Phase diagram of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability

Transcription

1 J. Fluid Mech. (), vol. 44, pp Printed in the United Kingdom c Cambridge Universit Press 95 Phase diagram of interfacial instabilities in a two-laer Couette flow and mechanism of the long-wave instabilit B FRANÇOIS CHARRU AND E. JOHN HINCH Institut de Mécanique des Fluides de Toulouse,, allée du Professeur C. Soula, 34 Toulouse, France Department of Applied Mathematics and Theoretical Phsics, Universit of Cambridge, Silver Street, Cambridge CB3 9EW, UK (Received 4 Januar 998 and in revised form 6 Februar ) A unified view is given of the instabilities that ma develop in two-laer Couette flows, as a phase diagram in the parameter space. This view is obtained from a preliminar stud of the single-fluid Couette flow over a wav bottom, which reveals three flow regimes for the disturbances created at the bottom, each regime being characterized b a tpical penetration depth of the vorticit disturbances and an effective Renolds number. It appears that the two-laer flow exhibits the same flow regimes for the disturbances induced b the perturbed interface, and that each tpe of instabilit can be associated with a flow regime. Tpical curves giving the growth rate versus wavenumber are deduced from this analsis, and favourabl compared with the existing literature. In the second part of this stud, we propose a mechanism for the long wavelength instabilit, and provide simple estimates of the wave velocit and growth rate, for channel flows and for semi-bounded flows. In particular, an explanation is given for the thin-laer effect, which is tpical of multi-laer flows such as pressure driven flows or gravit driven flows, and according to which the flow is stable if the thinner laer is the less viscous, and unstable otherwise.. Introduction Interfacial instabilities in multi-laer flows have received much attention in the recent literature, for their importance in engineering processes, such as coating, polmer extrusion, oil transportation, as well as for their basic scientific significance. Three fundamental sstems have been studied: gravit-driven flows such as multilaer films on an inclined plane (Kliakhandler & Sivashinsk 997), pressure-driven flows such as core-annular (Joseph et al. 997) and plane Poiseuille flows (Laure et al. 997), and shear-driven flows such as plane Couette flow. This paper is concerned with the linear stabilit of the latter flow of two superposed laers of immiscible fluids (figure ). Indeed, despite the numerous papers alread devoted to the subject, and although the problem is governed b the well-established Orr Sommerfeld equation and boundar conditions, several questions remain unanswered. Linear stabilit analses have identified three tpes of instabilit. Two of them, a long-wavelength instabilit (Yih 967; Hooper 985) and a short-wavelength instabilit (Hooper & Bod 983), are low-renolds-number instabilities and arise from an interfacial mode. This mode comes from the jump at the interface in the slope of the

2 96 F. Charru and E. J. Hinch µ µ x Figure. Two-laer Couette flow. velocit profile caused b the viscosit difference, and is neutral for equal viscosit. The mechanism for the short-wave instabilit has been given b Hinch (984): at the perturbed interface, the base flow velocities are discontinuous (figure ), and velocit disturbances must develop to satisf continuit. The resulting vorticit disturbances are in-phase with the deformed interface, but small out-of-phase components arise from advection b the base flow. The flow induced b these out-of-phase vorticit disturbances is such that the initial perturbation of the interface amplifies, at least for equal densities, corresponding to instabilit. The long-wave instabilit is linked to the presence of a wall close to the interface. Tpicall, when the thin laer is the less viscous, the flow is stable, and it is unstable otherwise. This thin-laer effect is not specific to the two-laer Couette flow and also appears in multi-laer liquid films flowing down an inclined plane (Wang, Seaborg & Lin 978) and in core-annular flows (Joseph et al. 997). Experimental evidence of this instabilit has been presented (Barthelet, Charru & Fabre 995; Sangalli et al. 995; Charru & Barthelet 999), but it has received no phsical explanation. Finall, the third tpe of instabilit arises from a wall mode (Hooper & Bod 987), and corresponds to the classical instabilit of shear flows. It tpicall occurs at high shear rates, with a wavelength of the order of the thickness of the fluid laers (Renard 985). However, it is not clear whether the above picture is complete, and the domains of existence of the above instabilities in the parameter space remain blurred. Indeed, the problem depends on six dimensionless parameters, including surface tension and gravit effects; man asmptotic expansions have been performed, as well as numerical studies, but the numerous lengthscales used to compute and discuss the results do not facilitate their comparison. As a result, it feels like having the pieces of a puzzle, whose global picture remains unknown. From the point of view of experimentalists, it is difficult to guess from the existing literature, for given flow conditions, what is the critical flow rate or pressure gradient, and the fastest growing mode. The purpose of this paper is to clear up the above points, and to propose a unified and more phsical view of the instabilities that ma arise in two-laer Couette flows. A clear picture of the lengthscales involved in multi-laer flows is gained from the much simpler problem of single-fluid Couette flow over a wav bottom ( ), which depends on two dimensionless lengths onl, the laer thickness α and a viscous lengthscale β. This problem reveals three different flow regimes, each characterized b the magnitude of inertia and penetration depth of the vorticit disturbances created at the wav bottom. Each regime lies in a particular region of the parameter space (α β), or phase diagram. Then, coming back to the two-laer Couette flow, we analse the existing literature with a unified set of scales, and show that the two-laer flow exhibits the same flow regimes as the single-fluid flow, and that each instabilit can be associated with one of these flow regimes ( 3). The second part of this work

