Rheol Acta (2001) 40: 467±484 Ó Springer-Verlag 2001

Chao-Tsai Huang Bamin Khomami

Received: 20 October 2000 Accepted: 11 January 2001

C.-T. Huang á B. Khomami (&) Department of Chemical Engineering The Material Research Laboratory Washington University, St. Louis, MO 63130-4899, USA e-mail: [email protected]

ORIGINAL CONTRIBUTION

The instability mechanism of single and multilayer Newtonian and viscoelastic ¯ows down an inclined plane

Abstract The instability mechanism of single and multilayer ¯ow of Newtonian and viscoelastic ¯uids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer ¯ows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Speci®cally, for viscoelastic ¯ows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base ¯ow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian ¯ows at ®nite Re, the driving force for the interfacial instability in the limit of longwaves

Introduction Free surface ¯ow of viscoelastic ¯uids is of great importance in many industrial applications where solids are coated with polymeric ®lms. For example, in slide coating operations the ¯ow of liquid ®lms down an inclined plane is a fundamental step in production of multilayer products. The viscoelastic nature of most polymeric ¯uids can signi®cantly a€ect ¯ow instability in ¯ows that are unstable due to capillary or inertial forces. In addition, viscoelastic forces can give rise to new mechanisms of instability that are absent in ¯ow of Newtonian ¯uids (Shaqfeh 1995). In the past decade a number of studies

depends on the placement of the less viscous ¯uid. If the less viscous ¯uid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic ¯uids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base ¯ow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of ¯ow down inclined plane is the same as plane Poiseuille ¯ows (Ganpule and Khomami 1998, 1999a, b). Key words Free surface instability á Interfacial instability á Viscoelastic ¯uids á Linear stability analysis á Energy analysis

have focused their attention on elastic instabilities in simple shear ¯ows such as those occurring in viscometric ¯ows (Joo and Shaqfeh 1991, 1992a, b). These studies have identi®ed various mechanisms for viscoelastic instabilities which hopefully can be used to provide insight about instabilities occurring in more complex geometries (Shaqfeh 1995; McKinley et al. 1996). Interfacial or free surface instabilities have not received as much attention as bulk viscoelastic instabilities. However, recent studies (Ganpule and Khomami 1998, 1999a, b) have clearly demonstrated that the most prevalent mechanism which gives rise to viscometric shear ¯ow instabilities (i.e., coupling between normal stresses and stream line curvature) does not play a

468

signi®cant role in interfacial instabilities that occur in multilayer pressure or drag driven unidirectional channel ¯ows. Taking into consideration the fact that multilayer free surface ¯ows can give rise to a very rich dynamics due to potential interactions of interfacial and free surface instabilities we have focused our attention on multilayer ¯ows that contain a free surface. Specifically, the instability mechanism of multilayer Newtonian and viscoelastic ¯ows down an inclined plane has been investigated. The simplicity of the base ¯ows kinematics in inclined ¯ows in combination with the fact that experimental evaluation of free surface and interfacial instabilities can be accurately made in this class of ¯ows makes them ideal for studying free surface and interfacial viscoelastic instabilities. Moreover, as mentioned above, these ¯ows are of great importance in various coating operations where a wide range of desired properties can be achieved by constructing a layer-wise structure of various polymers. Hence, an understanding of the mechanism that drives this class of interfacial/free surface instabilities is needed. The free surface instability in ¯ow down an inclined plane manifest itself in the form of traveling waves and has been the topic numerous investigations. The free surface stability of one layer ¯ow down an inclined plane to longwave disturbances has been studied by Yih (1963) and Benjamin (1957) for a Newtonian ¯uid and by Gupta (1967) for the Upper-Convected Maxwell (UMC) ¯uid. These investigators demonstrated that the free surface can become unstable above a critical Reynolds number. In addition, it was shown that reduction of the inclination angle and presence of surface tension increase the critical Reynolds number while increasing the ¯uid elasticity decreases the critical Reynolds number. The asymptotic technique used in the above studies is only applicable to longwave disturbances. However, in general the dominant mode of the instability can be due to disturbances of any wavelength. To study the e€ect of arbitrary disturbances, Lin (1967) constructed neutral stability diagrams for one layer Newtonian ¯ow down an inclined plane using a numerical approach. Based on his analysis he demonstrated that longwave disturbances are the dominant mode of instability. Shaqfeh et al. (1989) performed a similar study for ¯ow of an OldroydB ¯uid down an inclined plane and demonstrated that even in the presence of ¯uid elasticity longwaves are the most dangerous modes. Moreover, they demonstrated that elastic e€ects are always destabilizing in the limit of longwaves but at intermediate wavenumbers they could be stabilizing at moderate Reynolds numbers. The mechanism that drives the free surface instability in single layer ¯ow of a Newtonian ¯uid down an inclined plane has been studied by Kelly et al. (1989) using a rigorous energy analysis as well as Smith (1990)

