number. In Section 4, the influence of a moderate wavenumber on these stable regions is considered. The effect of surface tension S (which prevents a destabilization induced by short wave perturbations) on critical Reynolds number is also analyzed.

2. Governing equations 2.1. The geometries

The stability analysis will be developed in two different geometries. The first one is a two-layer flow (Fig. lea)), labelled as case A. In this case, each fluid is in contact with a wall. The first layer, located between y = 0 and y = db is designated as region 1 and the second one, located between y = d, and y = d, + d2 , as region 2. The second geometry is a three-layer flow with identical external fluids (Fig. l(b)) and is labelled as case B. The same fluid is in contact with both walls and the other one flows in the

y

y

~ I I I I I I I

dl +dz

dl +dz

I I I I

fluid 2

I

fluid 2

dl

: I I I

fluidl

-------r---------------------------> o

dl

I

x

I

I I I

fluidl -dl

---->

-~O-----------

I I I

I

,

fluid3=fluid2

I

x

- dl -dz : I

(a)

(b)

I

Fig. 1. Geometries: (a) two-layer geometry; (b) three-symmetrical layer geometry.

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5

core. This system is symmetric with respect to horizontal axis and the core layer (y E( - db d l )) is designated as region 1. The upper and lower layers are designated as region 2 and have the same depth d2 • This configuration is expected to give close results to the concentric case and it has also been studied experimentally in [21,22]. Each fluid has different rheological properties which are noted in the standard way; namely the viscosity Y/b the polymeric components IXk and the density Pk' Finally, the interface equations are Ys = sdl + hs(x, t) with s = 1 in the case A and s = ± 1 in the case B (obviously, we have hsCx, t) = 0 for the basic flow). 2.2. The equations and stationary solutions Let us denote Uk and !p,k the velocity field and viscoelastic extrastress in each layer. We assume that fluids are incompressible and obey an Oldroyd-B model. We choose the following space, velocity, time and stress scales, d* t*=-8 . U* '

U*

(J'*

= Y/I d* .

where U* is the value of the Poiseuille velocity at the interface. Finally, the dimensionless equations in fluid 'k' read as follows: rk Re ( !p,k +

aUk) + (Uk' V)Uk + VPk

at

Wek(a~~'k + Uk' V!p,k -

= mk(1

- IXk)Auk = V' !p,k -

VUk' 'rp,k - !p,k' VUf)

rk Re -pz e

y,

= IXkmk(VUk + VuD,

(1)

In the above equations, we exhibit the classical dimensionless parameters: the Reynolds number Re=P I U*dd1JI; the Weissenberg numbers Wek=AkU*ldl the Froude number F= U*I jid;; the ratio of viscosities mk = 1Jk/Y/I; the ratio of densities rk = PklPI; the layer thickness ratio € = d2ldl ; the surface tension number 8 = TI1JI U*. To simplify the notations, we will use also non-indexed parameters: We = Web r = r2, m = m2 and the ratio of relaxation times M). = (1X2A2)/(IXIAI)' Let us notice that the problem has ten dimensionless parameters (Re, We, M)., IXI. 1X2, m, r, 8, F, €) and that an infinite Froude number means that the vertical gravitational force is not taken into account. This system of equation is completed by non-slip boundary condition,

uix, s(1 + E)) =

V2(X,

s(1

+ E)) =

0;

s=

± 1,

and the interfacial conditions on the velocity and stress vector of a function I at the interface (s = 1 or s = ± 1)): [[U ]]s = 0,

[[ - pI + 2(1 -1X)m~[u] + !p]]s'I =

0) =

UI(X,

VI(X,

(lIf]s =

- 2H 8n,

0) = 0 Case A,

(2)

s(.!I(s) - lis)) is the jump (3)

where the vector n is a unit normal to the interface, 2H the sum of the principal curvatures. If we note Uk = (Ub Vk) the velocity components, the immiscibility condition can be written as

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6

(4)

Obviously in the above equation, the interface s = three-layer case.

-

I

IS

taken into account only for the

2.3. Basic flow

The multilayer Poiseuille flow is the x-invariant solution of Eqs. (1)-(3). The velocity field is in each region: Case A: uo(y) = Case B: uo(y) The extrastress

't'o

=-

E+ m (1 ) Yz mkE + E mkE

m (2

+E

m + 2E + E2 )y

+ mkE (I +

) y2 + 1 +

mkE

E

m (2

+E

+

mk - 1 , mk

(5) (6)

).

tensor and the pressure Po are respectively, for each fluid,

't'0(0" 0 TO) TO Yo To(y)

= mklXku~,

O"o(y) =

2mkIXk

(7)

Wek U~2,

(8)

Yo = 0,

(9)

) = - rk Re Po( X,y F2 y

+ mkuO x + & o·

(10)

1/

The constant &0 is an arbitrary pressure, and prime

I

designates

alay.

2.4. Perturbation equations and interfacial eigenvalue

Since we are interested in the stability of the basic flow under infinitesimal perturbations and because of the x invariance, we classically introduce perturbations in the following form: (Ub Pk,

!:p,b

hs) = (Ub fik, !p,b hs) exp[iqx + O"tl

(II)

The tilded variables, designating the amplitudes of disturbances, are (except the constant hs) functions of y only. The number q is the wave number in the flow direction and 0" is the frequency. Neglecting terms of order two in the amplitudes of the perturbations, we get the usual generalized Orr-Sommerfeld equations which can be found in [17,20]. Non-slip conditions are imposed on the two walls, whereas at the interface (y = Ys) we get the same set of boundary conditions already written in [17,20]. It is important to notice that the y ~ - y symmetry of the three-layer case, allows us to predict two possible unstable modes depending if this symmetry is preserved. Therefore, the symmetric solution is called the varicose mode (h _ I = - hI) whereas the anti-symmetric solution is the sinuous mode (h _ I = hI)' Let us note that Hinch et al. [26] show that the sinuous mode can grow more rapidly than the varicose mode in the concentric coextrusion.

