Effect of wall deformability on the stability of shear-imposed film flow past an inclined plane

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International Journal of Engineering Science 150 (2020) 103243

Contents lists available at ScienceDirect

International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci

Effect of wall deformability on the stability of shear-imposed ﬁlm ﬂow past an inclined plane Mahendra Baingne, Gaurav Sharma∗ Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247 667, India

a r t i c l e

i n f o

Article history: Received 13 January 2020 Accepted 3 February 2020

Keywords: Film ﬂows Linear stability Flow past soft solid Interfacial instabilities Wall deformability

a b s t r a c t The linear stability of shear-imposed liquid ﬁlm ﬂow past an inclined plane is examined when a soft, deformable solid layer is attached to the inclined plane. A liquid ﬁlm ﬂowing past a rigid inclined plane exhibits a long-wave gas–liquid (GL) interfacial instability which is modiﬁed by the presence of an imposed shear stress at gas–liquid interface. This GL interfacial mode does not become unstable in creeping ﬂow limit whether a GL interfacial shear is present or not. We demonstrate that the GL interface becomes unstable even at zero Reynolds number when the shear-imposed liquid ﬁlm ﬂows past an inclined plane which is coated with soft solid layer. This GL mode instability exists only when both shear and a deformable liquid–solid (LS) interface are simultaneously present. For non-zero Reynolds number, we show that there exists multiple unstable modes originating because of the presence of deformable LS interface. These unstable LS modes become important and practically realizable only for shear-imposed liquid ﬁlm ﬂow and become irrelevant for ﬁlm ﬂows in absence of imposed shear. We also show that this LS interfacial instability dominate the stability behavior of the composite ﬂuid ﬁlm-solid system in low Reynolds number regime. Our results suggest that the shear-imposed ﬁlm ﬂow can be made unstable by using a deformable solid coating in parameter regime where the ﬁlm ﬂow otherwise remains stable in rigid wall limit. Finally, we show that a deformable solid layer can be used to obtain a stable ﬁlm ﬂow conﬁguration for the parameter regime where the GL interfacial mode is unstable for shear imposed ﬁlm ﬂow over a rigid incline. Thus, we demonstrate the capability of a soft solid layer in manipulation and control of instabilities for shear-imposed ﬁlm ﬂows. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Gravity-driven liquid ﬁlm ﬂows are prevalent in various technological processes such as coating applications (Weinstein & Ruschak, 2004), microﬂuidic devices (Craster & Matar, 2009; Squires & Quake, 2005), environmental and physiological systems such as pulmonary ﬂows (Grotberg, 2011). Because of such widespread applications, studies related to stability of liquid ﬁlm ﬂows keep attracting increasing interest since the early works of Kapitza (1948). Liquid ﬁlm ﬂowing down a rigid inclined plane becomes unstable to long wavelength disturbances when Reynolds number increases above a critical value given as Rec = 5/4 cot θ , where θ is the inclination angle (Benjamin, 1957; Yih, 1963). This long wave instability occurs owing to the presence of a deformable gas–liquid (GL) interface, and it is usually referred as free-surface or GL or Yih’s mode ∗

Corresponding author. E-mail address: [email protected] (G. Sharma).

https://doi.org/10.1016/j.ijengsci.2020.103243 0020-7225/© 2020 Elsevier Ltd. All rights reserved.