3 Interfacial instabilities in a two-laer Couette flow 97 h ν U=a Figure. One-fluid Couette flow over a wav bottom. x is devoted to the mechanism of the long-wave instabilit ( 4). Simple estimates of the wave velocit and growth rate are provided, and an explanation for the thinlaer effect is given. Surface tension and gravit effects are considered elsewhere (Albert & Charru ) and are ignored throughout this paper. Finall, note that the consideration of two-dimensional disturbances is justified b the fact that Squire s theorem holds for two-laered flows (Hesla, Pranckh & Preziosi 986), as long as marginal stabilit conditions are studied.. Couette flow over a wav bottom Consider the Couette flow of a fluid laer with kinematic viscosit ν and thickness h, over a wav bottom ˆη dim cos kx, where superscript dim indicates dimensional (figure ). The aim of this section is (i) to find how far the disturbances induced b the wav bottom penetrate into the fluid, i.e. their penetration depth δ, and (ii) to estimate inertial effects, i.e. the effective Renolds number Re eff... The linearized problem The velocit u =(U,V) and the pressure P are considered as the superposition of a base flow U = a, V =, P =, corresponding to a flat bottom, and stationar disturbances u, v, p created b the wav bottom. Choosing the inverse wavenumber k as the unit length, and the inverse shear rate a as the unit time, the problem depends on two dimensionless parameters: the thickness α of the fluid laer and the viscous length β: ( ) k /3 ν α = kh, β =. (a) a The explanation for the power in the definition of β will appear below. For 3 convenience, we also define a shear Renolds number Re, which is a dimensionless measure of the shear rate: Re := ah ν = α β 3. Taking into account the smmetr x x +π of the problem, a stationar solution for the disturbances is searched for as: (u, v, p) = ((û(), ˆv(), ˆp())eix +c.c.) where c.c. means complex conjugate. For small slope waves (ˆη := k ˆη dim ), the solution can be searched for from the linearized mass and momentum conservation equations: iû + ˆv =, () (b)

4 98 F. Charru and E. J. Hinch. α =. β =.. (a) û 3 α = β =. 3. vˆ 3 ωˆ (b) 9 α = β = û 9.5 vˆ 9 5 ωˆ (c) û. vˆ Figure 3. Amplitudes of the disturbances. From left to right: û, ˆv and ˆω; from top to bottom: α =.,β =;α =,β =.; α =,β =. Bold line: real part (component in phase with the bottom), plain line: imaginar part (component with phase leading b 9 that of the bottom). ωˆ iû + ˆv = iˆp + β 3 ( û + û), (3a) iˆv = i ˆp + β 3 ( ˆv + ˆv), and the no-slip boundar conditions at the walls: (3b) û() = ˆη, ˆv()=, (4a) û(α) =, ˆv(α) =. (4b) Since the problem to be solved is linear, we impose the normalization condition ˆη =... Exact solution for the linearized problem and phase diagram The exact solution of (), (3), and (4) can be obtained from the vorticit equation: i ˆω = β 3 ( ˆω + ˆω), (5) which can be put in the standard Air equation z ˆω z ˆω = through the transformation z =e iπ/6 ( iβ 3 )/β. The solution of (5) is the linear combination of the Air functions Ai(z) and Bi(z): ˆω = C Ai(z)+C Bi(z). (6) Then, the streamfunction ψ can be obtained from the equation ˆψ ˆψ = ˆω, whose solution is: ˆψ() = α } {e e Y ˆω(Y )dy +e e Y ˆω(Y )dy + C 3 e + C 4 e. (7) The constants C,C,C 3 and C 4 can be determined from the boundar conditions (4) on û = ˆψ and ˆv = i ˆψ: the are given in Appendix A. A first idea of the penetration depth δ of the disturbances can be gained from figure 3, which displas the real and imaginar parts of the amplitude of velocit and

5 Interfacial instabilities in a two-laer Couette flow 99 û Figure 4. Definition of the penetration depth δ. vorticit disturbances for three tpical cases. Figure 3(a) shows that for α =. and β =, disturbances diffuse up to the upper wall. Thus, the penetration depth δ is the laer thickness α, and transverse gradients are O(/α). As proved below, these features are encountered more generall for α< and β>α, and this tpe of flow will be referred to as the shallow viscous regime. Figure 3(b) shows that for α = and β =., the penetration depth divides into two parts: a thin laer of rotational flow with strong gradients close to the interface, surrounded b a region of potential flow decaing exponentiall. These features are encountered more generall for α> and β<, and this tpe of flow will be referred to as the inviscid regime. Finall, figure 3(c) shows that for α = and β =, the thickness of the penetration depth is O(), i.e. roughl equal to one wavelength. This remains true more generall for α> and β>, and this tpe of flow will be referred to as the deep viscous regime. A first idea of the importance of inertial effects can also be gained from figure 3. Indeed, inertial effects are expected to shift the disturbances created b the wav bottom, so that the magnitude of inertial effects can be measured b the imaginar part of the disturbances. For the shallow viscous regime and the deep viscous regime, the velocit u and the vorticit ω are essentiall in phase with the interface: this corresponds to small inertial effects. The phase of the velocit v leads b 9 the phase of the bottom, and disturbances form cells rotating clockwise above the peaks and anticlockwise above the troughs. For the inviscid regime, the real and imaginar components have same order of magnitude: the rotating cells formed b the disturbances are shifted downstream b significant inertial effects. The above remarks can be made more precise b introducing the following definitions of the penetration depth of vorticit disturbances (figure 4) and of the effective Renolds number Re eff measuring inertial effects: δ := û() = ˆω(), (8a) δ Re eff := ˆω i() ˆω r (). (8b) Figure 5 displas the penetration depth δ and the effective Renolds Re eff number versus β, for several α. Figures 5(a) and 5(b) show that for α>, an asmptotic regime is reached from α = 5. In this regime, δ β and Re eff forβ<, and δ and Re eff β 3 for β>. Figures 5(c) and 5(d) show that for α<, an asmptotic regime is nearl reached from α =. In this regime, δ β and Re eff forβ<α, and δ α and Re eff (α/β) 3 = αre for β>α. Note that the transitions between the asmptotic domains are sharp and occur over less than one decade. These results are summarized in the (logβ, logα) plane (figure 6), which can be considered as the phase diagram of the Couette flow over a wav bottom. This plane is divided into three regions, corresponding to the shallow viscous regime, the inviscid regime, and

6 F. Charru and E. J. Hinch (a) (b) δ α = Re eff α = 3 β (c) (d) β δ α α =. Re eff α =. β/α 3 β/α Figure 5. (a) Penetration depth δ and (b) effective Renolds number Re eff versus β for α. (c) Penetration depth δ/α and (d) effective Renolds number Re eff versus β/α for α. Inviscid regime δ ~β Re eff ~ Inviscid log α Deep Deep viscous regime δ ~ Re eff ~/β 3 Viscous log β Shallow viscous regime δ ~α Re eff ~(α/β) 3 Shallow Figure 6. Phase diagram of the Couette flow over a wav boundar. the deep viscous regime. Each flow regime is defined b its penetration depth δ and effective Renolds number Re eff. Finall, note that in all regions δ and Re eff satisf: δ = min (,α,β), Re eff = min (, /β 3, (α/β) 3 ). (9a) (9b)