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7

So, if we denote by X the vector of coefficients of the Chebyschev expansion of each variable, we obtain, using a Tau method [4], a generalized eigenvalue problem of the following form: (12)

A(q)X = aB(q)X.

The matrices A and B are complex and B is singular. As usual, the stability of the basic flow is given by the sign of the real part of the generalized eigenvalues a. For a one-layer flow, the spectrum (the set of all the generalized eigenvalues) is well known. For example, the linear stability of Couette flow has been studied by Renardy and Renardy [27] for upper convected Maxwell fluid, whereas an Oldroyd-B model is considered for linear stability of Poiseuille flow by Sureshkumar and Beris in [3]. It is shown that the spectrum is constituted of a continuous spectrum and an infinite discrete spectrum. The destabilization comes from an eigenvalue belonging to the discrete part of the spectrum and always occurs at high Reynolds numbers (Re '" 103 ). For a two-layer flow, the spectrum has the same general aspect as for a one-layer flow. The main feature [6] is that a new eigenvalue (referred as the interfacial eigenvalue) directly related to the interfacial equations allows destabilization of the Poiseuille flow at small values of the Reynolds number. The critical Reynolds number is then a function of relevant parameters such as Weissenberg number, viscosity, elasticity and thickness ratios.

3'. Long waves asymptotic analysis In this section, the stability of the interface under longwave (q ~ 0) disturbances is examined by using an asymptotic analysis. A streamfunction Wk defined by (Uk, Vk) = (OWk/Oy, - iqWk) is used and hence the divergence free condition is directly satisfied at each order. Following standard procedure [6], the streamfunction wand the growth rate a are expanded in the form, (13) and a

= -

iq(co + qCl

+ "').

A sequence of equations, written in [7,13] for Newtonian fluid and [18] for Maxwell fluid in Case A, is obtained by substitution of these expansions into the whole system and identification at each order. As it is found that Co is real and Cl purely imaginary, the stability is given by the sign of Cj and qco is the period of the perturbation (this last is independent of the elastic properties). More precisely the coefficients Co and Cl are in the following simple form (obtained with the help of the symbolic calculator MAPLE): Co

= f( € ,m) with f real,

Cl = i

Re(Jl (€, m) + (r -l)(J2(€, m) +

(15)

J3(~2m»)) + iIXI We(J4(€, m) + (M). -

1)J5(€' m»,

(16)

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8

where M;. = (IX2A2)/(IX\A\). The real functions Ji> J 2 , J 3 , J4 and J s are given in Appendices A and B for the Cases A and B. It follows from relation (16) that c\ is the sum of a Newtonian part (Ji> J 2 and J 3 ) and an elastic part (J4 and J s). The density stratification (r =f:. 1) alters only the Newtonian part. As for given values of € and m, the stability is the result of the additive influence of the Reynolds number and the Weissenberg number, we consider successively in the following sections Re =f:. 0, We = and Re = 0, We =f:. 0. This additivity was used by Chen for the Couette flow [17] and the concentric coextrusion [18]. Nevertheless, this author has preferred to write the eigenvalue as the sum of a term proportional to [u~]s and another proportional to [l1ol In the Case A, these functions satisfy symmetry properties corresponding to an exchange of the fluid layers. The following relations are easily checked:

°

1 1 1) = 2m c\(€, m, r)

c\ ( -,-,€ m r

€

c{~,~,~J = c\(€, m, M;.)

. If IX; = We = 0,

if Re = 0,

(17) (18)

Consequently, we limit the discussion to M;. > 1 in the Case A. 3.1. Stability of purely Newtonian fluids (Re =f:. 0 and We = 0)

The Case A has been studied by numerous authors [6-8,13,28] and we obtain basically the same results for the Case B. Therefore, we only report here the main results. Eq. (16) becomes (neglecting the density stratification): (19) The Poiseuille flow is stable if the function J\ is negative. The expressions of J 1 given in Appendices A and B show that J 1 has the opposite sign to the jump [u~] in both cases. So the remark about stable velocity profile given in [8] for two-layer Poiseuille flow can be extended to the symmetrical three-layer case. That means that the process is stable when the basic velocity profile Uo is convex. Finally, results about stable configurations of Newtonian flows can be summarized by the two following rules: the two-layer Poiseuille flow is stable when the thin layer is less viscous (lubrication effect); the symmetrical three-layer Poiseuille flow is stable when the more viscous fluid flows at core. 3.2. Stability of viscoelastic fluids at zero Reynolds number (Re = 0 and We =f:. 0)

Due to the high viscosity of polymer melts, we have to pay a particular attention to flows stable at zero or vanishing Reynolds number. For matched viscosities (m = 1), the stability is given by the sign of (M;. - l)Js(€, 1) (because Ji€, 11=0). The function J s(€, 1) is positive if € < 1 for the two-layer flow (Case A) and if € <,J2 for the varicose mode (Case B) whereas it is always positive for sinuous mode. It is to notice that the corresponding value obtained by Chen [18] for the stability of the varicose mode concerning the stability of concentric two-layer flow is € = 1.7. If M;. > 1, the stable arrange-

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9

10'

s

10- 1 +--~-_~~~+-L---H-~-~~~~<_I 10- 1

10'

Fig. 2. Two-layer case: Neutral curves J4 + (M). - 1)J5 = 0 in the plane (E, m) for more elastic upper layer; M). = 1, 1.05, 1.4. The arrows indicate the displacement of the neutral curves with the relaxation time ratio M)..