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of instability in literature. This instability develops in the form of surface waves displaying a variety of complex spatiotemporal wave patterns (Chang & Demekhin, 2002). The wave formation on gas–liquid interface is desirable in processes involving heat and mass transfer where the complex ﬂow pattern enhances transport rates. On the other hand, in applications like coating, occurrence of surface waves is detrimental to the ﬁnal product quality. Thus, suppression or enhancement of GL instability in case of liquid ﬁlm ﬂow (and interfacial instabilities, in general) forms an important aspect in several processes. In the present work, we examine the stability of a shear-imposed liquid ﬁlm ﬂowing past an inclined plane which is coated with a deformable solid layer. The primary objective is to investigate the role of presence of a deformable wall (or deformable liquid–solid interface) in altering the stability characteristics of shear-imposed liquid ﬁlm ﬂow. There are several studies exploring the aspect of control and manipulation of free-surface GL mode instability using a variety of strategies. Lin, Chen, and Woods (1996) and Jiang and Lin (2005) examined the effect of imposing in-plane horizontal oscillations on the gas–liquid free-surface instability for the case of single and two liquid ﬁlm ﬂow down an inclined plane. They identiﬁed the parametric window in terms of amplitude and frequencies of imposed oscillations where the instabilities can be suppressed for ﬁlm ﬂows. They also showed that outside of this stability window, the interfacial instabilities get enhanced because of the presence of imposed oscillations. Wall heating in order to create a linear temperature gradient has also been used to control the free-surface instabilities for liquid ﬁlm ﬂowing down an inclined plane (Demekhin, Kalliadasis, & Velarde, 2006). A number of studies have investigated the effect of presence of surfactant at GL interface and demonstrated that the surfactant have a stabilizing effect on the GL mode instability (Blyth & Pozrikidis, 2004; Wei, 2005; Whitaker, 1964). Smith (1990) examined the stability of shear-imposed liquid ﬁlm ﬂow down an inclined plane in the limit of long wavelength perturbations, and concluded that the imposed shear can have a stabilizing or destabilizing effect on Yih’s GL mode depending on whether it acts in upstream or downstream direction. A similar conclusion for GL mode have been made by Wei (2005) for the case of shear-imposed surfactant-laden liquid ﬁlm ﬂow past a rigid inclined plane. Wei (2005) also discussed the effect of imposed shear on Marangoni mode which is present in addition to GL mode because of the presence of surfactant at GL interface. Samanta (2014) extended the analysis of Smith (1990) and Wei (2005) for a clean ﬁlm ﬂow (past a rigid incline) to moderate Reynolds numbers and ﬁnite wavenumbers. He demonstrated that the imposed shear stress acting in downstream (upstream) direction decreases (increases) the critical Reynolds number for long wave GL mode instability. In the non-linear regime, it was shown that the downstream (upstream) acting imposed shear stress strengthens (weakens) the evolution of spatially growing surface waves. The above studies illustrate that the shear stress at GL interface can be used to modify the GL mode interfacial instability. In the past one decade or so, a number of studies have examined the stability of liquid ﬁlm ﬂow past a ﬂexible inclined plane with a motivation to control free-surface instabilities by tuning the wall ﬂexibility/deformability. For example, the stability of liquid ﬁlm ﬂowing past an inclined plane coated with a deformable solid layer has been investigated by Shankar and Sahu (2006) and it was demonstrated that the soft solid layer has a stabilizing effect on the GL interfacial mode. However, for such composite ﬂuid–solid system, there exists an additional deformable liquid–solid (LS) interface which also becomes unstable. Indeed, there are many theoretical and experimental investigations establishing that the LS interface becomes unstable for ﬂuid ﬂow past a soft, deformable solid surface when the solid layer becomes suﬃciently deformable (Eggert & Kumar, 2004; Gaurav & Shankar, 2010; Kumaran, 20 0 0; Kumaran, Fredrickson, & Pincus, 1994; Kumaran & Muralikrishnan, 20 0 0; Neelamegam & Shankar, 2015; Verma & Kumaran, 2012; 2013). The conﬁguration considered in these studies include Couette ﬂow past soft solid layer and pressure-driven ﬂow in a tube or channel with deformable walls. Such instabilities arising becuase of the presence of a deformable liquid–solid interface are referred as LS mode instabilities in the present study. For liquid ﬁlm ﬂow past a soft solid layer, Shankar and Sahu (2006) observed that the GL mode instability is entirely suppressed when solid layer deformability increases above a threshold value. When solid deformability is further increased, the LS interface also becomes unstable, however, there exists a range of values of shear modulus of solid layer (characterizing the deformability of solid layer) where the ﬁlm ﬂow conﬁguration past a ﬂexible inclined surface remains stable. Shankar and Sahu (2006) employed a linear visco-elastic model to represent the dynamics in soft solid layer. Gkanis and Kumar, in a series of papers Gkanis and Kumar (20 03, 20 05, 20 06), pointed out that it is necessary to use a non-linear solid model to accurately capture the stability of ﬂow past soft surfaces even in the limit of linear stability analysis. They used a neo-Hookean solid model and demonstrated that the coupling between ﬁnite base state deformations in solid and displacements in perturbed state appear at several places in linearized governing stability equations for solid layer and boundary conditions at LS interface. These coupling terms remain absent for linear elastic solid and signiﬁcantly affect the stability behavior of composite ﬂuid–solid conﬁguration. In view of these observations related to use of solid model, Gaurav and Shankar (2007) used a neo-Hookean constitutive relation for solid layer and re-examined the stability of liquid ﬁlm ﬂow past a ﬂexible inclined plane. They demonstrated a complete stabilization of ﬁlm ﬂow conﬁguration, and concluded that a stability window (in terms of shear modulus) exists even with neo-Hookean solid model in a manner similar to as shown by Shankar and Sahu (2006) using a linear elastic solid model. Of course, the neutral curves corresponding to instability of LS interface were modiﬁed due to use of neo-Hookean solid model, however, the qualitative predictions of existence of a wide stability window remains unaltered. Jain and Shankar (2007) extended the work of Shankar and Sahu (2006) for visco-elastic liquid ﬁlm and demonstrated that the instability suppression for ﬁlm ﬂows using soft solid coatings remains independent of ﬂuid rheology. Recently, in context of ﬁlm ﬂows past soft solid surface, a deformable solid bilayer is used and it is shown that the solid bilayers could provide more options in controlling free-surface instabilities (Sahu & Shankar, 2016). There are few studies which investigated the stability of liquid ﬁlm ﬂow past a ﬂexible inclined plane when the GL interface is covered with a layer of surfactant (Baingne & Sharma, 2019; Matar & Kumar, 2004; Peng, Jiang, Zhuge, & Zhang, 2016;

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Fig. 1. Schematic showing dimensionless coordinates for liquid ﬁlm ﬂow past an inclined plane coated with a soft solid layer and with shear stress applied at gas–liquid interface.