based on examination of the eigenfunctions. These analyses clearly demonstrate that the instability is caused by perturbation shear stresses at the deformed free surface. Speci®cally, the instability is due to the competition between the stabilizing in¯uence of hydrostatic pressure and the inertial destabilization. In fact, the perturbation shear stress at the free surface can drive a perturbation vorticity that, depending on its phase di€erence with the motion of the free surface, could give rise to a free surface instability. To date, the mechanism of the free surface stability of viscoelastic ¯uids down an inclined plane has not been established. The above studies have clearly demonstrated the characteristics of free surface instabilities of one layer ¯ow down an inclined plane. Since the stability of multilayer ¯ows down an inclined plane with multiple interfaces and a free surface is of great importance due to the fact that most industrial coating ¯ows have multiple layers, there exists a clear need for understanding the stability characteristics of this class of ¯ows. However, studies concerned with the stability of multilayer ¯ows down an inclined plane are comparatively fewer in number than one layer ¯ows. This can be attributed to the fact that the parameter space determining the stability of multilayer ¯ows down an inclined plane is very large. Hence, even the limiting cases are numerous. To reduce the parameter space most of the earlier studies have considered two layer ¯ows. Kao (1965a, b, 1968) was the ®rst investigator to consider the stability of two-layer Newtonian ¯ows down an inclined plane. He used longwave asymptotic techniques to demonstrate that the stability of free surface and the interface depend not only on the magnitude of the Reynolds number but also the placement of the more viscous or denser ¯uid with respect to the inclined surface. More recently, Loewenherz and Lawrence (1989) have extended the work of Kao (1965a, b, 1968) by using numerical methods to examine the neutral stability of both the interface and the free surface to arbitrary disturbance wavelengths. Based on their linear stability analysis they demonstrated that, in the limit of zero Reynolds number (i.e., where the free surface is always stable) and in absence of a density strati®cation, the interface is always unstable when the viscosity of the upper layer is greater than that of the lower layer, regardless of the thickness of the layers (i.e., an antilubrication e€ect). Chen (1993) extended the studies of Loewenherz and Lawrence (1989) to consider ®nite Reynolds number e€ects and demonstrated that variation of Reynolds number plays a signi®cant role in determining the stability of the interface. Speci®cally, he has shown that as the Reynolds number is increased the interface can be stabilized when the less viscous ¯uid is adjacent to the solid wall (i.e., the lubrication e€ect is recovered). Moreover, he has demonstrated that when the more viscous ¯uid is adjacent to the solid surface,

469

inertial e€ects destabilize the interface. In addition, it was shown that critical conditions that determine the stability of the free surface are insensitive to presence of an additional layer. Hence, they are the same as those for a single layer ¯ow down an inclined plane. The e€ect of ¯uid elasticity on stability of two-layer ¯ows down an inclined plane has also been studied by Chen (1992) using both asymptotic and numerical techniques by utilizing the UCM constitutive equation. Speci®cally, Chen (1992) considered two UCM ¯uids with a matched viscosity and density and demonstrated that if the more elastic ¯uid is adjacent to the solid surface, the interface is stable in the limit of longwaves irrespective of the thickness of the layers. In addition, it was demonstrated that the critical conditions for the stability of the free surface are insensitive to presence of an additional layer. Although the above studies have considered the linear stability of one- and two-layer ¯ows of Newtonian and viscoelastic ¯uids down an inclined plane, a number of issues have not been addressed. Speci®cally, the mechanism for instability of single layer viscoelastic ¯ows as well as multilayer Newtonian and viscoelastic ¯uids down an inclined plane has not been investigated. Hence, the objective of this study is to determine the mechanism of both interfacial and free surface instabilities in Newtonian and viscoelastic ¯ows down an inclined plane. This paper is organized as follows. First, the problem formulation and the method of solution are summarized. Then the results of our analyses are presented. Finally, the conclusions are presented.

Problem formulation Figure 1 depicts the ¯ow geometry considered in this study. The parameter space that determines the stability

of multilayer ¯ows down an inclined plane is very large. Hence, we have concentrated our e€orts on one or twolayer ¯ow of Newtonian and viscoelastic ¯uids down an inclined plane.

Governing equations The governing equations are the equations of conservation of mass and momentum, and the constitutive equation. The equations of motion and continuity can be expressed as * rP

*

q

Du ˆ Dt

* * r
s‡qg ;

…1†

*

r
uˆ0 ; *

…2† *

where u; P; g; and s denote the velocity vector, isotropic pressure, gravitational acceleration, and the deviatoric stress tensor. Numerous constitutive equations may be identi®ed as candidates for studying the stability of viscoelastic ¯ows. However, since our aim in this study is to identify the e€ect of ¯uid elasticity on the stability of multilayer ¯ows down an inclined plane, we have selected a relatively simple constitutive equation, namely the Upper Convected Maxwell (UCM) model. The UCM model can be derived from a molecular theory in which the polymer molecules are modeled as non-interacting Hookean elastic dumbbells (Bird et al. 1987). Although this model gives rise to relatively simple material properties (i.e., constant viscosity and ®rst normal stress coecient), it retains the essential physics necessary for investigating the e€ects of various parameters such as viscosity strati®cation, elasticity strati®cation, and inertia on the instability mechanism of ¯ows under consideration. The upper convected Maxwell model (UCM) is given by sp ‡ ksp…1† ˆ

gp c_ ;

…3†

where k is the mean relaxation time of the polymer and gp is the polymer contribution to the shear viscosity. The subscript 1, denotes the upper convective derivative de®ned as X …1† ˆ

oX * * ‡ U
rX ot

** …r U †T
X

** X
rU ;

and the deformation rate tensor is given by   ** ** T ; c_ ˆ r U ‡ r U Fig. 1 Schematic of the two layer ¯ow down an inclined plane

…4†

…5†

The following set of dimensionless variables is introduced to nondimensionalize the above equations:

470

x1 x2 q g Sinb d 2 x ˆ ; y ˆ ; Rek ˆ k …k† k d1 d1 g ek ˆ

qk dk g…k† t  U0 ; dk ˆ ; mk ˆ …1† ; t ˆ ; q1 d1 d1 g

Pk P k ˆ g…1† U ; skij ˆ d1

0

skij g…1† U0 d1

; Sk ˆ

rk …1† g U

0

; Wek ˆ

…6† k1 U0 d1

skyy ˆ 0

where U0 is the velocity at the top of layer 1, d1 is the thickness of layer 1, q is the ¯uid density, r is the interfacial or surface tension, and b is the inclination angle. The subscripts and superscripts k ˆ 1, 2 denote the layer position with respect to the solid surface (see Fig. 1). Base ¯ow For steady one-dimensional two-layer ¯ow down an inclined plane the equation of motion in each phase reduces to oskxy oy

‡ qk g x ˆ 0

k ˆ 1; 2

  dUk k ˆ 1; 2 ˆ mk dy   dUk k k k k ˆ 1; 2 sxx ˆ 2…We †sxy dy skxy

…7†

and the nonzero components of the constitutive equation are given by  
dUk k k k ˆ 0 k ˆ 1; 2 …8† sxx 2 We sxy dy   dUk ˆ 0 k ˆ 1; 2 …9† skxy ‡ mk dy The base ¯ow velocity and stress ®elds are obtained by solving the above equations with the following boundary conditions, namely, no slip at the solid boundary:

…14† …15†

k ˆ 1; 2

…16†

The coecients in the above equations are given by the following expressions: 1 e2 d2 C1 ˆ 0 B 2 ˆ Ak ˆ 2 m2 ek 1 2 Ck ˆ d ‡ Bk 1 dk 1 ‡ ck 2 mk 1 k 1  ek 1 d k 1 ‡ Bk 1 mk Bk ˆ mk 1 mk 1

ek 2mk

A1 ˆ

1

kˆ2

…17†

Perturbation ¯ow The linear stability analysis is based on the classical normal mode expansion technique. Any ¯ow variable Z is expressed as the sum of Zss and a perturbation Zp with complex eigenvalue c ˆ cR+ici, wavenumber a, and an inhomogeneous dependence on the y-coordinate, i.e., Z ˆ Z ss ‡ Real …Z p …y† exp…i a…x

ct††

…18†

For the constitutive equation used in this study the Squire's theorem is valid (Talpa and Bernstein 1970). Hence we have only considered the stability of the ¯ow to two-dimensional disturbances, i.e., the perturbation vector Zp is de®ned as follows:     Z p ˆ u0 ;v0 ;p0 ;dij ˆ U …y†;V …y†;f …y†;Fij …y† exp…i a…x ct†† …19†

The linear stability equations are derived by substituting Eq. (19) into the governing equations and keeping only the terms that are linear with respect to the Continuity of velocity and shear and normal stresses perturbation quantities. To reduce the number of at the interface (i.e., at yk)1 ˆ dk)1 and yk ˆ 0): variables, the linear stability equations are recast in k 1 k Uk 1 …y ˆ dk 1 † ˆ Uk …y ˆ 0†; sxy …y ˆ dk 1 † ˆ sxy …y ˆ 0†; terms of the perturbation stream function F de®ned as follows: … p ‡ syy †k 1 ˆ … p ‡ syy †k k ˆ 2 U ˆ /…y† exp…i a…x ct†† …20† …11† where and vanishing of shear stress at the free surface: oU oU …12† u0 ˆ sxy …y ˆ d2 † ˆ 0 and v0 ˆ …21† oy ox Although the constitutive equation is nonlinear, for the ¯ow under consideration one can obtain an analyIn turn the pressure terms in the x and y component tical solution for the base ¯ows velocity and stress ®elds: of the linearized equation of motion are eliminated by cross di€erentiation and the stability governing equa…13† tions are obtained: Uk ˆ Ak yk2 ‡ Bk yk ‡ Ck k ˆ 1; 2 * U1 ˆ 0

…10†

471



 2   d /k 2 i aek Re1 …Uk c† a /k /k U k dy 2 ( 2 ! )   n  o Fxyk
o 2 k k k F F k ˆ 1; 2 ˆ ‡ a ‡ i a F xy yy xx oy 2 …22†
Fxxk 1 ‡ i a…Uk c†Wek     2 dskxx d/k k k dUk k d /k k ‡ 2sxx i a ‡ 2Fxy ‡ 2sxy ˆ We i a/k dy dy dy dy 2   2  d/ dUk d/ 2 2k ‡ 2a2 /k k ˆ 1; 2 2mk i a k dy dy dy …23†  Fxyk 1 ‡ i a…Uk "

 c†Wek # dskxy k k dUk 2 k ‡ Fyy ‡ a /k sxx ˆ We i a/k dy dy  2  d /k 2 ‡ a /k k ˆ 1; 2 mk dy   Fyyk 1 ‡ i a…Uk c†Wek   d/ k ˆ 1; 2 ˆ 2Wek a2 /k skxy ‡ 2mk i a k dy

…24†

and

d/1 ˆ0 dy

…25†

…26†

Continuity of the x and y components of velocity at the interface (i.e., at yk-1=dk-1 and yk=0): / k 1 ˆ /k ;    d/k 1 d/k dUk ‡ dy dy dy

1

 dUk hk dy

dy

dFxyk

1

!

dy

 d/ dUk ‡ i aFxxk i aFyyk ‡ i aUk Rek k i aRek /k dy dy  d/ dUk 1 i aUk 1 Rek 1 k 1 i aRek 1 /k 1 dy dy 
‡i aFxxk 1 i aFyyk 1 i a FF …k 1† ‡ a2 S…k 1†



 d/ ˆ i c aRek k dy

aRek

d/k 1 dy

1

 ; kˆ2

…29†

and balance of tangential and normal stresses at the free surface (i.e., at yN=dN): FxyN ‡

dsNxy dy

hs

 i a sNxx

 sNyy hs ˆ 0;

…30†

N ˆ2

 dUN ‡ i a FxxN dy dy
d/ ‡i a FF …N † ‡ a2 S…N † ˆ i aReN c N ; N ˆ 2 dy

dFxyN

The boundary conditions for the perturbation ¯ow problem are as follows: No slip at solid boundary: /1 ˆ 0

dFxyk

‡ i aReN UN

d/N dy

i aReN /N

FyyN



…31† In the above equations, /N ˆ …c

Un †hs

and /k

1

ˆ …c

Uk 1 †hk

1

KˆN ˆ2 …32†

where hk)1 denotes the amplitude of interface deviation from its unperturbed position and hs denotes the amplitude of free surface deviation from its unperturbed position. In addition, FF(K) ˆ (ek‡1 ek )Cotb and FF(N) ˆ eN (Cotb) represent the e€ect of gravitational * acceleration g. Solution procedure

1

ˆ 0;

kˆ2 …27†

Balance of stress normal to the interface (i.e., at yk)1=dk)1 and yk=0): ! k 1 k   ds ds xy xy Fxyk 1 Fxyk ‡ hk 1 dy dy kˆ2 h   i k 1 k 1 k k sxx syy ˆ 0 i ahk 1 sxx syy …28† Balance of stress tangential to the interface (i.e., at yk)1=dk)1 and yk=0):

The stability of the multilayer ¯ow down an inclined plane is governed by the modi®ed Orr-Sommerfeld equation, perturbation stress equations, and boundary conditions. To determine the stability of the perturbation ¯ow problem, a generalized eigenvalue problem, * * Ax ˆ cBx, with the complex wave speed c as the eigenvalue and the disturbance amplitudes /, Fxx, Fxy, and Fyy as the eigenfunctions must be solved. A multidomain spectral-Tau method coupled with the QR algorithm has been used to determine the eigenvalues and eigenfunctions. The details of the methods of solution are given in detail in our earlier publications (Ganpule and Khomami 1998, 1999a, b; Su and Khomami 1991, 1992a, b); hence they are not being reproduced here.