ments correspond to a larger upper layer for the Case A, but the three symmetrical layer configuration is never stable because of the sinuous mode. If M). < I, the outer layer has to be thinner for the Case B. In these stable configurations, the layer close to the wall is less elastic and thinner. When the viscosities are not matched (m =1= I) we have to analyze the competition between viscous and elastic stratifications by means of functions J 4 and J 5 • This effect is analyzed hereafter by means of Figs. 2-4 in which the neutral curves of function J 4 + (M). - I)J5 are plotted for various values of M).. The Case A has widely been studied by Su and Khomami [20]. If the two fluids have the same elastic properties (M). = 1), we have once again four regions around the point m = E = 1, bounded by the curves m = I and m = e2 , with the same stability as in the purely Newtonian 10'

10'

U

U

m 10° -r:!::-=----t====~

.....::::;;.-==='1==

m 10°

S 10-' 10- 1

10- 1 +--~-~~~~-+--~-~~~~<-I 1 10- 1 10° 10

(a)

E

(b)

10°

10'

E

Fig. 3. Varicose mode for symmetrical three-layer case: neutral curves J4 + (M). -1)J5 = 0 in the plane (E, m) for various values of elastic stratification M).: (a) the core layer is less elastic, M). = 1, 1.05, 1.4; (b) the core layer is more elastic, M). = 0.5, 0.99, 1. The arrows indicate the displacement of the neutral curves with the relaxation time ratio MA.-

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

10

U M)..=O.99

m

M)..=l

m

10°

10°

S

S 10,1 10'1 (a)

10° E

M)..=l

10

10,1 10,1

1

(b)

10°

10

1

E

Fig. 4. Sinuous mode for symmetrical three-layer case: neutral curves J 4 + (M;. - I)J5 = a in the plane (E, m) for various values of elastic stratification M;: (a) the core layer is less elastic, M; = 1, 1.05, 1.4; (b) the core layer is more elastic, M;. = 0.5, 0.99, 1. The arrows indicate the displacement of the neutral curves with the relaxation time ratio M;-

case. Moreover, there exist two other stable domains: the first one for small E and large m and the second for large E and small m. Therefore, the existence of these stable regions implies that the simple Charru's criterion on Poiseuille profile does not apply for Oldroyd-B fluids. Note also that these new domains can be eliminated by increasing the Reynolds number up to a critical one (as shown in Fig. 3(a,b) of [20]). The influence of elastic stratification (M). =I- 1) can be easily taken into account since J 5 has the sign of m - E 2 for all reasonable values of m and E. If M; > 1, the elastic stratification acts as the Reynolds number in the upper mid-plane (m> 1) and the new stable region (observed for small E and large m) can disappear. On the other hand, the elastic stratification increases the stable region in the lower-mid plane (m < 1) and the stability becomes independent of m for sufficiently large E). In the Case B and for varicose perturbations, J4 is negative for m < 1 or 1 + (1 - E)3 /2 < m. Consequently, a configuration with a more viscous fluid in the core layer (m < 1) is again stable when M). = 1. Moreover, there exists a new stable region when the more viscous layer is outside (m> 1 + (1 + E)3/2). SO, if M). > 1, the elastic stratification acts again as the Reynolds number and the new stable region in the upper half plane m > 1 can disappear. Moreover, the half plane m < 1 can become unstable for small values of thickness ratio E. If M; < 1, there exists a region for small values of E which is stable for all values of viscous stratification m. This means that if the outer layer is less elastic and thin enough, the Poiseuille profile is stable. This last result is similar to Chen's result in the case of concentric coextrusion (Fig. 5 in [18]). So, it seems that the two-layer interfacial mode and the varicose mode have the same behavior. The action of the sinuous mode is simpler. If M; = 1, there is no difference with the Newtonian case (the mid-plane m> 1 is unstable, whereas the mid-plane m < 1 is stable). As seen in Fig. 4, the mid-plane m < 1 is destabilized if M; is increased from 1 whereas the mid-plane m > 1 becomes more stable as M). is decreased from 1. Experimental results are given in [22] for three and five layers. Concerning the three-layer flow and in agreement with the stability analysis, irregular interfaces are not observed if the more

P. Laure et at. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

11

viscous fluid is sandwiched between the less viscous one (the LDPE/PS/LDPE system). On the contrary, unstable processing conditions are reported if the more viscous fluid is in contact with the wall (the PS/LDPE/PS system). Fig. 15 and 16 in [22] show that unstable interfaces are obtained for a layer thickness ratio hps/hLDPE less than 0.8 when the viscosity ratio '1PS/'1LDPE"'" 10 and a normal stress difference ratio (0'1 - O'z)PS/(O'I - 0'2)LDPE "'" 1. With our parameters, that means that when m = 10 and M). = 10, the flow will be unstable if E < 1.6. It is found from the longwave analysis at zero Reynolds number that the basic flow is unstable with respect to varicose perturbation if E < 6.7 whereas the sinuous perturbations is always destabilizing. We could explain the difference between experimental and our asymptotic analysis by a wrong choice of rheological behaviors. Both fluids have a shear-thinning or pseudoplastic behavior which corresponds to a power-law model '1 = '1o(Oyu)n-1 with nps = 0.33 and nLDPE = 0.45. Moreover, the authors compute the viscosity ratio by taking the viscosity at the wall for a single fluid. For a better comparison, more suitable values should be the viscosity ratio evaluated at the interface. 3.3. Asymptotically stable arrangements at Re = 0