7 Interfacial instabilities in a two-laer Couette flow The fact that the penetration depth of the inviscid regime scales as β can be understood from the vorticit equation (5). Indeed, when the penetration depth is much thinner than the wavelength, transverse gradients dominate and (5) reduces to i ˆω = β 3 ˆω. Setting δ and /δ in the above equation leads to δ β, which justifies the definition (a) ofβ..3. Asmptotic solutions in the three regimes All the above results can be confirmed from asmptotic expansions. In the shallow viscous regime (α and β>α), the asmptotic solution of (3) and (4) can be found b rescaling the thickness of the laer to unit b the change of variable Y = /α, so that the gradients appearing in (3) are O(). Expanding the disturbances in powers of α and assuming Re = O(), the solution for the first two orders is found to be: û =(Y )( 3Y )+iα Re 3 Y ( Y ) ( Y 3Y )+O(α ), ˆv =iαy (Y ) + α Re 6 Y ( Y ) 3 ( + Y )+O(α 3 ), ˆp =i 6 Re α 5 + O(α ). Dominant terms correspond to Stokes flow, whereas correction terms take into account small inertial effects. (Note that the amplitude of the dimensional pressure is ˆp ρ(a/k) (k ˆη dim )=6(ˆη dim /h) µa/k ˆη dim : for fixed ˆη dim /h, it diverges for small wave slope as /(k ˆη dim ), which corresponds to the classical result for the lubrication pressure (Batchelor 967).) Thus, the vorticit at the wav bottom is: ˆω() = 4 α ire + O(α) () 3 from which the penetration depth and the effective Renolds number are obtained: δ = 4 α + O(α ) (a) Re eff = αre + O(α ). (b) These results agree with the exact results displaed in figure 5: the penetration depth is O(α) and inertial effects are O(αRe). In the inviscid regime (α >βand β ), the upper wall is expected to have no effect, and the thickness α, which appears in the no-slip conditions at the upper wall onl, disappears from the problem. Transverse gradients are expected to be O(/β), and are rescaled to O() b the change of variable Y = /β. Expanding the disturbances in powers of β, the solution at dominant order for û is found to be: û = Y Ai (e iπ/6 Y )dy Ai (e iπ/6 Y )dy + O(β) with Ai (e iπ/6 Y )dy.336 e iπ/6, with similar expressions involving integrals of Air functions for ˆv and ˆp. Thus, the

8 F. Charru and E. J. Hinch vorticit at the wav bottom is: ˆω() = β Ai() Ai (e iπ/6 Y )dy + O(β ).6 β eiπ/6 + O(β ), () and the penetration depth and the effective Renolds number are: δ β.6 + O(β ), (3a) Re eff O(β). (3b) These results agree with the exact results shown in figures 3 and 4: the penetration depth is O(β) and inertial effects are found from dominant order and are O(). Finall, in the deep viscous regime (α and β ), the upper wall is expected to have no effect, and the thickness α disappears from the problem as in the inviscid regime. Transverse gradients are expected to be O(). Expanding the disturbances in powers of /β 3, the solution of (3) and (4) for the first two orders is found to be: û =( )e + i β 3 (6 )e + O(/β 6 ), ˆv =ie + β 3 (3 + )e + O(/β 6 ), ˆp =iβ 3 e ( + + )e + O(/β 3 ). As for the shallow viscous regime, dominant terms correspond to Stokes flow, and correction terms account for small inertial effects. (Again, ˆp scales as β 3 which corresponds to the lubrication pressure.) Thus, the vorticit at the wav bottom is: ˆω() = i ( ) β + O (4) 3 β 6 and the penetration depth and the effective Renolds number are: δ = + O ( β 3 ), (5a) Re eff = ( ) 4β + O. (5b) 3 β 6 Again, these results agree with the exact results shown in figures 3 and 4: the penetration depth is O() and inertial effects are O(β 3 ). 3. Two-laer Couette flow The aim of this section is to provide a unified view of the stabilit results previousl obtained for the two-laer Couette flow, from the construction of phase diagrams similar to that brought out in the previous section.

9 Interfacial instabilities in a two-laer Couette flow The linearized problem Guided b the single-fluid problem discussed in the previous section, we define for each fluid a dimensionless thickness and a dimensionless viscous length: ( ) k /3 ν j α j := kh j, β j =. (6a) Together with the viscosit ratio m or the densit ratio r: m := µ, r := ρ, (6b) µ ρ the above lengthscales completel define the problem, when gravit and surface tension are ignored. The six parameters (6) are not independent: continuit of shear stress at the interface for the base flow, µ a = µ a, gives: β 3 β 3 a j = m r. (7) For convenience, we also define the shear Renolds number of the lower fluid as: Re := a h = α. (8) v β 3 Taking k as the unit length and a as the unit time, the base flow is given b U =, U = /m, and the linearized conservation equations for the amplitudes of the normal modes exp {i(x ct)} are: iû j + ˆv j = (j =, ), (9a) i(/m j c)û j + ˆv j /m j = iˆp j r j + β3 j m j ( û j + û j ) (j =, ), (9b) i(/m j c)ˆv j = ˆp j r j + β3 j m j ( ˆv j + ˆv j ) (j =, ), (9c) where r =,r = r, m =,m = m. The no-slip conditions at the walls are: û ( α )=, ˆv ( α )=, (a) û (α )=, ˆv (α )=, (b) the linearized equations for continuit of velocit and stress at the interface = are: ˆη + û = ˆη m + û, (c) ˆv = ˆv, û +iˆv = m( û +iˆv ), (d) (e) (ˆp ˆp )+ rβ3 m ˆv β 3 ˆv =, (f) and the linearized no-mass transfer at the interface (or kinematic condition) is: iˆηc + ˆv j =. (g) Note that the equations for the single-fluid problem addressed in the previous section