472

Energy analysis To investigate the mechanism of the interfacial and/or free surface instability of multilayer ¯ows down an inclined plane, the method of disturbance energy analysis can be used. Speci®cally, this approach allows one to examine the energy transfer between the mean ¯ow and the disturbance ¯ow through a disturbance energy balance. The method of energy analysis used in this study was originally developed by Joo and Shaqfeh (1991, 1992a, b) and then modi®ed by Ganpule and Khomami (1998, 1999a, b). The disturbance energy equation is obtained by multiplying the linearized perturbation momentum equation by the perturbation velocity and integrating over a unit volume in space (i.e., from y1 to yN, y1 ˆ 0
 Z  * *0 0 ˆ r
I p
u dV

…36†

where I is the unit tensor, Z  2  * *0 *0 r u
u dV /vis ˆ mk

…37†

…38†

/p is given by the following expression:   Z * *0 * *0 /p ˆ We r
u
r ~sp
u dV   Z * * * 0 *0 ~ ‡ We r
U
r sp
u dV "   # Z * **0 T **0 *0 We r
r u
~sp ‡ ~sp
r u
u dV We

* r


"

** rU

T

and dVD d ˆ We mk dt dt

   # Z " ** 0 ** 0 T ru : ru dV

…40†

…41†

is the rate of change total viscous dissipation. In the above equations, dKE signi®es the rate of change of dt kinetic energy with time, /rey corresponds to Reynolds stresses, /pre is the energy associated with perturbation pressure, and /vis is the viscous dissipation energy term. /el is the viscous component of the power associated with the polymeric stress and /jump is the term representing jump in physical properties across an interface or a free surface. /p is the energy associated with the coupling between base ¯ow and perturbation ¯ow velocities and stresses and their gradients. In their studies, Ganpule and Khomami (1998, 1999a, b) have veri®ed the accuracy of their approach by comparing the results of their analysis in a TaylorCouette ¯ow with those of Joo and Shaqfeh (1991, 1992a, b). To demonstrate the accuracy of our simulation code we have compared our results with the earlier results of Kelly et al. (1989) for one-layer Newtonian ¯ow down an inclined plane. The speci®cs of these comparisons will be discussed in the next section.

Results and discussion One-layer ¯ow down an inclined plane Newtonian ¯uids

where mk is the viscosity ratio, P Z  d p *0
u dV /el ˆ We r
dt

Z

/jump

2 0 2 0

0o u

0 o v

‡v ˆ We mk u otoy otoy

# ** *0
~sp ‡ ~sp
r U
u dV 0

0



…39†

The mechanism of instability for one-layer Newtonian ¯ow down an inclined plane in the limit of longwave disturbances (i.e., the most dangerous disturbances) has been previously investigated by Kelly et al. (1989) using a rigorous energy analysis and by Smith (1990) using examination of the eigenfunctions. Their results indicate that the free surface instability is caused by the perturbation shear stress at the free surface. Speci®cally, the instability is due to the competition between the stabilization provided by the hydrostatic pressure and the inertial destabilization. Moreover, Kelly et al. (1989) have shown that the perturbation shear stress drives a perturbation vorticity that can either enhance or reduce the surface deformations. In fact, they clearly demonstrated that the stability of the free surface can be determined by examining the phase shift between the disturbance vorticity and the free surface shape. The method of energy analysis used in this work is signi®cantly di€erent than that used by Kelly et al.

473

Fig. 2a, b Results of the energy analysis of one-layer Newtonian ¯ow down an inclined plane, a ˆ 0.05, b ˆ 45.3°: a terms of the disturbance energy equation as a function of Re; b the interfacial and bulk contributions to the term which represents the mechanism of instability in a

(1989). Hence, to check the accuracy of our technique a disturbance energy analysis is conducted. The result of this analysis is shown in Fig. 2. Figure 2a clearly shows that in the limit of longwave disturbances the Fvis term is the driving force for the instability. Further, Fig. 2b shows that the free surface contribution is the driving mechanism for the instability. These results are in complete agreement with those of Kelly et al. (1989). Moreover, our method of analysis allows us to di€erentiate between bulk and surface contributions. Hence, a more complete picture of the instability mechanism can be obtained with our methodology. Clearly one can express these perturbation velocities in terms of perturbation vorticities. The phase di€erence between the perturbation vorticity and the interface position can be used to illustrate the stability of the ¯ow. If one focuses at the peak of the perturbed interface (see Fig. 3a), the maximum value of the vorticity at the free surface lags that of the perturbed free surface (i.e., downstream), hence the vorticity will induce an upward motion of the interface resulting in growth of disturbances. If the maximum in vorticity leads the maximum amplitude of the perturbed free surface, disturbances will decay. Clearly the critical Re must correspond to a