Tomar, Baingne, & Sharma, 2017; Tomar & Sharma, 2018). The studies by Tomar et al. (2017), Tomar and Sharma (2018), and Baingne and Sharma (2019) examined the effect of wall deformability on the stability of surfactant-laden liquid ﬁlm ﬂow and demonstrated that the presence of soft wall signiﬁcantly alters the stability behavior of surfactant-loaded ﬁlm ﬂows. It was shown that while the soft wall has stabilizing affect on GL interfacial mode, it can destabilize the Marangoni interfacial mode when the wall becomes suﬃciently soft. The above discussion shows that the presence of deformable wall signiﬁcantly affects the stability behavior of gravitydriven ﬁlm ﬂows down an inclined plane both in terms of instability suppression and enhancement. The application of shear stress at GL interface also modiﬁes the free-surface instability for liquid ﬁlm ﬂowing down a rigid inclined plane. In the present work, we undertake a comprehensive study of stability of shear-imposed liquid ﬁlm ﬂow past an inclined plane when the inclined plane is coated with a soft, deformable solid layer. The present work shows the emergence of GL and LS mode interfacial instabilities in low Reynolds number regime because of simultaneous presence of imposed shear at GL interface and a deformable liquid–solid interface. These GL and LS mode interfacial instabilities occur because of interaction between the imposed shear and deformable ﬂuid–solid interface and these instabilities vanish (or become practically irrelevant) if either of the two (imposed shear or deformable LS interface) is absent. The rest of the paper is organized as follows: The ﬂow conﬁguration description, governing equations and boundary conditions, base state and linearized stability equations are given in Section 2. A glimpse of long-wave asymptotic procedure to capture the GL mode instability is provided in Section 3. We only provide a brief outline of the long wave asymptotic procedure and mainly discuss the results obtained therein. The numerical results for arbitrary wavelength disturbances are discussed in Section 4. In this section, we start by ﬁrst giving parameter estimates which could be experimentally realizable in Subsection 4.1. This is followed by results in creeping ﬂow limit (Section 4.2), results in low Reynolds number limit (Section 4.3), discussion on critical mode of instability for this composite ﬂuid–solid system in Section 4.4, and ﬁnally the results showing instability suppression in Section 4.5. A summary of all the results and concluding remarks are given in Section 5. Appendix B and Appendix C provide details about the liquid–solid modes observed in low Reynolds number regime. 2. Problem formulation 2.1. Governing equations We consider a Newtonian liquid ﬁlm ﬂow (thickness R, density ρ , viscosity μ) past an inclined plane (inclination angle θ with the horizontal) coated with an incompressible and impermeable soft solid layer (thickness HR, and shear modulus Es ). The conﬁguration is shown in Fig. 1. A constant shear stress τs∗ is applied parallel to the plane at gas–liquid interface z∗ = 0. Positive/negative values of τs∗ refer to the shear stress acting in positive/negative x direction. An asterisk over a variable denotes dimensional quantity. The liquid layer is governed by Navier–Stokes mass and momentum conservation equations:

∇ ∗ · v∗ = 0 , ρ [∂

∗ ∗ t v

where

v∗

(1a) ∗ ∗

∗

+ v · ∇ v ] = ∇ · T + ρ g, ∗

∗

is the velocity ﬁeld, g is the gravity vector, and ∇ ∗

∗

(1b) ∗

= eˆ i ∂∂x∗ is the gradient operator. The stress tensor in liquid i

layer is given by T∗ = −p∗ I + [∇ v∗ + (∇ v∗ )T ], where p∗ is the pressure. The GL interface remains ﬂat in the steady unperturbed base state. In perturbed state, the GL interface is located at z∗ = h∗ (x∗ , t ∗ ), and the stress continuity condition at GL interface in presence of an imposed shear stress is given as: ∗

n · T∗ + τs∗ t = γ ∗ n(∇ · n ).

(2)

Here n and t are the unit (inward) normal and tangent vectors, γ ∗ is the GL interfacial tension. The kinematic condition at GL interface is: ∂t∗ h∗ + v∗x ∂x∗ h∗ = v∗z .

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An Eulerian framework is used in writing down the governing equations for liquid layer and GL interfacial conditions in which the spatial coordinates with respect to origin (x∗ = (x∗ , y∗ , z∗ )) form the independent variables. We use a Lagrangian framework to describe the deformation ﬁeld in soft solid layer and the soft solid is modeled using a simple non-linear neoHookean constitutive relationship. The position vector of a material point in solid layer is denoted by X∗ = (X ∗ , Y ∗ , Z ∗ ) in the initially undeformed state. The stress exerted by ﬂuid ﬂow over solid layer results in creation of deformation ﬁeld in solid layer. The solid layer undergoes a uni-directional displacement ﬁeld in the ﬂow direction as a result of imposition of steady state basic ﬂow in liquid layer. Following Patne, Giribabu, and Shankar (2017), this uni-directional displacement/deformation ﬁeld generated in solid layer is referred as pre-stressed base state or simply pre-stressed state in the present work. Each ∗ ∗ ∗ material particle in solid layer assumes a new position denoted as x = X∗ + u (X∗ ), where u (X∗ ) represents the Lagrangian displacement of the material point from undeformed position in the pre-stressed base state. To carry out the linear stability analysis, inﬁnitesimal perturbations are imposed on this pre-stressed/base state, and the current position vector in perturbed ∗ ∗ ∗ state can be written as: x∗ (x ) = x + u∗ (x , t ∗ ). Here, u∗ is the Lagrangian displacement of the material particle from the pre-stressed state. Note that the pre-stressed state coordinates are used as independent variable to express the current perturbed state in solid layer. The mass and momentum conservation equations for solid layer are:

det(F ) = 1,

ρ

∂ x ∂ t ∗2

2 ∗

(3a)

X∗

= ∇X∗ ∗ ref · P∗ + ρ g.