The previous section was devoted to the stability of the Poiseuille flow in the particular case Re = 0 according to longwave disturbances (q -. 0). We have now to extend this analysis to weak values of the Reynolds number and moderate or large values of the wavenumber q. Because of the increasing complexity due to the large number of parameter, we will test our results on five configurations (two or three-layer) described in Fig. 5 whose stability at Re = 0 and q = 0 has different origin. The persistence of the stability for non zero value of Reynolds number and moderate or large wavenumber is studied in next sections. Our goal is to describe 'generically' stable flows in the range of polymer processing. For the two and three-layer flows of Fig. 5(a-c), stability is due to the viscosity stratification (Section 3.1) and is not modified by the elastic stratification. As seen in Figs. 2-4, it means generally that the more elastic layer has to be the larger one. The two-layer flow of Fig. 5(a) is stable if E > ~ and M). > 1. Similarly, the three-layer flow of Fig. 5(b) (respectively Fig. 5(c)) is stable if parameter E is large enough (respectively small enough). On the contrary, for the arrangements depicted in Fig. 5(d,e), the stability is due to elastic properties overcoming the instability due to viscous stratification ((M). - I)Js(E, m) > - J4CE, m)). For the two-layer flow, it can be seen in Fig. 2 that the larger and less viscous layer has to be more elastic (Fig. 5(d)). For the three-layer flow, the core layer is larger, less viscous and more elastic (Fig. 5(e)). As noted above, these flows are unstable above an asymptotic critical Reynolds Reo(q = 0) obtained from the Eq. (16) (the function J I is positive): -0)- R eo(q -

Ct]

W J4CE,m)+(M).-I)Js(E,m) e J () . ] E,m

(20)

In the next section, the stability of these configurations according to small and moderate wavelength disturbances is studied for zero or small Reynolds number.

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4. Stability analysis for moderate wavelength

The eigenvalue problem for any disturbance wavelength q is discretized by using a Tau method [4] and the vector X is expanded in terms of Chebyschev polynomials of degree N. This

a)

b) m

m>1

M

A

11 11

2

>

11

!X 2 1.. 2 > !X] A]

1

< 1

MA

>1

>1 <

2

11]

- _ _-

!X 2 1.. 2 > !XI Al

._------_

.•..

...

_-

11, A]

11 1

.~~_.-

11, Al

11 1

11

<

2

11

1

!X 2 1.. 2 > !X] AI

c) m <1 <

11 2

11 I

MA

!X 2 1.. 2 < !X] AI

11, A]

11 1

11

<

2

11 (

d) MA~ 1

>

!X 2 1.. 2 < !XI A]

2

11 1

11 I

!X 2 1.. 2 < !XI A]

e)

m >1 11

<1

11, Al

m >1 11 2

>

11 I

>

<1

!X 2 1.. 2 < !X] 1..(

11 ,AI

11]

11 2

MA

11 I

!X 2 1.. 2 < !XI AI

Fig. 5. Sketches of asymptotic stable situations: (a) two-layer, m> 1, M;. > 1, large E; (b) three-layer, m < 1, M;. > 1, large E; (c) three-layer, m < 1, M;. < 1, small E; (d) two-layer, m> 1, M;. ~ 1, small E; (e) three-layer, m> 1, M;. < 1, small E; flk' Ak and (J.k are respectively the viscosity, the relaxation time and the polymeric component of fluid k.

P. Laure et al. / J. Non-Newtonian Fluid Meeh. 71 (1997) 1-23 (a)

13

(b) 10'

10'

S q 10·

q 10·

---,----" ---, ..

I

U

I

U _'" "," "

j'

"/

If

" ""

S

U

S 10" 10"

10·

101

10,1 10"

10·

10'

Fig. 6. Moderate wavelength disturbances of two-layer flow. neutral stability contours for Oldroyd-B fluids and We = 0.01 (-), 0.1 (j.) and 1 (- -). Parameters: Re = 0, S = 0, r = 1, F = 00, IX; = 0.9, m = 4: (a) M) = 1.2; (b) M) = 0.8. Stable regions are denoted by S, unstable regions by U.

leads to a generalized eigenvalue problem (A - s B)X = 0 where A and B are complex matrices of size l2*N + 1. The eigenvalue of largest real part is found by using an Arnoldi decomposition and an inverse iterative method [29]. It was checked that the numerical and asymptotic results are in excellent agreement. 4.1. Stability analysis at zero Reynolds number

Following the classical methodology, we look for stable and unstable regions in the (E - q) plane for given rheologies (given values of parameters m, MJ..' We, !Xl and !X2 and zero Reynolds number Re). As this study is focused on the competition between viscous and elastic effects, we consider a null surface tension S, absence of density stratification (r = 1) and an infinite Froude number F. In each case, the value of MJ.. and m are fixed and computations are made for three different values of Weissenberg number (We = 0.01, 0.1 and 1). Neutral curves are plotted in Figs. 6 and 7. The two-layer (Case A) arrangements described in Fig. 5(a,d) are first considered. The contour plotted in Fig. 6(a) (m = 4, !Xk = 0.9 and MJ. = 1.2) shows that the stable asymptotic region (E >..j;) obtained for q = 0 remains stable for q > 0 and hence configuration of 5(a) is stable according to moderate wavelength disturbances. Considering configuration in Fig. 5(d) with the more viscous and less elastic fluid in the thinner layer (m = 4, !Xk = 0.9 and MJ. = 0.8), it is found on the contrary from Fig. 6(b) that this flow, stable if E < 0.5 and We .::;; 0.1 (for values of the other parameters considered in Fig. 6), is destabilized ~ a wavelength q around 1 at We = 1. The second stable region of Fig. 6(b) found for E >.Jm is not itemized in Fig. 5. The stability of this arrangement is not insured for a smaller elastic stratification MJ. as shown on the asymptotic neutral contours plotted in Fig. 2.