Fig. 3a, b Phase di€erence between the perturbation vorticity and the motion of the free surface in single layer ¯ow of Newtonian ¯uid down an inclined plane, a ˆ 0.05, b ˆ 45.3°: a a positive phase di€erence; b phase di€erence as a factor of Re

phase di€erence F ˆ 0 and for F>0 the ¯ow is unstable while for F<0 the ¯ow is stable due to the facts described above. Indeed, this is what is observed (see Fig. 3b). Viscoelastic ¯uids Following the strategy of Smith (1990), the mechanism of the instability is investigated by a careful examination of the eigenfunctions. For purely elastic instabilities (i.e., Re ˆ 0), the equation of motion is reduced to   oFxy x-component : 0 ˆ iap0 ; …42† iaFxx ‡ oy   oFyy op0 iaFxy ‡ ; …43† y-component : 0 ˆ oy oy The boundary conditions are given by at y ˆ 0;

u0 …0† ˆ 0; v0 …0† ˆ 0 ;

at y ˆ 1;

Fxy ‡

dsxy 0 h ˆ0 ; dy

…44† …45†

474

at y ˆ 1;

p0 ‡ Fyy ‡

at y ˆ 1;

v0 ˆ ia…U

dp0 0 h ˆ0 ; dy

…46†

c†h0 ;

…47†

Solving the above set of equations in the limit of longwave disturbances, a ˆ 0, (i.e., the most dangerous disturbances) one obtains ou0 ; oy

Fxy0 ˆ

o2 u0 ˆ 0; oy 2

Fyy0 ˆ 0 ; ou0 …1† ˆ2 ; oy

u0 …0† ˆ 0;

op0 ˆ 0; oy

…48† …49†

p0 …1† ˆ 2 cot b ;

u0 …y† ˆ 2y;

v0 ˆ 0;

…50†

p0 …y† ˆ 2 cot b ;

…51†

At order a, the following equations are obtained: ov1 ‡ iu0 ˆ 0; v1 …0† ˆ 0; c0 ˆ U …1† ‡ iv1 …1† ; …52† oy v1 …y† ˆ 0ˆ

iy 2 ;

i p0

…53†

c0 ˆ 2 ; oFxy1

iFxx0

oy

  dU dy

; ou0 ; oy

2Fxy0

Fxy1

   ou1 1 dU ‡ i 2We Fyy dy oy

ˆ

o2 u1 ˆ ip0 oy 2

i We

1 Wecrit ˆ cot b ; …63† 2 A close examination of the relationship between perturbation shear stresses, pressure and elastic forces at O(a) (i.e., Eq. 57) reveals that the balance between the hydrostatic pressure due to the displacement of the interface and elastic forces determine the sign of the perturbation velocity in the mean ¯ow direction. It should be noted that there are two separate elastic contributions. ®rst elastic contribution, i.e.,   The  0 We dU ou , is due to the coupling between the oy dy base ¯ow shear stress  the perturbation velocity  and dU and ou0 are always positive in gradient. Since oy dy the domain, this elastic term is always negative and is proportionalto the  We. The second elastic contribu2 tion, i.e., We u0 d U2 , is due to the coupling between dy the perturbation velocity and the base ¯ow shear stress 2 gradient. Since d U2 is always negative and u0 is always dy

…54†

Fxx0 ˆ

2sxy

Therefore

…55† …U

    dU ou0 dy oy

ou1 ˆ0 ; oy  2  y u1 …y† ˆ ip0 y i We 2y 2 2

 dsxy We v1 ; dy …56†

c0 †Fxy0

u0

d2 U dy 2

 ;

u1 …0† ˆ 0;

…57† …58†

4y




;

…59†

At order a2, the following equations are obtained: ov2 ‡ iu1 ˆ 0; v2 …0† ˆ 0; c1 ˆ iv2 …1† ; oy  3  y 2 We y 3 ‡ 2We y 2 ; y2 v2 …y† ˆ cot b 3 3   2 4 cot b ‡ We ; c1 ˆ i 3 3

…60† …61† …62†

Fig. 4a±c The physical mechanism for purely elastic free surface instability of one-layer UCm ¯ow down an inclined plane in the limit of longwaves: a hydrostatic pressure b the ®rst elastic  contribution;  ou dU 0 ; c the second elastic contribution which is We oy dy    2 contribution which is We u0 d U2 dy

475

positive, this term is also always negative in the domain. Hence, if one considers a perturbed interface, the hydrostatic pressure term tends to push ¯uid away from under the crest while the elastic terms tend to introduce ¯uid under the crest (i.e., the elastic terms are destabilizing while the hydrostatic pressure is stabilizing). Therefore, when the contribution of the elastic terms that are proportional to the We exceeds a certain critical value the ¯ow becomes unstable (see Fig. 4). The magnitude of each term can be examined by measuring the power associated with it. The average power is de®ned: Z 1      dU ou0 d2 U Xˆ p0 We u0 2 dy dy dy dy 2 0 ˆ 2 cot b

2We

2We :

…64†

This expression clearly shows that in the limit of longwaves, elastic e€ects are always destabilizing and proportional to  addition, the ®rst elastic  the We.In ou dU 0 stress, We , and the second one,   dy oy 2 We u0 d U2 , have the same magnitude as depicted in dy Fig. 4b,c. A rigorous energy analysis has also been performed to investigate the instability mechanism for one-layer viscoelastic ¯uid down an inclined plane. Figure 5a, b shows the variation of various terms in the disturbance Fig. 5a±d Results of energy analysis for one-layer viscoelastic ¯uid down an inclined plane, a ˆ 0.05, b ˆ 45.3°: a, b Re ˆ 0, instability is due to the (Fp)jump term; c, d We ˆ 0.1, in presence of inertia, instability is due to the (Fvis)jump term

energy equation as a function of We. Clearly, the Fp term (i.e., the coupling between base ¯ow and the perturbation velocities and stresses and their gradients due to the polymeric stresses) is the mechanism of instability. Furthermore, by investigating the bulk and free surface contributions of this term, it can be shown that the free surface component of (Fp)jump is the main driving mechanism for the instability. As expected as one increases the Reynolds number while keeping the We constant the instability mechanism for the Newtonian ¯ow down an inclined plane, namely, the

Fig. 6 The phase di€erence between the perturbation vorticity and the motion of the free surface for one-layer viscoelastic (UCM) ¯ow down an inclined plane, Re ˆ 0, a ˆ 0.05, b ˆ 45.3°