(3b)

ref

∗

Here F = ∂∂Xx ∗ = ∇X∗ ∗ x∗ is the deformation gradient tensor, X∗ ref is the position vector of a material particle in reference conﬁguration, and P∗ is the ﬁrst Piola-Kirchhoff stress tensor deﬁned as force per unit area of the reference conﬁguration. Initial undeformed state is treated as the reference conﬁguration while deriving the base state solid deformations and pressure proﬁles. However, while deriving the linearized governing equations in solid layer and the liquid–solid interfacial conditions, a consistent formulation requires that the pre-stressed base state must be considered as the reference conﬁguration ∗ and the base state coordinates (x ) must be treated as independent variables (Patne et al., 2017). The ﬁrst Piola–Kirchhoff stress tensor is related to Cauchy stress tensor as: P∗ = F−1 · σ ∗ so that P∗ represents current force per unit undeformed state area. The Cauchy stress tensor for a neo-Hookean elastic solid is given as (Beatty & Zhou, 1991; Destrade & Saccocmandi, 2004; Fosdick & Yu, 1996; Hayes & Saccocmandi, 2002):

σ ∗ = −p∗s I + Es (F · FT − I ),

(4)

where p∗s is the solid pressure distribution. The conditions of continuity of velocities and stresses hold at any point on the LS interface.

v∗ =

∂ x∗ , ∂t∗

(5a) ∗

n · T∗ + γls∗ n(∇ · n ) = n · σ ∗ ,

(5b)

γls∗

where is the LS interfacial tension. At = (1 + H )R, the soft solid layer is assumed to be strongly attached to the rigid boundary and thus zero displacement condition holds at rigid boundary: u∗ = 0. The aforementioned system of equations is made dimensionless using the following scales: R for lengths and displacez∗

2 ments, stress-free base ﬂow velocity at gas–liquid interface V = ρ gR2μsin θ for velocities, μV/R for stresses and pressure.

2.2. Base ﬂow Both the GL and LS interfaces remain ﬂat in the unperturbed steady base state. The steady state, laminar fully developed dimensionless velocity proﬁle and pressure distribution are given as:

vx (z ) = 1 − z2 + τs (1 − z ),

vz = 0,

p(z ) = (2 cot θ )z.

(6)

As mentioned earlier, the base state deformation in the soft solid layer is: x = X + u(X ). In the base state, the deformation gradient tensor is F = ∂∂Xx , and the governing equations can be written as: det(F ) = 1, and ∇X · P + 2g/(g sin θ ) = 0. The −1

T

relation between Piola-Kirchhoff stress tensor and Cauchy stress tensor is: P = F · σ , and σ = −ps I + G1 (F · F − I ), where G = μV/Es R is the dimensionless solid/wall deformability parameter and it represents the ratio of viscous shear stresses in liquid to elastic stresses in the solid layer. Rigid solid limit can be obtained by G → 0 and higher magnitude of G implies softer solid layer. The base state governing equations along with stress continuity condition at liquid–solid interface and zero displacement at rigid boundary are used to obtain the deformation and pressure ﬁelds in solid layer:

x(X, Z ) = X + ux (Z ),

ux (Z ) = G (1 + H )2 − Z

z(Z ) = Z,

2

ps (Z ) = (2 cot θ )Z,

+ G τs ( 1 + H − Z ) .

(7a) (7b)

Patne et al. (2017) have shown that the solid layer variables should be expressed in terms of the pre-stressed state coordinates (x, z ) in order to have a consistent Eulerian–Lagrangian (in ﬂuid and solid, respectively) formulation while deriving

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equations governing the stability of the composite ﬂuid–solid system. Thus, (x, z ) are used as independent variables to write down the base state displacement and pressure ﬁeld, linearized governing stability equations in solid layer, and solid variables appearing in conditions at LS interface. Using transformation Z (z ) = z, X (x, z ) = x − ux (z ), the base state displacement and pressure ﬁeld in terms of pre-stressed state coordinates are:

u x ( z ) = G [ ( 1 + H ) 2 − z ] + G τs ( 1 + H − z ) , 2

uz = 0 ,

ps (z ) = (2 cot θ )z.

(8)

2.3. Linear stability analysis A standard temporal linear stability analysis is performed assuming two-dimensional perturbations and these perturba tions are expressed in the form of Fourier normal modes. For liquid layer variables f = f˜(z ) exp[ik(x − ct )], and for solid variables f = f˜(z ) exp[ik(x − ct )]. Here, f is the inﬁnitesimal disturbance to a dynamical variable, f˜ is the complex amplitude function of the disturbance, k is the (real) wavenumber of perturbations and c is the complex wave speed. Following the standard procedure of linearization, the non-dimensional governing stability equations for liquid layer are:

dv˜ z + ikv˜ x = 0, dz

(9a)

Re [ik(vx − c )v˜ x + (dz vx )v˜ z ] = −ik p˜ + d p˜ ikRe(vx − c )v˜ z = − + dz

2

2

d − k2 v˜ x , dz2

(9b)

d − k2 v˜ z , dz 2

(9c)

VR where Re = ρμ is the Reynolds number.