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

14

Concerning the three-layer flows (Case B) of Fig. 5(b,c,e), stable for q = 0, our computations exhibit the same kind of results. Neutral stability curves are plotted respectively in Fig. 7(a,b) for a more viscous and less elastic core layer (m = 0.4, M). = 1.2, Fig. 5(b)) and a less viscous and more elastic core layer (m = 4, M). = 0.8, Fig. 5(e)). It is observed on these figures that the first curve bounding the asymptotic stable area (corresponding to critical sinuous modes) depends weakly on the Weissenberg number. However, the Fig. 7(b) shows that a new unstable domain due to varicose modes occurs for We = 1 and therefore the asymptotic stable arrangement sketched in Fig. 5(e) does not remain for moderate wavelength. Our computations show also that a situation with a more viscous and more elastic core layer (m = 0.4, M). = 0.8, Fig. 5(c)) remains stable for 0.1 < q < 10 and We = 0.01, 0.1, 1, whatever the thickness ratio E is. Finally, only the stable arrangements depicted in Fig. 5(b,c) remain stable for all moderate wavelength perturbations, whereas the arrangement of Fig. 5(e) can be destabilized. Finally, we point out two different behaviors: 1. if the asymptotic stability is due to viscous stratification (that means that the Newtonian part (11) and the elastic part (J4 and J 5 ) of asymptotic eigenvalue (Eq. (16)) are both negative), (a)

10'

q lO"

(b)

s

(c)

s q lO"

Fig. 7. Moderate wavelength disturbances of three-symmetrical layers. Neutral stability contours for Newtonian and Oldroyd-B fluids for We=O.Ol (-),0.1 (A) and 1 (- -). Parameters: Re=O, 8=0, r= 1, F= 00, cx;=0.9: (a) m=O.4, MJc= 1.2; (b) m=4, M Jc =0.8; (c) Re=O.1, 8=0, r= 1, F= 00, CXk , We=O, m=O.4. Stable regions are denoted by S, unstable regions by U.

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

15

then the eigenvalue (J is negative for all wavelengths q. Moreover, the asymptotic analysis gives also the critical thickness ratio. 2. in the other case, the asymptotic analysis does not allow to forecast the stability with respect to moderate wavelength. More precisely, if the asymptotic stability is due to elastic stratification (the Newtonian part J] is positive and the elastic part is negative), the eigenvalue (J does not always remain negative. Therefore, the three first asymptotically stable configurations depicted in Fig. 5 remain stable with respect to moderate wavelength disturbances when Re = O. From these very specific computations, it seems possible to conclude that the asymptotic stability at vanishing Reynolds number induces a full stability when this asymptotic stability corresponds to situations in which the elastic stratification does not change the stability arising from the viscous stratification. As illustrated in the next section, this result does not hold when Re #- 0 and for purely Newtonian fluids. 4.2. Effect of Reynold number Re and surface tension

Let us first consider the influence of Reynolds number on the stability of the two-layer Poiseuille flow (Case A). For Newtonian fluids without surface tension, the flow becomes unstable if the wavenumber is greater than a critical value and this value does not depend on the Reynolds number (Fig. 4 in [7] and Fig. 8(a)). Nevertheless, Fig. 8(a) shows that if S#-O (it is generally true), the flow is stable if the Reynolds number is smaller than a critical one and this is important because of the low values of Reynolds number in polymer processing. Concerning two-layer Poiseuille flow of viscoelastic fluids, it was shown by Figs. 5-7 in [20] that, for a sufficiently high Reynolds number, the flow with a larger layer of the more viscous fluid of Fig. 5(a), is destabilized with a critical wavenumber of order unity (this instability is not detected by the longwave analysis). Fig. 8(b) shows that this critical Reynolds number is largely increased by effects of dimensionless surface tension. It is a result of the longwave analysis that the two-layer flow of Fig. 5(d) with larger and less viscous fluid becomes unstable with respect to longwave disturbances at a neutral asymptotic Reynolds number given by relation (20). Fig. 8(c) points out that the asymptotic analysis gives a good approximation of the most critical Reynolds number (reached for q = 2.15) of the arrangements of Fig. 5(d) and shows that the influence of the surface tension is moderate in this case. Results are globally the same for three-layer Poiseuille flow (Case B). For Newtonian fluids without surface tension, Fig. 7(c) plotted for Re = 0.1 shows that the flow becomes unstable if the wavenumber is greater than a critical value depending on the thickness ratio E. It decreases from 10 to 0.1 as E increases from 0.1 to 10 and this is a notable difference with respect to the two-layer case. Moreover, in this last figure the instability is due to varicose mode. Fig. 9(a,b) show that the three-layer Poiseuille flow of viscoelastic fluids represented in Fig. 5(b,c) are unstable if the Reynolds number is greater than a critical one (critical wavenumber and Reynolds number are given in the captions of these figures). The critical wavenumber decreases with the thickness ratio for the three-layer system and the Reynolds number is largely influenced by the dimensionless surface tension. It is shown in Fig. 9(c) that for the arrangements of Fig. 5(d), influence of wave number q and surface tension S are moderate (the asymptotic analysis gives a good approximation in this case).

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

16 (al

(b)

5

5 8=.01

4

4

3

3

Reo

Reo

2

2 8=0

8=0

o t---'-":::=t====:::;:==~+-----+------I 10 6 8 4 2 o

0 0

2

4

6

8

10

q

(el

q

2.0

1.5 Reo

1.0

0.5

0.0 0

2

4

6

8

10

q

Fig. 8. Neutral Reynolds number Reo versus wavenumber q for Newtonian and Oldroyd-B two-layer flow. Parameters: r = I, F = 00, m = 4: (a) Newtonian fluids, E = 3, unstable if q > 1.09 for S = 0 and (qc> Rec ) = (1.62, 0.52) for S = 0.01; (b) E = 3, M). = 1.2, (Xk = 0.9, We = 0.01, (qc> ReJ = (2.5, , 0.175) for S = 0 and (qc> Rec ) = (1.7,1.267) for S = 0.01; (c) E = 0.3, M). = 0.8, (Xk = 0.9, We = 0.1, Reo(q = 0) = 0.319, (qc> Rec) = (2.15, 0.282) for S = 0 and (qc, Rec ) = (2, 0.292) for S = 0.1.