476

interfacial contribution of Fvis is recovered (see Fig. 5c,d). As mentioned before, one can express these velocity perturbations in terms of perturbation vorticities. Then the phase di€erence between the perturbation vorticity and the motion of the free surface can be used to illustrate the stability/instability of the ¯ows. Figure 6

depicts the phase di€erence between the perturbation vorticity and the free surface motion. Speci®cally, it is shown that F ˆ 0 corresponds to the critical We. Two-layer ¯ows down an inclined plane Newtonian ¯uids Free surface mode. The instability mechanism for the free surface mode in two-layer Newtonian ¯uid down an inclined plane is similar to that of single layer ¯ow. Figure 7 clearly shows that the free surface contribution of the Fvis term provides the driving force for the instability. Moreover, a similar behavior in the phase di€erence between the vorticity and interface position is observed (see Fig. 7c). Overall, these results indicate that presence of the interface does not greatly alter the stability characteristics of the free surface in two-layer Newtonian ¯ows down an inclined plane. This fact is further con®rmed based on an eigenfunction analysis. Speci®cally, the eigenfunctions clearly demonstrate that the hydrostatic pressure does provide stabilization. Hence, below the critical Re, the perturbation stresses at the free surface are not sucient to cause an instability (see Fig. 8).

Interfacial mode Longwavelength disturbances. In the limit of Re ˆ 0 the free surface mode is always stable. However, perturbation at the interface could lead to de¯ection of the free surface. This is particularly important in the limit of longwaves where the magnitudes of perturbations are signi®cant throughout the ¯ow domain. Speci®cally, when the less viscous ¯uid is adjacent to the inclined plane, the disturbance at the interface travels in the opposite direction of the mean ¯ow and the free surface disturbance travels along the ¯ow direction while being

Fig. 7a±c Stability mechanism of the free surface mode of two-layer Newtonian ¯ows down an inclined plane, m ˆ 2.0, d ˆ 0.75, a ˆ 0.01, b ˆ 11.5°: a terms of the energy equations as a function of Re; b the interfacial and bulk contributions of the term which represents the instability in a; c the phase di€erence between the disturbance vorticity and motion of the perturbed free surface

Fig. 8 Interface and free surface shapes for two-layer Newtonian ¯ows down an inclined plane at Re ˆ 0, a ® 0

477

out of phase with the interface motion by p. When the more viscous ¯uid is adjacent to the inclined plane the disturbance velocities at the interface and the free surface travel exactly in the opposite direction of the case mentioned above. Moreover, they are in phase. It should be noted that in both cases the forward and reverse traveling waves have identical speeds irrespective of the depth and the viscosity ratio. Hence, the instability is clearly caused by the interactions of the free surface and the interface in a very similar fashion to resonant instability observed in three-layer ¯ows down an inclined plane (Weinstein and Kurt 1991). Figure 9 shows a typical plot of the disturbance vorticity x at the peak of the longwave disturbance. As expected, interfacial perturbations persist all the way to the free surface; moreover, the magnitude of the disturbance vorticity is more signi®cant in the less viscous layer. To examine the mechanism of the instability a careful examination of the disturbance eigenfunctions has been performed. Speci®cally, in the limit of longwaves and absence of inertia, the stability of two-layer Newtonian ¯ow is determined by …I†

Layer 1 :

o2 u1 p0 ˆi 2 oy lI

Layer 2 :

o2 u1 p0 ˆi oy 2 lII

…II†

Fig. 9a, b The perturbation interfacial vorticity x at the peak of the disturbance for two-layer Newtonian ¯ows down an inclined plane. b ˆ 11.5°, Re ˆ 0: a m ˆ 0.5; b m ˆ 2.0

Fig. 10a, b Interface and free surface shapes for two-layer Newtonian ¯ows down an inclined plane at Re ˆ 0, a ® 0: a m>1 (out-of-phase by p); b m<1 (in phase)

…65† …66†

p0 ˆ 2Cotb

…67†

where u1(k) is the ®rst order perturbation velocity of layer k, and lk is the viscosity of the k-layer. Figure 10 depicts the con®gurations between the interface and free surface associated with the interfacial mode eigenvalue at zero Re. When the less viscous ¯uid is  lII adjacent to the solid wall m ˆ l > 1 , the free I surface and interface are out-of-phase by p (see Fig. 10a). In this con®guration, the hydrostatic pressure of the bottom layer (layer I) tends to push ¯uid from under the crest (it is stabilizing) while the hydrostatic pressure of the upper layer (layer II) tends to push ¯uid from under the trough (it is destabilizing). Hence, the disturbance at the free surface grows, while the disturbance at the interface should decay. However, since the motion of the interface and the free surface are highly coupled, and the disturbance vorticity in the less viscous ¯uid is larger, this results in an unstable ¯ow. Moreover, Fig. 10b shows the con®guration of the free surface and interfacial modes when the more
viscous ¯uid is adjacent to the solid wall m ˆ llII < 1 . I Clearly, since the hydrostatic pressure terms can push ¯uid from under the crest of the interface and free surface, the ¯ow is stable. As mentioned earlier, a simple way to visualize these instabilities is to examine the disturbance vorticity in each layer. Figure 11a shows that when the less viscous ¯uid is adjacent to the inclined plane, the