As mentioned in Section 2.1, the position vector of a material particle in perturbed state is written as x(x ) = x + u (x, t ). The deformation gradient tensor F = ∂∂Xx = ∂∂ xx · ∂∂Xx = F · F, and using the multiplicative properties of determinants, the mass conservation equation becomes det(F ) = 1. The non-dimensional momentum conservation equation considering the prestressed state as the reference conﬁguration is written as:

∂ 2x Re ∂t2

= ∇x · P + 2g/(g sin θ ),

(10)

x

−1

−1

where P = F · σ . Note that the dot product is taken between F and Cauchy stress tensor so that the current area is mapped to pre-stressed state conﬁguration area. Following the linearization procedure, the governing stability equations in solid layer are:

du˜z + iku˜x = 0, dz

(11a)

2ik du˜ k 1 d (dz ux ) x + i(d2z ux ) − k(dz ux )2 u˜x + − k2 u˜x , G dz G G dz2

(11b)

d p˜ s 2ik du˜ k 1 d − (2 cot θ )iku˜x + (dz ux ) z + i(d2z ux ) − k(dz ux )2 u˜z + − k2 u˜z . dz G dz G G dz2

(11c)

−k2 c2 Reu˜x = −ik p˜ s + (2 cot θ )iku˜z +

−k2 c2 Reu˜z = −

2

2

The linearized kinematic condition, tangential and normal stress balances at GL interface are obtained by Taylor expansion of ﬂuid variables about the mean position of GL interface (z = 0):

˜ = v˜ z , ik(vx |z=0 − c )h

2 dv˜ x ˜ = 0, + ikv˜ z + dz vx |z=0 h dz − p˜ + 2

(12a) (12b)

dv˜ z ˜ = 0, − 2 cot θ + 2ik(dz vx |z=0 ) + k2 gl h dz

(12c)

where gl = γ ∗ /μV is the non-dimensional GL interfacial tension parameter. The conditions at LS interface incorporate both Eulerian (ﬂuid) and Lagrangian (solid) variables. As mentioned in Section 2.2, the independent variables for liquid layer are spatial coordinates with respect to origin (x, y, z), and the independent variables for solid layer are coordinates in pre-stressed state (x, y, z ). Following Patne et al. (2017) which states that while Taylor expansion is performed for ﬂuid variables in both tangential and normal directions, the same is not required for solid (Lagrangian) variables because of pre-stressed state coordinates (x, y, z ) being used as independent variables. Thus, the linearized conditions at LS interface (z = 1) are:

v˜ z = −ikc u˜z ,

(13a)

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v˜ x + (dz vx |z=1 )u˜z = −ikcu˜x , 2 dv˜ x 1 du˜x + ikv˜ z + dz vx |z=1 u˜z = + iku˜z , dz G dz dv˜ z 2 du˜z − p˜ + 2 − 2[cot θ + ik(dz vx |z=1 )]u˜z = − p˜ s + − k2 ls u˜z , dz

(13b) (13c) (13d)

G dz

where ls = γls∗ /μV is the non-dimensional LS interfacial tension parameter. Finally, no deformation (or zero displacement) conditions hold at rigid boundary (z = 1 + H):

u˜z = 0,

u˜x = 0.

(14)

Eqs. (9a) –(14) govern the stability of the shear-imposed ﬂuid–solid coupled ﬂow considered in the present study. This set of linear ordinary differential equations and boundary conditions form an eigen-value problem, which is solved to evaluate the eigenvalue c as a function of all other parameters i.e. c = f (Re, τs , gl , H, G, ls , θ ). We use a numerical shooting procedure (Drazin, 2002), and a pseudo-spectral collocation method (Weideman & Reddy, 20 0 0) to numerically obtain the eigenvalues and neutral stability curves. 3. Long-wave asymptotic analysis The GL interface is known to exhibit a long wave interfacial instability in the imposed shear stress (Yih, 1963). Thus, we ﬁrst solve the problem in the long-wave limit to examine the role of presence of both shear and deformable liquid– solid interface on the stability of GL interfacial mode. The low wave number (or long-wave) approximation remains valid when the perturbation wavelength is much greater than the total ﬁlm and solid layer thickness. This gives k 1/(1 + H ) in general, and k 1 for H ~ O(1). All other dimensionless parameters (e.g. Re, G, gl etc.) are considered to be O(1) quantities, i.e., they do not scale with k in any manner for GL interfacial mode. The complex wave speed (c) is expanded in an asymptotic series in k: c = c (0 ) + kc (1 ) + · · · , where c(0) is the leading order wave speed, and c(1) is the ﬁrst correction to wave speed. All the dynamical variables in both liquid and solid layer (v˜ z , v˜ x , p˜, u˜z , etc.) are expanded, according to their relative scalings with k as determined from the governing equations, in the following manner:

1 1 1 1 1 v˜ z , v˜ x , p˜, h˜ , u˜z , u˜x , p˜ s = v˜ z(0) , v˜ x(0) , 2 p˜(0) , h˜ (0) , u˜z(0) , u˜x(0) , 2 p˜(s0) k k k k k (1 ) 1 (1 ) 1 (1 ) 1 ˜ (1 ) (1 ) 1 (1 ) 1 (1 ) + k v˜ z , v˜ x , 2 p˜ , h , u˜z , u˜x , 2 p˜ s + ... k

k

k

k

k

These expansions are used in the governing equations (Eqs. (9a)–(11c)) and boundary conditions (Eqs. (12a)–(14)), and the resulting equations are solved at each order in k. The details of the low-k procedure are given in Appendix A and it shows that calculations up to O(k) are required to capture the stability of GL mode in k 1 limit. The leading order and ﬁrst correction to wave speed for GL mode are given as:

c ( 0 ) = 2 + τs , gl

2 4 c (1 ) = i (2 + τs )Re − cot θ − 2(2 + τs )GH . gl 15 3

(15)