In Fig. 9, the critical Reynolds number is reached for the sinuous mode. Therefore, the sinuous mode seems to be the first unstable mode when the Reynolds number increases. This feature is enforced by the surface tension which increases the difference between the two neutral Reynolds number associated to each mode. However in Fig. 9(c), the neutral Reynolds number is associated to a varicose mode at small value of q while it is associated to a sinuous mode at moderate or large value of q.

5. Discussion and conclusion

There is a relatively large number of papers concerned with the stability of multilayer Poiseuille flows. They have already examined the interfacial stability of two-layer or three-layer parallel channel flows by means of longwave asymptotic analysis and computation of growth rate for disturbances with moderate wavelength. In this paper, we have followed the same

P. Laure et al.jl. Non-Newtonian Fluid Mech. 71 (1997) 1-23

17

methodology, but in addition to previous studies, we have revisited the asymptotic analysis by using more extensively the additivity of Newtonian and elastic terms in the expression of the interfacial eigenvalue (relation (16». It was also extended to the symmetrical three-layer flow which has been studied experimentally [21,22] and can give a good and easy guideline on the stability of concentric coextrusion [18]. In particular, the influence of the sinuous modes was not completely studied in the previous works. These modes slightly increase the unstable region of the Poiseuille flow. For example in Fig. 7(a,b), the sinuous mode is the first unstable when an alteration of the thickness ratio E destabilizes stable configurations itemized in Fig. 5. We have summarized in Fig. 5 some asymptotic stable situations at zero Reynolds number for both two-layer and symmetrical three-layer Poiseuille flow. So, we find that for symmetrical three layer: 1. if the high viscosity component is between the low viscosity one (m < 1) and for any elastic stratification M A, it is possible to adjust the thickness ratio E in order to reach a configuration stable with respect to long and moderate wavelength.

(a)

II

V'-"

1.0

Re. 0.5

0.0

0

4

2

6

10

q

(b)

10

(c)

1.0

5=.01

Re. 0.5

Re. 5

O+-~--+----+----+---f--~--i

o

2

4

6 q

8

10

q

10

Fig. 9. Neutral Reynolds number Reo versus wavenumber q for an Oldroyd-B symmetrical three-layer flow. Parameters: r = 1, F = 00, (Xi = 0.9, We = 0.01: (a) € = 3, m = 004, M A = 1.2, (qc> Rec ) = (0.8, 0.07) for S = 0 and (qc, ReJ = (0.54, 0.64) for S = 0.01; (b) € = 0.3, m = 0.4, M A = 0.8, (qc, ReJ = (7.6,5.08) for S = 0.01; (c) € = 0.3, m = 4, M) = 0.8, Reo (q = 0) = 0.067 and (qc, Rec ) = (0.6, 0.06) for S = 0.0 and 0.01.

18

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

2. if the outer layer is less elastic (M). < 1), there exists a thickness ratio E under which the system is asymptotically stable whatever the viscosity ratio m is. 3. if the outer layer is more elastic (M). > 1) and more viscous (m> 1), due to the sinuous mode the situation is always unstable whatever the thickness ratio E is. Finally, we point out that the stable arrangements are related to Poiseuille velocity profile and have two different origins: 1. the Poiseuille velocity profile is convex. The stability comes from viscous stratification (in the meaning that this arrangement is stable for Newtonian fluids). The elastic stratification is chosen in order to save this stability (the more elastic fluid has to be in the larger layer). However, if the elastic stratification does not preserve the stability, a stable configuration can be obtained for higher value of Reynolds number. 2. the Poiseuille velocity profile is not convex. The stability is due to the elastic properties. In fact, the elastic stratification has to balance the destabilization due to the viscosity stratification. Therefore, if the Poiseuille velocity profile is not convex and the more elastic fluid is in the thinner layer the arrangement is always unstable. Of course asymptotic stability does not ensure moderate wavelength stability in the general case. Nevertheless, it was found that at zero Reynolds number, the asymptotically stable configurations (1) remain stable with respect to moderate wavenumber perturbations. This is an important difference with respect to the purely Newtonian fluid case. A particular attention was payed in this study to parameters inducing loss of stability. Important kinematic parameters in the experiments are the flow rates at the exit of each extruder (namely Q! and Q2 for each fluid). The elasticity or viscosity ratios also depend on the melt extrusion temperature in the feedblock, but this sensitivity varies from polymer to polymer and does not easily allow all possible combinations of viscosity and elasticity ratios. For fixed viscosity and elasticity ratios, a specific thickness ratio E can be chosen by imposing a suitable flow rate ratio Q2/ Q!. Then, the Reynolds number is modified by changing the flow rates Q, and Q2 keeping the ratio Q2/Q, constant. Consequently, it is possible to deal with systems which are destabilized by a Reynolds number increase. In this way, we are in the classical framework where the Reynolds number is the bifurcation parameter. On the contrary, Anturkar et al. [13] and Chen [18] rather determine the critical value of the flow ratio or the volume ratio. As the layer thickness may be mandated by a product specification and therefore one may not be free to change it, we have considered in this paper loss of stability due to a Reynolds number increase. Let us note that the destabilization of stable configurations by changing the thickness ratio leads to the neutral curves as those plotted in Figs. 6 and 7. In this way, we have found that if the Reynolds number is small enough and if the Poiseuille velocity profile is convex, the critical thickness ratio is given by the longwave analysis. We have found that all tested configurations (determined by asymptotic analysis at vanishing Reynolds number) are destabilized at a critical Reynolds number in two different ways: 1. if the Poiseuille is convex, the critical Reynolds number is reached for a non zero critical wavenumber (q is 0(1) in the two-layer case and q depends on E in the three-layer case) and the most critical Reynolds number highly depends on the surface tension. The nonlinear behavior is described by the classical amplitude equation hI = J..lh + bIh 12h and travelling wave