10 4 F. Charru and E. J. Hinch Shallow viscous regime (Yih 967) Deep viscous regime (Hooper & Bod 983) d m d c dim ( m) = + O(α a h d ) σ dim a = α rd Re( m) 6m + O(α4 ) c dim = m + O(α a h 4 ) σ dim = α 7( r) Re( m) + O(α 4 a 96 ) c dim a h =+O(/β 6 ) σ dim a = β 3 ( m) m (r m ) ( + m) + O(/β9 ) Inviscid regime, r = (Hooper & Bod 983) c dim a m k m σ dim ( + m)β 3 a Table. Dimensional wave velocit c dim and growth rate σ dim of the interfacial mode for channel flows. Since m = O(), d = O() and r = O(), then β β and α α, and both flows are in the same regime. For the wall mode, see Appendix B. are recovered from the above equations with lower fluid of infinite densit and viscosit, after the unit time has been changed to a. Finall, extending the definitions (8) to the two-fluid flow, we define the penetration depth δ j of the vorticit disturbances and the effective Renolds number Re eff j in each fluid as: δ j := û j () = ˆω j () (j =, ), (a) Re eff j := ˆω ji() ˆω jr (j =, ), (b) () where ˆω jr and ˆω ji are the real part and imaginar part of the vorticit disturbance in fluid j, respectivel. 3.. Phase diagram for channel flows (finite α and α ) Two different cases have been considered in the literature. The first case corresponds to channel flows, i.e. finite α and α (d = h /h = α /α = O()), and the second case, to semi-bounded flows, i.e. finite α and α = (d = ). The asmptotic results for the wave velocit and growth rate are summarized in this subsection for channel flows (table ), and in the next subsection for semi-bounded flows (table ). All studies assume fluid properties to be of the same order of magnitude, i.e. m = O() and r = O(), so that β β (for air over water, β a /β w =.6, and for viscous oil over water with µ o /µ w =3,β o /β w ). The stabilit of channel flows against long-wavelength disturbances has been studied b Yih (967), who performed a long-wave expansion similar to that performed here

11 Interfacial instabilities in a two-laer Couette flow 5 in.3, with: α, Re = O(). () The wave velocit and the growth rate have complicated the expressions given in Appendix C. Table gives simpler expressions for two limits: (i) similar laers (m and d ), and (ii) laers with ver different thickness (d, or more precisel α α ). The most important feature is that the growth rate is proportional to α Re( m): instabilit is entirel due to the viscosit difference of the laers, and arises however small the shear Renolds number is. Finall, the penetration depth is δ j α j and the effective Renolds number is Re eff j α 3 j /βj 3 : the flow within each laer is in the shallow viscous regime defined in for the single-fluid Couette flow. The stabilit against short-wavelength disturbances has been studied b Hooper & Bod (983) from a short-wave expansion similar to that performed here in.5, with: β 3, α = α =. (3) It appears that for equal densit, short waves are alwas unstable when surface tension is ignored. The mechanism for this instabilit has been given b Hinch (984) and was outlined in the introduction of this paper. Eigenfunctions deca exponentiall awa from the interface with penetration depth δ j, and the effective Renolds number is Re eff j /βj 3 : the flow within each laer is in the deep viscous regime defined in. The effect of the walls (finite thicknesses) has been taken into account b Hooper (989), assuming equal densit. This stud gives more accurate results in the deep viscous regime for α α close to unit, and shows that a wall has no effect as soon as the laer thickness is greater than the wavelength (kh j > 6). However, this expansion is unable to give the right growth rate and wave velocit for long waves, because the limits β 3 and α are not interchangeable. The third case for channel flows corresponds to: β /α with β. (4) It has been studied b Hooper & Bod (983) for unbounded fluid laers (α = α = ) from an asmptotic analsis for β of the exact dispersion relationship. All eigenmodes are found to be stable, and table gives the wave velocit and growth rate of the least stable mode. The form of the eigenfunctions indicates that the penetration depth is O(β j ) and inertial effects are O(). For bounded laers, these results are expected to remain valid as long as β /α, and this is confirmed b numerical studies (Renard 985; Hooper & Bod 987; Albert & Charru ). Thus the flow within each laer is in the inviscid regime defined in. The above discussion is snthesized on the phase diagram (figure 7a), similar to that of the single-fluid Couette flow (figure 6). The boundaries separating the various regimes are the same for the two fluids. The scaling laws for the penetration depths δ j and for the effective Renolds number Re eff j are the same as for the single-fluid Couette flow. The growth rate is O(α Re eff )=O(α Re eff ) in the shallow viscous regime, and is O(Re eff )=O(Re eff ) in the deep viscous regime. In the inviscid regime, the interfacial mode is alwas stable and its damping rate is O(β 3)=O(β3 ) Phase diagram for semi-bounded flows (finite α,α = ) For the deep viscous and inviscid regimes discussed above, the walls pla no role, so that there is no difference between channel flow and semi-bounded flow (in the deep viscous regime, the asmptotic expansion in powers of /β 3 performed b Hooper & Bod (987, 4.4) onl gives more accurate results for α close to unit

12 6 F. Charru and E. J. Hinch (a) logα logα (b) logα and : inviscid and : deep viscous and : inviscid and : deep viscous logβ logβ logβ logβ and : shallow viscous : shallow viscous : inviscid : shallow viscous : deep viscous Figure 7. Phase diagram of the two-laer Couette flow, (a) channel flow; (b) semi-bounded flow. Shallow viscous/inviscid regime (Hooper 985) c dim =.46( m)α /3 a h σ dim =.7( m)α 4/3 a ( rre m ( rre m ) /3 + O(α /3 ) ) /3 + O(α 5/3 ) Deep viscous regime (Hooper & Bod 987) c dim α ( m) a h (cosh α + m sinh α ) + α ( m ) No expression available for σ dim ;forα, see table. Table. Dimensional wave velocit c dim and growth rate σ dim for semi-bounded flows (α = ). Since m = O() and r = O(), then β β. For the wall mode, see Appendix B. (table )). Differences appear onl when the lower fluid is in the shallow viscous regime (α and α β ). This case has been studied b Hooper (985), matching the long-wave expansion of Yih in the lower fluid to the exact solution in the upper fluid, with: α, α =, Re = O(). (5) Waves are dispersive and the growth rate is stronger than in the channel flow case: it scales as k 4/3 rather than k (table ). The upper fluid is in the inviscid regime for β β, and in the deep viscous regime for β β. The case of fluids with similar viscosit and densit was tackled b Renard (987) as a perturbation of the single-fluid problem. Unfortunatel, no significant simplification arises from this case, and phsical information seems hard to obtain from the complicated expression of the perturbed eigenvalue, unless some additional long- or short-wave limit is taken. The above discussion is snthesized on the phase diagram shown in figure 7(b). The regions for the inviscid regime and deep viscous regime are the same as for channel flows, but the former region of the shallow viscous regime is now divided into two parts, depending on whether β β orβ β. Finall, note that the penetration depth and the effective Renolds number satisf: δ j = min (,α j,β j ), (6a) Re eff j = min (, /β 3 j, (α j /β j ) 3 ). (6b)