The leading order wave speed (c (0 ) ) is found to be real, and exactly identical to the leading order wave speed obtained by gl Smith (1990) who analyzed the stability of Newtonian liquid ﬁlm falling down a rigid inclined plane under the action of an imposed shear stress in long wave limit. The leading order wave speed is affected by the presence of imposed shear stress, however, it is unaffected by the presence of soft solid layer. The ﬁrst correction to wave speed (c (1 ) ) is purely imaginary, and gl therefore, determines the stability of the GL interfacial mode. The above expression of c (1 ) recovers Yih’s (1963) result for gl τs = 0 and G or H = 0, and Smith’s expression of c(1) (Smith, 1990) in rigid plane limit (i.e. G and/or H = 0). Eq. (15) shows gl that stabilizing/destabilizing nature of the terms proportional to Re (inertial contribution) and GH (solid contribution) depends on the sign of (2 + τs ). The term proportional to Re is destabilizing and the term proportional to GH is stabilizing for (2 + τs ) > 0. Such destabilizing role of inertia in presence of imposed shear has been demonstrated by Smith (1990), and more recently by Wei (2005) and Bhat and Samanta (2019) for surfactant-covered liquid ﬁlm ﬂowing past a rigid inclined plane. The stabilizing effect of soft solid layer on Yih’s free-surface instability in absence of any imposed shear stress has also been demonstrated by several earlier studies (Gaurav & Shankar, 2007; Shankar & Sahu, 2006; Tomar & Sharma, 2018). On the other hand, the qualitative nature of the terms proportional to Re and GH get reversed when (2 + τs ) < 0, and as a consequence, new feature appears with respect to the role of deformable solid layer on the stability of GL interfacial mode. For (2 + τs ) < 0, the term proportional to Re becomes stabilizing while the term proportional to GH becomes destabilizing. Such destabilizing effect of soft solid layer on GL interface has not been observed in any of the previous studies related to ﬁlm ﬂows (without any imposed shear) past a ﬂexible solid surface (Baingne & Sharma, 2019; Gaurav & Shankar, 2007; Gkanis & Kumar, 2006; Jain & Shankar, 20 07; 20 08; Shankar & Sahu, 2006; Tomar et al., 2017; Tomar & Sharma, 2018). This destabilizing contribution of soft solid layer occurs because of presence of both deformable wall and imposed shear stress. Thus, for a given Reynolds number, angle of inclination, and shear stress satisfying the condition (2 + τs ) < 0; the GL mode

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instability can be activated by appropriately selecting the values of solid deformability parameter G and solid layer thickness H. On the other hand, for the case of (2 + τs ) > 0, the term proportional to GH is negative and hence, the soft solid layer parameters (G and H) can be chosen to suppress the GL instability which becomes operative above a critical Reynolds number. 4. Results and discussions 4.1. Estimation of parameters The primary aim of the present work is to understand the interaction between wall deformability and imposed shear stress at gas–liquid interface in modifying the stability characteristics of ﬁlm ﬂow past an inclined plane. The results presented in this work demonstrate that it is possible to manipulate the stability behavior of shear-imposed liquid ﬁlm ﬂow by tuning the wall deformability (or equivalently, shear modulus of soft solid layer) and soft solid thickness. In this section, we provide estimates of various parameters (in particular, wall deformability parameter or shear modulus) which are practically realizable. If we ﬁx ﬁlm thickness R ~ 1 mm, ρ ~ 103 kg/m3 , and g ~ 10 m/s2 , the Reynolds number can be estimated as Re ∼ R3 ρ 2 g/μ2 ∼ 10−2 /μ2 . Since we are interested in investigating the GL mode instability which occurs at Re ~ O(1) and instabilities that could occur at low Reynolds number due to presence of a deformable liquid–solid interface, we have used Re ∼ 0.01 − 1. These estimates of Re will hold for viscous liquids with μ ∼ O(10−1 ) − O(1 ) Pa s. The values of non-dimensional imposed shear stress used in presenting the results varies between O(10−1 ) − O(1 ). This estimate is made by using the expression of interfacial tangential shear stress at GL interface given in a recent study by Lavalle, Li, Mergui, Grenier, and Dietze (2019). They used both Orr-Sommerfeld formulation as well as experiments to examine the stability of falling liquid ﬁlm in interaction with a gas ﬂow over liquid ﬁlm in a narrow inclined channel. There are several other studies related to gas ﬂow over a liquid ﬁlm (Alekseenko, Aktershev, Cherdantsev, Kharlamov, & Markovich, 2009; Ju, Liu, Brooks, & Ishii, 2019), and the estimate of dimensionless shear stress from these studies as well is τs ∼ O(10−1 ) − O(1 ). The air–liquid dimensional interfacial tension is σ ∗ ~ 0.01 N/m for commonly used liquids, and this corresponds to gl ~ 1. The value of dimensionless GL interfacial tension parameter is ﬁxed as gl = 0 or 1 while the LS interfacial tension parameter