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

19

pattern is expected on the extruded sheet [30]. Moreover, in the three-symmetrical layer case, it seems that the critical mode is sinuous for viscoelastic case whereas it is varicose for purely Newtonian fluid. Nevertheless, the effective value of the critical Reynold number of order unity is difficult to reach in polymer processing (it could be observed in pipe coextrusion of less viscous fluids). 2. if the Poiseuille profile is not convex, there exists a zero critical wavenumber and therefore the asymptotic analysis gives an estimate of the most critical Reynolds number. Moreover, the surface tension does not play an important role. In this case, the nonlinear behavior is described by the well known Kuramoto-Sivashinsky equation hI + hh x + dh xx + sh xxxx = 0 and very irregular interfaces are expected in this case [31]. Concerning symmetrical threelayer, the longwave critical mode is most of the time the varicose mode. In this case, the critical Reynolds number can be very small and this destabilization mechanism could take place in polymer processing condition.

Acknowledgements

This research has been partially supported by the GDR CNRS 901 'Rheologie pour la transformation des polymeres fondus'. P.L. and YD. have also been supported by grants awarded by Elf-Atochem (Serquigny, France). The authors also thank the referees for their relevant remarks.

-fi(

Co-

2

E )_12 + (m-Edo)(m-l)

Em,

with

J _ (m - E2)(m -l)/I(E, m) 1420m2d6(E + 1)2

11= (6E + l)m 7 + (- 2E - 9E 2 - l20E 3 - 88E 4 )m 6 + (32E 2 + 344E 3 + 82lE 4 + l426E 5 + 1096E 6 + 224E 7)m 5 + (408E 4 + l642E 5 + 3667E 6 + 3424E 7 + l240E 8 + 224E 9)m 4 + (224E 5 + l240E 6 + 3424E 7 + 3667E 8 + l642E 9 + 408E IO)m 3

20

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

+ (224E 7 + lO96E 8 + 1426E 9 + 821E IO + 344E ll + 32E 12)m 2 + (- 88E IO -120E II - 9E I2 - 2E 13)m + EI4 + 6E 13

J _ E3(m - E2)(m -l).fz(E, m) 2420m2d6(E + 1)2

12 =

(224 + lO43E + 1486E 2 + 672E 3)m S + (392E + 2086E 2 + 3021E 3 + lO80E 4 - 272E S)m 4 + (224E 2 + 1512E 3 + 2344E 4 + 646E S - 444E 6 + 16E 7)m 3 + (224E 4 + 424E S - 60E 6 + 6E IO + Ell

-

222E 7 + 120E 8 + 32E 9 )m 2 + (- 88E 7 -120E 8

-

9E 9

-

2E IO)m

J_ E3 (m+E) 3d o

3 -

J _ - 2E(m - E2)(m -l)(m + E)h(E, m) 4 3md6(E + 1)2

h=

m 3 + ( - 2 - 5E - 8E 2 - 2E 3)m 2 + ( - 2E 2 - 5E 4 - 8E 3 - 2E S)m + ES

J ~ (m - E2)/s(E, m) 3md6(E + 1)2

S -

Is = (3 + 16E + 11E 2 -12E 3 - 12E 4 )m 6 + (8E + 99E 2 + 242E 3 + 287E 4 + 184E 5 + 60E 6)m 5 + ( - 32€2 - 94E 3 - 142E 4 - 96E 5 - 26E 6 + 16E 7 - 4E 8)m 4 + (-16E 3 - 88E 4 - 216E 5 - 254E 6 -188E 7 - 66E 8 - 40E 9 - 12E IO)m 3 32E 6 + 176E 9 + 28E 7 + 79E 10 + 12E 11 - 16E 5)m 2 + (- 8E 8 -16E 9 + HE IO + lOE ll + 3E 12)m + 2E II + (l23E 8

-

Appendix B. Functions

Co,

J h J 2 , J4 and Js in the Case B

B.l. Varicose mode _"'( Co - l ' E,

)-1

m -

+

E(3+2E)(m-1) (2 + E)d o

with

(m - l)fl(E, m) I

J = 420m(2

+ E)2d6

P. Laure et al. / J. Non-Newtonian Fluid Mech. 71 (1997) 1-23

21

j; = (32 + 96E)m 4 + (272E 2 + 96E - 464E 3 - 352E 4 )m 3 + (252E 2 + 1428E 3 + 980E 4 + 940E s + 224E 7 + 896E 6)m 2

+ (882E 4 + 212E 8 + 642E 6 + 702E 7 + 714E S)m + 378E S + 120E 9 + 342E 8 + 438E 7 + 504E 6 + lOE IO J _ (m - I)E'iiE, m) 2 420m(2 + E) 2d6

12 = (672 + 2212E + 2172E 2 + 672E 3)m 2+ (882E + 3234E 2 + 3477E 3 + 1262E 4 + 72E S)m + 378E 2 + 1449E 3 + 1698E 4 + 762E s + 120E 6 + 10E 7 J _ - E(m - 1)(4m + 3E)(2m - 2 - (1

+ E)3)

3(2 + E) 2d5

4 -

J _ fs(E, m)

S- (2 + E) 2d6 fs = (4 + 16E + 4E 2 - 8E 3 - 4E 4)m 3+ (8E + 78E 2 + 112E 3 + 72E 4 + 24E s + 4E 6)m 2 + ( - 24E 2 - 27E 3 - 20E 4 - 19E s - 15E 6 - 6E 7 - E 8)m - 9E 3 - 18E 4 - 21E S - 15E 6 - 6E 7

B.2. Sinuous mode

CO=

E(m -1) 1+----(2 + E)(E + m)

2

J = (m - I)E (1 + E)/I(E, m) I 30m(2 + E)2(E + m)3

= (m - I)E 3(1 + E)(E 2 + 7Em + 16m 2)