13 Interfacial instabilities in a two-laer Couette flow 7 (a) (b) S U S U α α U S β β Figure 8. Marginal stabilit curves for semi-bounded Couette flow, from figures 3, 6 and 7 of Hooper & Bod (987). (a) m =.5, asmptotes: α/β =.47 and β =.438; (b) m =, asmptotes: α/β =3.,α =.8 and β =.76. These curves follow the boundaries of the flow regimes defined in figure Relevance of the phase diagram from numerical results Along with asmptotic studies, numerical studies have been performed for α = O() or Re (Renard 985; Hooper & Bod 987; Albert & Charru ). Marginal stabilit curves are generall shown in the (Re,α )-plane or in the ((α /β ),α )-plane; however, plotting these curves in the (β, α )-plane is more informative: each tpe of instabilit is clearl isolated, and the relevance of the phase diagram (figure 7) can be confirmed. As an example, figure 8 displas figures 6 and 7 of Hooper & Bod (987) redrawn in the (β,α )-plane. Densities are equal, so that short waves are unstable in both cases. For m =.5 (figure 8a), long waves are unstable (thin-laer effect), so that there is no marginal curve between the shallow viscous and the deep viscous regimes; the marginal stabilit curve follows the boundar of the inviscid regime. For m = (figure 8b), one of the two marginal curves follows the boundar of the shallow viscous regime: long waves are stable, in agreement with the thin-laer effect; the vertical part of the second marginal curve separates the unstable deep viscous regime from the stable inviscid regime. However, for β and α = O(), i.e. for strong shear Renolds number (Re = α /β3 ), another unstable region appears, the nature of which is discussed below. The small-re eigenmodes discussed until now are clearl interfacial modes in the sense that the exist because of the presence of the interface, and that disturbances are essentiall localized in the vicinit of the interface. Another signature of an interfacial mode can be found from the kinetic energ equation, which shows that instabilit arises from the small difference between the rate of energ dissipation (alwas negative) and the net rate of work done at the interface b shear stress disturbances (Hooper & Bod 983; Goussis & Kell 988; Albert & Charru ). However, tracing the interfacial mode for increasing Re shows that the nature of the instabilit changes for high Re: the maximum of the eigenfunctions lies near the walls, and the kinetic energ equation shows that instabilit now arises from energ transfer from the mean flow to disturbances via the Renolds stresses; for Couette flow, there is no crossing of the interfacial mode with a shear mode as it might occur for Poiseuille flow (Albert & Charru ). This mode of instabilit, which corresponds to the classical wall mode of single-fluid shear flows, has been studied b Hooper & Bod (987) from a singular perturbation method matching an inviscid solution in the bulk of each fluid to viscous boundar laers at the walls and at the interface. The solution is found as series expansions in powers of (α Re) / =(β /α ) 3/, which

14 8 F. Charru and E. J. Hinch (a) logα logα k k (b) logα k k Strong shear Re >> Weak shear Re << logβ logβ Strong shear Re >> Weak shear Re << logβ logβ Figure 9. (a) channel flow; (b) semi-bounded flow. Arrows are lines of increasing k at constant shear rate (slope 3 ). Left-hand line: strong shear rate (Re = a h /ν = α /β3 ); right-hand line: weak shear rate (Re ). is the order of magnitude of the thickness of the boundar laers. The wave velocit and growth rate are given in Appendix B for the case of equal densit (r = ). Note that the growth rate is of order β 3/ = k(ν /a ) / and that instabilit arises when the thin laer is the less viscous (m >), unlike the thin-laer effect which holds for small Re The experimental point of view From the experimental point of view, there are two important questions. (i) For given flow conditions, i.e. for given fluids and shear rate, what is the most amplified wavenumber and its growth rate? More precisel, what does the curve σ(k) look like? (ii) How does this curve change as the shear rate changes? For given fluid properties, constant shear rate corresponds to constant Re := a h /ν = α /β3, i.e. to the oblique lines drawn in figure 9. Along one of these lines, the dimensional wavenumber varies from zero (bottom left-hand quadrant) to infinit (top right-hand corner). Increasing the shear rate shifts the line to the left. Tpical curves σ(k) can then be obtained easil. First, consider small shear rate (Re ). Small wavenumbers (α ) lie in the shallow viscous regime, with growth rate scaling as k for channel flow and k 4/3 for semi-bounded flow; high wavenumbers (α ) lie in the deep viscous regime, with growth rate scaling as ±k. For unstable short waves, the corresponding curves σ(k) are sketched in figure (a), for stable and unstable long waves. For strong shear rate (Re ), an intermediate region appears between the long- and short-wave regions for α [Re,Re / ], which corresponds to the stable inviscid regime. Figure (b) displas two tpical curves σ(k), for stable and unstable long waves. For stronger shear rate, a new wall instabilit, not shown on the figure, arises with α = O(). 4. Mechanism of the long-wave instabilit Wh are long waves stable if the thinner laer is less viscous, and unstable if more viscous, however small the shear Renolds number is? The aim of this section is to provide a phsical explanation for this thin-laer effect, and to understand how the presence of the walls modifies the short-wave mechanism given b Hinch (984). The stle is now changing a little to a phsical argument supported b mathematical descriptions. For this purpose, dimensional quantities are used. Three