is ﬁxed as ls = 0. This is because our numerical results show that the effect of interfacial tension at GL or LS interfaces is insigniﬁcant on the stability behavior of the system under consideration. Finally, the dimensionless wall deformability parameter is G = μV/Es R ∼ ρ gR/Es , and for ﬁlm thickness R ~ 1 mm, g ~ 10 m/s2 , ρ ~ 103 kg/m3 , the wall deformability parameter is estimated as G ~ 10/Es . Our results are presented in the form of neutral curves in G vs. k plane which help us in predicting the values of G for which the ﬂow conﬁguration remains stable/unstable. This implies that the shear modulus, which is one of the tunable parameter, is Es ~ 10/G. Soft solid coatings with varying modulus can be fabricated using (poly)dimethylsiloxane (PDMS) by changing the percentage of cross-linker. For example, several studies have fabricated soft channels/tubes (Neelamegam & Shankar, 2015; Verma & Kumaran, 2012; 2013; 2015) and coating (Neelamegam, Shankar, & Das, 2013) using PDMS with Es ~ 103 –104 Pa. These values of shear moduli imply that the dimensionless wall deformability parameter varies between G ∼ O(10−3 ) − O(10−2 ). A value of G ~ 0.1 or 1 yields a shear modulus Es ~ 100 Pa or 10 Pa and such low values of shear moduli correspond to extremely soft solid layer which is unlikely to be of any practical importance. 4.2. Creeping ﬂow limit The low-k results show that when (2 + τs ) < 0, the GL interface can be made unstable in low wavenumber limit by tuning the deformability and thickness of the soft solid layer. We continue these low-k results to ﬁnite wavenumbers in Fig. 2(a) and (b) which show the variation of imaginary part of wave speed vs. wavenumber for two inclination angles. We ﬁrst present the results in creeping ﬂow limit, i.e. for Re = 0, to suppress any stabilizing contribution arising because of Re in k 1 limit (refer Eq. 15 when (2 + τs ) < 0). Fig. 2(a) shows that for a given solid thickness, the GL mode remains stable at lower G values and becomes unstable as G increases, and this observation is in agreement with the low-k results presented in previous section. For vertical inclined plane and in creeping ﬂow limit, the expression of c (1 ) reveals that GL gl mode is unstable for any non-zero value of G for a given solid thickness. This feature is depicted in Fig. 2(b) which shows that GL mode remains unstable for all values of G. The growth rate and the band of unstable wavenumbers increase with increase in deformability parameter for both inclination angles. We have veriﬁed that there are no other unstable modes present in the system for the parameter sets used in Fig. 2. It is important to mention that there exists a short wave LS mode instability for ﬂow past a neo-Hookean solid surface in creeping ﬂow limit when G ࣡ O(1) (Baingne & Sharma, 2019; Gkanis & Kumar, 20 03; 20 05; 20 06; Patne et al., 2017). While this short wave LS mode instability remains absent for the parameter values used in Fig. 2, we next show neutral stability curves in G vs. k plane depicting both the GL mode and short wave LS mode instabilities. Fig. 3 shows neutral stability curves for θ = 45◦ and θ = 90◦ in creeping ﬂow limit for τs = −3. We have exhaustively scanned the region in G vs. k plane, and observed that there exist only two sets of neutral curves for the present ﬂow conﬁguration: one corresponding to GL mode instability due to deformability of solid layer in presence of imposed shear stress, and the other corresponding to short-wave instability of LS interface. As shown in Fig. 2(a) and low-k results, the GL interface becomes unstable when G increases above a threshold value for θ = 45◦ . This ﬁgure also clearly shows that the

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Fig. 2. Imaginary part of wave speed vs. wavenumber data depicting the destabilization of GL mode due to deformability of soft solid layer: ci vs. k data for Re = 0, τs = −3, gl = 1, ls = 0.

Fig. 3. Neutral stability curves showing the destabilization of GL mode due to soft solid layer: Data for Re = 0, τs = −3, gl = 1, ls = 0.

threshold value of G for a given solid thickness exists in low wavenumber range, and hence, this threshold value can be determined from the expression of c (1 ) given in Eq. (15). The critical value of G decreases with increase in solid thickness. gl For vertical inclined plane and Re = 0, the GL mode is unstable for any non-zero value of G for a given solid thickness, again in agreement with the previously discussed results. However, for a given solid thickness, the growth rate increases with increase in solid deformability parameter G (as depicted in Fig. 2(b)). Both Fig. 3(a) and (b) shows that the GL mode remains the critical mode of instability in creeping ﬂow limit for shear-imposed ﬁlm ﬂow past a neoHookean deformable solid surface. To the best of our knowledge, this is the ﬁrst instance where the gas–liquid interfacial mode becomes unstable in creeping ﬂow limit for a Newtonian liquid ﬁlm ﬂowing past an inclined plane. This long-wave GL interfacial instability exists because of presence of both soft solid layer and imposed shear stress, and vanishes if either of the two components remains absent. It is worth to recall at this point that the deformable solid layer has a stabilizing effect on GL mode when imposed shear stress is positive (or zero). The GL interfacial instability becomes operative only in presence of inertia for ﬁlm ﬂows past a rigid incline, and earlier studies have demonstrated the potential of deformable solid layer in suppressing the GL instability in absence of any imposed shear stress (Baingne & Sharma, 2019; Gaurav & Shankar, 2007; Jain & Shankar, 2007; 20 08; Shankar & Sahu, 20 06; Tomar et al., 2017; Tomar & Sharma, 2018). We discuss this aspect of instability suppression with imposed shear stress in the later part of this work. 4.3. Results at low Reynolds number Figs. 2 and 3 presented results when Reynolds number is identically set to zero. For Re = 0, the characteristic equation is cubic in complex wave speed c, and we observed that out of the three roots of the characteristic equation, two roots become