J

30m (2 + E)2(E + m)3

2

(m - l)E(1 + E)(E + 4m) J4 = --------,:-------;:-3(2 + E)2(E + m)2

J = (1 + E)/sCE, m) s 3(2 + E)2(E + m)3

Is =

-

E3 + ( - 8E 2 + E3)m + (8E + 8E 2 )m 2 + (12 + 4E)m 3

22

P. Laure et al. / J. Non-Newtonian Fluid Meeh. 71 (1997) 1-23

References [1] e.D. Han, Rheology in Polymer Processing, Chapter 10, Academic Press, New York (1976). [2] D.D. Joseph and Y.Y. Renardy, Fundamentals of two-fluid dynamics, Part I: Mathematical Theory and Applications, Springer-Verlag (1992). [3] R. Sureshkumar and A.N. Beris, Linear stability analysis of viscoelastic Poiseuille flow using Arnoldi-based orthogonalization algorithm, J. Non-Newtonian Fluid Mech., 56 (1995) 151-182. [4] S. Orszag, Accurate solution of the Orr-Sommerfeld Equation, J. Fluid Mech., 50(4) (1971) 689-703. [5] Y. Saad, Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl., 34 (1980) 269-295. [6] C.S. Yih, Instability due to viscosity stratification, J. Fluid Mech., 27 (1967) 337-352. [7] S.G. Yiantsios and B.G. Higgins, Linear stability of plane Poiseuille flow of two superposed fluids, Phys. Fluids, 31 (1988) 3225-3238. Erratum Phys. Fluids AI, (1989) 897. [8] F. Charru and J. Fabre, Long waves at the interface between two viscous fluids, Phys. Fluids, 6 (1994) 1223-1235. [9] B. Khomami, Interfacial stability and deformation of two stratified power law fluids in plane Poiseuille flow, Part I. Stability analysis, J. Non-Newtonian Fluid Mech., 36 (1990) 289-303. [10] B. Khomami, Interfacial stability and deformation of two stratified power law fluids in plane Poiseuille flow, Part II. Interface deformation, J. Non-Newtonian Fluid Mech., 37 (1990) 19-36. [II] A. Pinarbasi and A. Liakopoulos, Stability of two-layer Poiseuille flow of Carreau-Yasuda and Bingham-like fluids, J. Non-Newtonian Fluid Mech., 57 (1995) 227-241. [12] Y. Renardy, Viscosity and density stratification in vertical Poiseuille flow, Phys. Fluids, 30 (1987) 1638-1648. [13] N.R. Anturkar, T.C. Papanastasiou and J.O. Wilkes, Linear stability analysis of multilayer plane Poiseuille flow, Phys. Fluids A, 2 (1990) 530-541. [14] e.-H. Li, Stability of two superposed elasticoviscous Liquids in plane Couette flow, Phys. Fluids, 12 (1969) 531-538. [15] N.D. Waters and A.M. Keeley, Stability of two stratified non-Newtonian liquids in Couette flow, J. Non-Newtonian Fluid Mech., 24 (1987) 161-181. [16] N.R. Anturkar, J.O. Wilkes and T.e. Papanastasiou, Stability of multilayer extrusion of viscoelastic liquids, AIChE J., 36 (1990) 710-724. [17] K.P. Chen, Elastic instability of the interface in Couette flow of viscoelastic liquids, J. Non-Newtonian Fluid Mech., 40 (1991) 261-267. [18] K.P. Chen, Interfacial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids, J. Non-Newtonian Fluid Mech., 40 (1991) 155-175. [19] Y. Renardy, Stability of the interface in two-layer Couette flow of an upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 28 (1998) 99-115. [20] Y.Y. Su and B. Khomami, Interfacial stability of multilayer viscoelastic fluids in slit and converging channel die geometries, J. Rheol., 36 (1992) 357-387. [21] C.D. Han and R. Shetty, Studies on multilayer film coextrusion I. The rheology of flat film coextrusion, Polym. Eng. Sci., 16 (1976) 697-705. [22] e.D. Han and R. Shetty, Studies on multilayer film coextrusion II. Interfacial instability in flat film coextrusion, Polym. Eng. Sci., 18 (1978) 180-186. [23] N.R. Anturkar, T.e. Papanastasiou and J.O. Wilkes, Estimation of critical stability parameters by asymptotic analysis in multilayer extrusion, Polym. Eng. Sci., 33 (1993) 1532-1539. [24] N.R. Anturkar, T.C. Papanastasiou and J.O. Wilkes, Stability of coextrusion through converging dies, J. Non-Newtonian Fluid Mech., 41 (1991) 1-25. [25] B. Khomami and G.M. Wilson, An experimental investigation of interfacial instability in superposed flow of viscoelastic fluids in a converging-diverging channel geometry, J. Non-Newtonian Fluid Mech., 58 (1995) 47-65. [26] E.J. Hinch, O.J. Harris and J.M. Rallison, The instability mechanism for two elastic liquids being co-extruded, J. Non-Newtonian Fluid Mech., 43 (1992) 311-324.

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23

[27] M. Renardy and Y. Renardy, Linear Stability of plane Couette Flow of an upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 22 (1986) 23-33. [28] B.S. Tilley, S.H. Davis and S.G. Bankoff, Linear stability theory of two-layer fluid in an inclined channel, Phys. Fluids, 6 (1994) 3906-3922. [29] N. Nayar and J. Ortega, Computation of selected eigenvalues of generalized eigenvalue problems, J. Comput. Phys., 108 (1993) 8-14. [30] Y. Renardy, Weakly nonlinear behavior of periodic disturbances in two-layer plane channel flow of upper-convected Maxwell liquids, J. Non-Newtonian Fluid Mech., 56 (1995) 101-126. [31] H.-C. Chang, Wave evolution on falling film, Annu. Rev. Fluid Mech., 26 (1994) 103-136.