15 Interfacial instabilities in a two-laer Couette flow 9 (a) σ Re << ~ ± k n ~k kh (b) σ Re >> ~ ± k n ~k Re Re / kh Figure. Tpical curves for the growth rate σ versus wavenumber k. (a) Small shear rate (Re ); (b) strong shear rate (Re ); n = for channel flow and n = 4 for semi-bounded flow; the upper 3 (lower) curve corresponds to unstable (stable) long waves. tpical situations are considered: channel flow with one laer much thinner than the other ( 4.), channel flow with nearl equal laer thickness ( 4.), and semi-bounded flow over a thin laer ( 4.3). For each case, the estimates for the wave velocit and growth rate are compared with available asmptotic results. The shear Renolds number Re is assumed to be small, and the viscosit and densit ratios m and r are O(). 4.. Channel flow: case of a thin laer (α α ) Consider a small-amplitude disturbance of the interface, η = ˆη cos k(x ct) (figure (a)). Owing to the viscosit difference, the base velocities are not equal at the disturbed interface, and velocit disturbances must develop for the continuit of longitudinal velocit to be satisfied, according to: a ˆη + û () = a ˆη + û (). (7) When µ >µ, the lower fluid slows down and the upper fluid speeds up, at both the peaks and troughs of the wave. Thus, the disturbance û is negative and û is positive, at least near the interface. When µ <µ, these velocit disturbances are reversed. For small shear Renolds number and long waves, disturbances diffuse awa from the interface up to the walls, and a linear shear flow might be expected in both fluids; but linear flows generall do not satisf the requirements of no net flow (mass conservation) and continuit of the -velocit at the interface, i.e. û d = ˆv () h ik = ˆv () ik = h û d. (8) Hence, pressure disturbances develop (figure (b)). However, the pressure driven flow is much smaller in the thin laer than in the thick laer, so that the flow in the thin laer is ver close to a linear shear flow. Then, continuit of shear stress, µ û () = µ û (), (9) requires onl small velocities in the thin laer. Thus, continuit of x-velocit (7) gives at the leading order: U := û ()=(a a )ˆη = µ µ µ a ˆη, (3)

16 F. Charru and E. J. Hinch (a) µ > µ µ (b) c Figure. (a) Continuit of x-velocit at the deformed interface creates velocit disturbances. (b) Sketch of the profile of the velocit disturbances, and sign of the vorticit disturbances. Mass conservation in the lower fluid imposes wave velocit to the left when µ >µ. and the velocit field is a combination of a linear shear flow U( /h ) and a pressure driven flow x ˆp (h )/µ. The pressure gradient is that which is needed to make the total net flow of the two laers equal to zero, i.e. at leading order, that which assures no net flow in the thick laer. Indeed, the flow in the thin laer is O(h /h ) smaller than that generated b the pressure gradient in the thick laer; thus onl a small change in that pressure gradient is needed to make the total net flow of the two laers equal to zero. Hence, ) û = U ( )( h 3 h, (3) with pressure gradient 6µ U/h. This upper laer flow exerts a shear stress µ û () = 4µ U/h on the interface which drives the flow in the thin laer. This flow is then: û = 4U µ ) /µ (+ h. (3) h /h Wave propagation is found from mass conservation in the thin laer: the volume û ()h δt, leaving the control volume [ x λ, < ] during a small time interval δt, must be balanced b a shift of the interface (figure b), according to: û ()h δt = λ/ which gives the wave velocit: ˆη[cos k(x cδt) cos kx]dx =ˆηc dim δt, (33) c dim a h = µ /µ h /h. (34)

17 Interfacial instabilities in a two-laer Couette flow (Note that the wave velocit ma alternativel be obtained from the kinematic condition at the interface, the small vertical flow being calculated from the mass conservation equation.) The results (3) and (3) for the fluid velocit, and (34) for the wave velocit completel agree with Yih s results when the leading terms for d = h /h are retained, see (A ) in Appendix C and table. The effects of inertia ma be found from either the momentum equations or the vorticit equation. Calculations from the momentum equations are complicated b the vertical advection of the base momentum and b pressure gradients; on the contrar, because the base vorticit is uniform, its advection b the disturbances has no net effect. Thus, the use of the vorticit equation seems preferable; in addition, it allows us to understand how the short-wave mechanism of Hinch (984) is modified. The leading-order vorticit disturbances, ˆω = û = U ( 4 6 ), (35a) h h ˆω = û =4 µ U, (35b) µ h are of same order of magnitude. Since the wave velocit is smaller than the base flow a h b O(h /h ), unsteadiness due to wave propagation can be ignored. Let ˆψi be the streamfunction of the inertial correction in the thick laer; it is governed b the problem: µ ˆψ i = ρ a ik ˆω (36a) with ˆψ i = ˆψ i = on =, = h. (36b) This flow is induced b a torque (the right-hand side of (36a)) acting as if on a viscous fluid (the left-hand side, viscous because inertia is a small correction in the long-wave limit). The zero-velocit condition (36b) comes from the fact that velocit in the thick laer is much larger than in the thin laer, so that, at the leading order, continuit at the interface reduces to the no-slip condition. Integrating, we find: û i = ikρ a U (3 + h 3h h )(h ). (37) Figure displas the above flow together with the exact eigenfunction, for two values of the viscosit ratio; it turns out that (37) gives a fairl good approximation of the eigenfunction, except in a region of thickness O(h ) near the interface. For µ >µ (figure a), the positive inertiall induced couple in the middle half of the laer creates a positive flow near the interface, whereas the negative inertiall induced couple near the upper wall reduces the velocit to zero. For µ <µ (figure b), all velocities are reversed. In the thin laer, the inertial correction is not driven b the inertia there, but b the shear stress exerted b the inertial correction flow in the thick laer which gives the flow in the thin laer: µ û i () = i 3 kρ a Uh, û i = ikρ a Uh 3µ (h + ). (38)

18 F. Charru and E. J. Hinch (a) (b) h h Advection Advection x Stable Unstable Figure. Eigenfunctions û j for long waves, for d =,r =,Re =.(a) m =;(b) m =.5. At x = : dominant order x-velocit in phase with the interface; at x = λ and x = 3 λ: out-of-phase 4 4 inertial correction. Dotted lines, equation (37). Stabilit (or instabilit) arises from mass conservation: the volume û i ()h δ t, entering (or leaving) the control volume [ x/λ 3, <] during a small time interval 4 4 δt, can be balanced onl b a deca (or growth) of the perturbed interface, according to: iû i ()h δt = which gives the growth rate: 3λ/4 σ dim λ/4 ˆη(e σδt ) cos kx dx = ˆησdim δt (39) k = µ /µ a 6 (kh ) ρ a h µ. (4) Again, the results (37) and (38) for the fluid velocit, and (4) for the growth rate completel agree with Yih s results when the leading terms for d = h /h are retained, see (A 5) in Appendix C and table. Most of the features of the short-wave mechanism found b Hinch (984) are present in the long-wave mechanism described above: generation of velocit disturbances in order to satisf continuit of velocit at the perturbed interface, and advection of the vorticit disturbances b the base flow creating out-of-phase components midwa between the peaks and troughs. The important difference for the long waves is the nearness of the lower wall, which reduces everthing in the thin laer: the leading-order velocit is reduced there b a factor h /h, the out-of-phase vorticit produced b inertia acting there is reduced b h 3 /h3, and so the locall induced vertical flow (and so the growth rate) is reduced b h 5 /h5. Hence, inertia acts effectivel onl in the thick laer. Rather subtl, inertia acting in the thick laer does not produce an immediate vertical flow because of the constraint of no net flow. So instead, the thick laer drags the thin laer along with it, which produces a linear flow within the thin laer, and it is the horizontal divergence of this flow which causes growth. Hence, the growth depends on the sign of the flow in the thick laer, which depends on whether the thin or thick laer is the less viscous.