478

Fig. 11a, b The disturbance vorticity and the position interface as a function of dimensionless time for two-layer Newtonian ¯ows down an inclined plane. Re ˆ 0, a®0: a m>1.0 (unstable); b m<1.0 (stable)

phase di€erence between the disturbance vorticity and the motion of the interface is out-of-phase by p. Since the disturbance vorticities in both phases are in phase (i.e., they are both either clockwise or counter clockwise), then in order to determine which one plays a more dominant role their magnitude must be examined. As mentioned earlier, the disturbance vorticity is always larger in the less viscous ¯uid. Hence, as shown in Fig. 11a the net e€ect of the disturbance vorticity will induce a downward motion of the interface leading to destabilization of the interface where the less viscous ¯uid is adjacent to the interface. Since in the longwave regime the motion of the interface and free surface are highly coupled, this destabilization of the interface will lead to de¯ection of the interface from its unperturbed position such that mass is conserved. When the more viscous ¯uid is adjacent to the inclined surface, the reverse is true and the interfacial mode is stable (see Fig. 11b). As expected these results are consistent with the results of the eigenfunction analysis. The mechanism of instability as the Re is increased can also be examined by a careful examination of the disturbance eigenfunctions. When the less viscous layer is adjacent to the wall (m>1), inertia will reduce the destabilization due to the interaction of the free surface and interface (see Fig. 10) by shifting the position of the maximum pressure from under the

Fig. 12a, b The inertial stabilization mechanism in the limit of a®0 for two-layer Newtonian ¯ows down an inclined plane when the less viscous ¯uid is adjacent to the solid wall, i.e., m ˆ 2.0, b ˆ 11.5°: a terms of the disturbance energy equation as a function of Re1; b the interfacial and bulk contributions to the term which represents the mechanism of instability in a

trough (see Fig. 10a). Using the same argument, it can be seen that when the more viscous layer is adjacent to the solid wall, inertia is always destabilizing (see Fig. 10b). To examine further the mechanism of instability in presence of inertia, we have performed a disturbance energy analysis. In the limit of longwaves, when the less viscous ¯uid is adjacent to the solid plane, the presence of inertia will stabilize the interfacial disturbance. Figure 12 shows the terms of the disturbance energy equation as a function of Re1. Clearly, the Fvis term is the mechanism of this stabilization. Furthermore, the interfacial contribution of Fvis (i.e., the combination of pressure and perturbation viscous stresses) is the source of the stabilization. Moreover, when the more viscous ¯uid is adjacent to the solid wall, inertial e€ects are always destabilizing. Figure 13 depicts the terms of disturbance energy equation as a function of Re. Clearly, the FRey term is the source of this destabilization. The bulk contribution of the Reynolds stress, i.e., (FRey)Bulk, gives rise to destabilization. Shortwavelength disturbance. In the limit of shortwaves, the eigenfunctions decay away from the interface at an exponential rate, hence their magnitude becomes negligible in the bulk of each ¯uid. Therefore, the mechanism of interfacial instabilities is independent of the layer depth ratio and the coupling of interfacial and free

479

Fig. 14a, b The mechanism of interfacial stability in the limit of shortwaves. a ˆ 40, Re ˆ 0.0, b ˆ 11.5° Fig. 13a, b The inertial destabilization mechanism in the limit of a ® 0 for two-layer Newtonian ¯ows down an inclined plane when the more viscous ¯uid is adjacent to the solid wall, i.e., m ˆ 0.5, b ˆ 11.5°: a terms of the disturbance energy equation as a function of Re1; b the interfacial and bulk contributions to the term which represents the mechanism of instability in a

surface motion. Hence, the mechanism of interfacial instabilities is the same as that observed by Ganpule and Khomami (1998, 1999a, 1999b) in the channel ¯ows. Figure 14a shows Fvis (the viscous term) is the mechanism of instabilities at shortwaves. Speci®cally, the interfacial friction term is the cause of the instability as shown in Fig. 14b. Viscoelastic ¯uids Free surface mode. The results of the disturbance energy analysis for two-layer viscoelastic (UCM) ¯ows is shown in Fig. 15. Clearly, the Fp term (i.e., the coupling between base ¯ow and the perturbation velocities and stresses and their gradients due to the polymeric ¯uid) gives rise to the instability. Speci®cally, the interfacial contribution of the (Fp)jump is the main driving force for the instability. Hence, the physical mechanism of the instability of the free surface mode in two-layer viscoelastic ¯ows down an inclined plane is

the same as that of one-layer viscoelastic ¯ow down an inclined plane. As mentioned before, free surface de¯ection leads to creation of perturbation shear stresses and these perturbation shear stresses give rise to a perturbation vorticity. Figure 15c depicts the phase di€erence between the perturbation vorticity and the motion of the free surface under stable or unstable conditions. Interfacial Mode Longwave disturbance. In the absence of viscosity strati®cation and ¯uid inertia, depending on the position of the more elastic ¯uid, the phase di€erence between the interfacial and free surface disturbances (i.e., based on the interfacial eigenvalue) could be p2 or 3p 2 (see Fig. 16). Clearly in the limit of Re ˆ 0, elastic e€ects are sucient to give rise to a phase di€erence between the motion of the free surface and the interface. However, to examine how this phase di€erence a€ects the stability of the interface one must examine the disturbance eigenfunctions. Figure 17 depicts the terms of disturbance energy equation as a function of We1 for longwave distur-

480

Fig. 15a±c Results of energy analysis for the free surface mode of two-layer viscoelastic (UCM) ¯ows down an inclined plane, m ˆ 2.0, d ˆ 0.75, a ˆ 0.01, b ˆ 11.5°, We1 ˆ 0.1, Re ˆ 0.0: a terms of the energy equation as a function of We; b the interfacial and bulk contributions to the term which represents the instability mechanism a; c the phase di€erence between the perturbation vorticity and motion of the free surface

Fig. 16 Interface and free surface shapes for ¯ow of two-layer UCM down an inclined plane. Re ˆ 0, m ˆ 1, a®0

bances. Clearly, the Fp term (i.e., the coupling between base ¯ow and the perturbation velocities and stresses and their gradients due to the polymeric stresses) is the mechanism of instability. Furthermore, the (Fp)jump provides the main driving force for the instability. As mentioned before, the phase di€erence between the disturbance vorticity at the interface and the motion of the interface can be used as a tool to illustrate the stability of the interface. Figure 18a depicts the phase di€erence between the vorticity and the interface when the more elastic layer is adjacent to the inclined plane. This phase di€erence, in addition to the fact that the disturbance vorticity in the more elastic and less viscous ¯uids have di€erent signs (i.e., one is counter clockwise and the other is counter clockwise ± see Fig. 19), will induce an upward motion of the peak and downward motion of the trough. Hence, the deformations of the interface will be enhanced and the ¯ow becomes unstable. In Fig. 18b, the con®guration when the less elastic ¯uid is adjacent to the inclined plane, the opposite trend is observed; hence the disturbance is