19 Interfacial instabilities in a two-laer Couette flow 3 (a) h (b) x (c) h h x Figure 3. Dominant order x-velocit û () j for m =4.(a) d =4=m; (b) d =<m; (c) d = >m. 4.. Channel flow: case of laers with similar thickness (α α ) As mentioned in the previous section, pressure disturbances develop in order to satisf the requirement (8) of no net disturbance flow. However, linear profiles fulfil this requirement, i.e. verif h û () = h û (), and also satisf continuit of shear stress if (h /h ) = µ /µ. Hence, the presence of the factor (d m) in the pressure disturbance (see Appendix C). Thus, when viscosities and laer thicknesses are such that d m, the pressure disturbance plas a negligible role and velocit profiles are linear in both fluids (figure 3a): with û = U µ /µ h /h ( + /h ), û = U( /h ), (4) U := û () = µ µ h /h a ˆη. µ µ /µ + h /h Otherwise, the pressure disturbance induces negative velocit curvature when d <m (figure 3b) or positive curvature when d >m(figure 3c), with negligible curvature effects in the lower fluid when d. The wave velocit can be calculated from mass conservation as in the previous subsection, leading to: c dim = µ /µ. (4) a h µ /µ + h /h When d = m, (4) and (4) are exactl Yih s results; when d m, i.e. for laers with nearl equal thickness and viscosit, (4) and (4) corresponds to (A 3) in Appendix C and to the wave velocit given in table. Inertial effects ma be taken into account as previousl. However, the velocities x

20 4 F. Charru and E. J. Hinch have the same order of magnitude in both fluids, and the momentum (or vorticit) equations cannot be uncoupled, leading to less simple calculations. A simple estimate of the growth rate ma be obtained when d m as follows. For nearl equal viscosities, the wave velocit is small, and unsteadiness can be neglected in the inertia terms. First, consider the case ρ =, i.e. the lower fluid with negligible inertia; figure 4(a) displas the corresponding eigenfunctions for long waves. In the upper laer, advection of the vorticit disturbances ˆω U/h creates an out-of-phase component ˆω i ire eff ˆω, where Re eff := α 3 /β3 =(kh )(ρ a h /µ ) is the effective Renolds number defined in 3. This out-of-phase vorticit corresponds to velocit û i ˆωi h ire eff U. In order to satisf the no net flow condition, a counter-flow û i ûi must develop in the lower laer, which is created b the pressure gradient ikˆp i µ ( û i /h ), i.e. b the pressure ˆpi ρ a h U. Since ˆp i = ˆpi, this pressure modifies slightl the flow in the upper laer but does not change the magnitude of the velocit there. This out-of-phase flow gives rise to amplification or deca of the interfacial disturbance according to the mass conservation equation (39), leading to the growth rate: σ dim =ikh û i () ˆη = a ( µ /µ )(kh ) ρ a h µ. Figure 4(b) displas the eigenfunctions when ρ /ρ =.5, and shows that the above reasoning for ρ = holds as long as ρ <ρ, with flows dominated b inertia in the upper fluid and b the pressure gradient in the lower fluid. Consider now the opposite case ρ = (figure 5a); a similar reasoning shows that (i) the flow in laer is driven b inertia with velocit û i ire eff U, and (ii) the flow in laer is driven b the pressure ˆp i ρ a h U, leading to growth rate σ dim /a =( µ /µ )(kh ) (ρ a h /µ ). Figure 5(b) displas the eigenfunctions when ρ /ρ =, and shows that the reasoning for ρ = holds as long as ρ <ρ. Finall, for fluids with similar inertia, the resulting flow can be considered as the superposition of the above two flows, leading to the following pressure disturbance and growth rate: ˆp i = ˆp i ( µ µ ) (ρ + ρ )a h ˆη (43) ( σ dim =(kh ) µ )( ρ ) ρ a h. (44) a µ ρ µ The pressure (43) and growth rate (44) agree with the long-wave results, giving the right dependence with all parameters, see (A 6) in Appendix C and table Semi-bounded flow (α and α ) When the upper wall is at a distance from the interface greater than the wavelength, it plas no more role, the vertical gradients in the upper fluid are no longer of order /h, and the analsis performed in 4. must be slightl modified. Rather than following exactl the same approach as in 4., we use in this section the results obtained in 3 on the penetration depth δ j and the effective Renolds number Re eff j to derive estimates of the wave velocit and growth rate. The lower laer is assumed to be in the shallow viscous regime (α and α β ), with dimensional penetration depth δ dim = δ /k h and effective Renolds number Re eff α Re; the upper laer is assumed to be in the inviscid regime (α β and β ), with dimensional penetration depth δ dim = δ /k β /k and effective Renolds number Re eff. Since β β for fluids with similar viscosit and

21 Interfacial instabilities in a two-laer Couette flow 5 (a) Advection Unstable h (b) Advection h Unstable Advection x Figure 4. Eigenfunctions û j for long waves, for d =,m =.,Re =.(a) ρ =;(b) ρ =.5ρ. At x = and x = λ: dominant order x-velocit in phase with the interface; at x = 4 λ and x = 3 4 λ: out-of-phase inertial correction. (a) h Stable (b) h Advection Advection Stable Advection x Figure 5. Eigenfunctions û j for long waves, for d =,m =.,Re =.(a) ρ =;(b) ρ =.5ρ. At x = and x = λ: dominant order x-velocit in phase with the interface; at x = 4 λ and x = 3 4 λ: out-of-phase inertial correction. densit, we have δ dim δ dim, with: δ dim δ dim h (k µ /ρ a ) /3 /k =(kh ) /3 Thus, continuit of shear stress at the interface, ( ) /3 rre. (45) m µ û () δ dim û () µ, δ dim

22 6 F. Charru and E. J. Hinch implies that û is much smaller than û. Then continuit of x-velocit (7) gives: U := û () µ µ a ˆη. µ Hence, in the thin laer, the linear shear flow in phase with the interface is such that: û () δdim µ µ a δ dim ˆη, µ which, together with mass conservation (33), gives the wave velocit: c dim δdim a h δ dim µ µ µ. (46) Together with (45), (46) is exactl the wave velocit obtained b Hooper (985), except for the numerical factor.46 (see table ). Note that, unlike the case of channel flow, waves are dispersive, because the penetration depth in the thick laer depends on the wavelength. Advection of the vorticit disturbances ˆω U/δ dim b the base flow in the upper fluid creates an out-of-phase vorticit component ˆω i ire eff ˆω, which induces a shear stress µ ˆω i on the lower fluid. This shear stress creates in turn a linear flow in the thin laer such that û i () µ /µ h ˆω i, which, together with mass conservation (33), gives the growth rate σ: σ dim = iûi ()kh µ µ δ dim kh Re a a ˆη µ δ dim eff. (47) With Re eff = and δ dim /δ dim given b (45), this is exactl the growth rate obtained b Hooper (985) for semi-bounded flows, except for the numerical factor (see table ). Note that the reasoning leading to (47) holds for channel flows with Re eff = (kh )(ρ a h /µ ) and δ dim /δ dim = h /h, giving (4) back. Finall, the mechanism for the long-wave instabilit for semi-bounded flows is the same as for channel flows, the difference in the growth rate arising from the difference in the penetration depth and effective Renolds number in the thick laer. 5. Summar and conclusion In order to gain better phsical insight into the numerous linear stabilit results for the two-laer Couette flow, we have first considered the much simpler problem of the single-fluid Couette flow over a wav solid boundar. Taking the inverse wavenumber as the unit length, this problem depends on two parameters onl: the dimensionless thickness α and a diffusion length β. Three different flow regimes have been exhibited: the shallow viscous, the deep viscous, and the inviscid regimes. Each regime occupies a well-defined region in the (β, α)-plane, defining a phase diagram, and is characterized b a penetration depth δ of vorticit disturbances, and an effective Renolds number Re eff measuring the importance of inertial effects on disturbances (figure 6). Then, armed with the idea of the phase diagram, we came back to the two-laer Couette flow. Ignoring gravit and surface tension, this problem depends on four parameters: two dimensionless thicknesses α j and two viscous lengths β j. Analsing the eigenfunctions from the existing literature, it appears that on considering the penetration depth δ j of vorticit disturbances induced b the slightl deformed interface, as well as the effective Renolds number Re eff j, several flow regimes can be

23 Interfacial instabilities in a two-laer Couette flow 7 defined. Remarkabl, these flow regimes are the same as those of the single-fluid flow, with the same scalings for the penetration depth and the effective Renolds number, allowing the construction of phase diagrams of the flow regimes, one for channel flows and one for semi-bounded flows. Then, each tpe of instabilit is associated with one flow regime: the long-wave instabilit found b Yih (967) is tpical of the shallow viscous regime, and the short-wave instabilit found b Hooper & Bod (983) is tpical of the deep viscous regime. No interfacial instabilit arises in the inviscid regime. However, in the latter regime, which corresponds to strong shear rates, a wall mode of instabilit ma appear (Hooper & Bod 987). Analsis of the available numerical results confirms the relevance of the phase diagram (figure 8). Moreover, it appears that the domain of existence of an instabilit extends beond the validit domain of the asmptotic expansion b which it was primaril discovered. For instance, the long-wave instabilit is tpical of all situations when the walls bound the diffusion of vorticit, i.e. when the wavelength and viscous lengths are greater than the laer thicknesses. Similarl, the short-wave instabilit tpicall arises for wavelengths smaller than the laer thicknesses and viscous lengths. Finall, on considering the lines of constant shear Renolds number in the phase diagram (figure 9), each wavenumber appears to fall into a particular flow regime, and tpical curves giving the growth rate versus wavenumber can be depicted (figure ). The basis of this unified view of interfacial instabilities is the comparison between the three lengthscales involved in each fluid laer, namel the wavelength, the laer thickness and the diffusion length: the penetration depth of vorticit disturbances must scale with one of these lengths, leading to the three flow regimes (when fluid properties are assumed to be of the same order of magnitude). This picture is significantl different and simpler than that proposed b Hooper & Bod (987), who found four regimes (their regime (iv) must be merged partl with their regime (i), and form the deep viscous regime, and partl with their regime (ii), and form the shallow viscous regime). The second part of this paper was devoted to the mechanism for the long-wave instabilit, and to an explanation of the thin-laer effect. As for the short-wave instabilit, the initiating mechanism is that the base velocities do not match on the disturbed interface, and velocit disturbances must develop in order to satisf continuit: this instabilit is a velocit-induced instabilit, as opposed to the stressinduced instabilit tpical of free-falling films (Smith 99). Three tpical situations have been studied: (i) channel flow with a thin laer, (ii) channel flow with nearl equal laer thickness, and (iii) semi-bounded flow over a thin laer. For all cases, the requirement of no net flow for the disturbances creates a pressure gradient, and, because of the presence of the wall, mass conservation in each fluid implies a shift of the interface with the less viscous fluid, which corresponds to the wave velocit. In each case, good estimates have been obtained for the wave velocit and the dominant Stokes flow of the disturbances. Instabilit then arises from small inertial effects. For a thin laer, inertial effects in the thick laer are much greater than in the thin laer, and create an out-of-phase flow which exerts a shear stress on the thin laer; this shear stress drives a small out-of-phase flow in the thin laer which is responsible for the growth or deca of the initial interfacial disturbance. The sign of the viscosit difference imposes the sign of the velocit disturbances, which impose in turn the sign of the growth rate, which originates the thin-laer effect. Moreover, the growth rate appears to be proportional to the ratio δ /δ of the penetration depths, which explains wh it scales as k for channel flow and as k 4/3 for semi-bounded flow. For laers with nearl equal thickness