Fig. 17a, b Results of energy analysis for two-layer UCM ¯ows down an inclined plane (i.e., associated with the interfacial mode), Re ˆ 0, d ˆ 0.75, We2 ˆ 0.5, m ˆ 1.0, b ˆ 11.5°: a terms of the energy equation as a function of We; b the interfacial and bulk contributions of the term which represents the instability mechanism in a

481

dissipated. It should be noted that changes in depth ratio do not a€ect the mechanism of the instability. It has been noted that the instability mechanism in presence of both viscosity and elasticity strati®cation is very dicult to ascertain based on an energy analysis because of the interaction of viscous and elastic forces (i.e., the purely viscous and elastic instabilities have di€erent mechanisms; hence when both viscosity and elasticity strati®cations are present both mechanisms

are important and it is very dicult to separate their individual contributions) (Ganpule and Khomami 1998, 1999a, 1999b). Therefore, we have chosen to examine the eigenfunctions since they contain all the necessary information. Speci®cally, to illustrate the mechanism of interfacial instabilities for two-layer viscoelastic ¯uids we have examined the eigenfunction as well as the phase di€erence between the vorticity and the motion of the interface. Figure 20a shows that, in the absence of viscosity strati®cation if the more elastic ¯uid is in the top layer, a phase di€erence of p2 between the disturbance vorticity and the interface position exists that results in a stable interface. As the viscosity ratio is increased from 1.0 to 10, the phase di€erence between the vorticity and the interface position is increased to Dh ˆ p and the interface is destabilized. This clearly shows the competition between stabilizing e€ects of elasticity strati®cation and destabilizing e€ects of viscous strati®cation. The opposite trend is observed when the viscosity ratio is decreased from 1.0 to 0.1. In fact, the phase di€erence between the disturbance vorticity and the motion of the interface is gradually reduced from p2 (i.e., 0 < Dh < p2) and the interface is stabilized. Shortwave disturbance. For the reasons mentioned above, the mechanism of the instability is expected to be the same as channel ¯ows studied earlier by Ganpule and Khomami (1998, 1999a, 1999b). Indeed this is the case. As shown in Fig. 21a, for a ®xed viscosity ratio the instability is due to Fpv2, which signi®es the coupling between the base ¯ow stresses and the gradient of the perturbation velocity. However, if the elasticity strati®cation is held ®xed and the viscosity ratio is enhanced, the instability is caused by the Fvis term and the interfacial component of this term is the cause of the change in the stability characteristics (see Fig. 21b). Hence, the interplay between the viscosity and elasticity strati®cation is the cause of the instability.

Fig. 18a, b The disturbance vorticity and the interface position as a function of dimensionless time for two-layer UCM ¯ows down an inclined plane. Re ˆ 0, a®0, m ˆ 1.0, d ˆ 1.0. x1 is the vorticity of the lower layer; x2 is the vorticity of the upper layer: a We1 ˆ 1.0, We2 ˆ 0.1; b We1 ˆ 0.1, We2 ˆ 1.0

Conclusions

Fig. 19a, b Perturbation vorticity (i.e., associated with the interfacial mode) for two-layer UCM ¯ows down an inclined plane. Re ˆ 0, m ˆ 1, a®0: a We1 ˆ 0.1, We2 ˆ 5.0; b We1 ˆ 5.0, We2 ˆ 0.1

We have performed linear stability and disturbance energy analysis to investigate the mechanism of insta-

482

Fig. 20a±e The disturbance vorticity and the interface position as a function of dimensionless time for two-layer UCM ¯ows down an inclined plane. Re ˆ 0, a®0, d ˆ 1.0: a, b m ˆ 1.0; c, d m>1.0; e m  1.0

bility in single and multilayer ¯ows of Newtonian and viscoelastic ¯uids down an inclined plane. For one-layer Newtonian ¯ows down an inclined plane, inertia is destabilizing and the main driving force for instability are the perturbation shear stresses at the free surface. In the case of one-layer viscoelastic ¯ows down an inclined plane, the elastic forces are destabilizing and the main driving force for instability is the coupling between the base ¯ow and the perturbation velocity and stresses and their gradients at the free surface. For two-layer ¯ows down an inclined plane, there exists two modes of instability. In general, the instability mechanism for the free surface mode in these ¯ows are the same as one layer ¯ows down an inclined plane. The

driving force for the interfacial instability of Newtonian ¯uids, in the limit of Re ˆ 0, is hydrostatic pressure. At ®nite Re, when the less viscous ¯uid is adjacent to the solid plane, the driving force for interfacial instability is interfacial friction (i.e., (Fvis)jump). When the more viscous ¯uid is adjacent to the solid plane, the driving force for interfacial instability is the bulk contribution of Reynolds stresses. For viscoelastic ¯uids, in the limit of Re ˆ 0, the driving force of interfacial instability is the coupling between the base ¯ow and perturbation velocity and stresses and their gradients across the interface. Furthermore, in the limit of shortwaves, the interfacial stability mechanism of ¯ows down an inclined plane is the same as Plane Poiseuille ¯ows. Speci®cally, for Newtonian ¯uids the driving force for interfacial instability is interfacial friction. For viscoelastic ¯uids the driving force is the coupling between the base ¯ow and perturbation velocity and stresses and their gradients across the interface.

483

Acknowledgement This work has been supported in part by a grant from the National Science Foundation CTS-9612499.

Fig. 21a, b The mechanism of viscoelastic interfacial instability in the limit of shortwaves, Re ˆ 0.0, b ˆ 11.5°, a ˆ 40: a m ˆ 2.5, We2 ˆ 0.05; b We1 ˆ 0.05, We2 ˆ 0.005

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