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-100

Fig. 4. Eigenspectrum showing emergence of upstream and downstream unstable modes with increase in deformability parameter at low-Re and τs = 0: Data for H = 10, k = 0.1, Re = 0.1, gl = 1, ls = 0, θ = 45◦ .

unstable as G is varied. Fig. 3 shows that the critical root of instability corresponds to GL interfacial mode, the second most unstable root corresponds to short wave LS mode instability while the third root was never observed to become unstable for Re = 0. For non-zero Reynolds number, the characteristic equation admits multiple solution to c, and it is possible that the other roots which emerge at non-zero Re could also become unstable. Previous studies related to ﬂuid ﬂow past ﬂexible surfaces (Gaurav & Shankar, 2009; 2010; Shankar & Kumaran, 2001) have demonstrated the proliferation of multiple unstable modes at non-zero Reynolds number. These unstable modes exist only in presence of a deformable ﬂuid–solid interface and were referred as a particular type of liquid–solid (LS) modes based on the Reynolds number regime in which they were observed (Gaurav & Shankar, 2009; 2010; Shankar & Kumaran, 2001). In this section, we examine the possibility of existence of such unstable LS modes or any other instability modes (roots) of the characteristic equation for small but non-zero Re in addition to the unstable roots observed in creeping ﬂow limit. Fig. 4 shows the eigenspectrum for k = 0.1, Re = 0.1, τs = 0 and different values of deformability parameter G. It is clear from this ﬁgure that for any given value of G, there exists multiple roots for c among which approximately half have positive real part (downstream traveling waves) and other half have negative real part (upstream traveling waves). All the eigenmodes (upstream or downstream) remain stable up to G = 0.6. As G increases to 0.7 and 1, one downstream and one upstream traveling modes become unstable. These two modes form the ﬁrst downstream and ﬁrst upstream modes based on the smallest magnitude of the real part of wave speed among all eigenmodes (except GL mode). Similarly, the downstream (upstream) mode with next higher real part of wave speed forms the second downstream (upstream) mode and so on for other eigenmodes. Fig. 4 demonstrates that as G is progressively increased, more and more eigenmodes (both upstream and downstream) start becoming unstable. For example, four eigenmodes (two upstream and two downstream) are unstable at G = 5 and six eigenmodes (ﬁrst three upstream and ﬁrst three downstream) become unstable at G = 10. Fig. 4 also depicts the GL interfacial mode (refer inset) which remains stable at low Reynolds numbers and the increase in wall deformability parameter further stabilizes this GL mode. We next examine the effect of non-zero values of τ s on the eigenspectrum shown in Fig. 4. We computed eigenspectrums for non-zero positive and negative values of τ s and two such eigenspectra are shown in Fig. 5(a) and (b). Both the eigenspectra are shown for G = 1, varying values of τ s , and other parameters kept same as in Fig. 4. Recall that for G = 1 and τs = 0, only the ﬁrst downstream and ﬁrst upstream modes are unstable as is also shown in Fig. 5 (refer data represented by circle in Fig. 5). Fig. 5(a) shows that for positive τ s , the imaginary part of ﬁrst upstream traveling eigenmode remains positive and keeps on increasing with increase in τ s . On the other hand, for the ﬁrst downstream traveling eigenmode, ci which remains positive at τs = 0, decreases with increase in τ s and ﬁnally becomes negative when τ s is increased to 0.6. Thus, increasing (positive) τ s have a destabilizing effect on the ﬁrst upstream traveling eigenmode and stabilizing effect on the ﬁrst downstream traveling eigenmode. In fact, a similar trend is observed for all other higher downstream and upstream modes i.e. the upstream traveling modes become more and more unstable on increasing τ s while the downstream modes become more stable with increase in (positive) τ s . For example, in Fig. 5(a), the second upstream mode which remains stable at τs = 0, becomes unstable as τ s increases to 0.4. At the same time, the ci value for second downstream mode (as well as for other higher downstream modes) becomes more and more negative with increase in τ s . In contrast, when imposed shear stress is negative (refer Fig. 5(b)), the stabilizing/destabilizing effect of increasing magnitude of τ s is opposite on downstream and upstream modes. Fig. 5(b) shows that the downstream traveling eigenmodes become more and more unstable with increasing magnitude of (negative) τ s while the upstream modes become more stable with increase in τ s . We have veriﬁed that the features depicted in Figs. 4 and 5 are present for a wide variety of parameter sets. Therefore, multiple roots originate at non-zero Reynolds number and as shown in Fig. 4, these roots could become unstable at low Reynolds number on increasing the wall deformability parameter G. These unstable modes remain absent in creeping ﬂow

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Fig. 5. Eigenspectrum showing the effect of variation of imposed shear stress on the stability of upstream and downstream eigenmodes: data for H = 10, G = 1, k = 0.1, Re = 0.1, gl = 1, ls = 0, θ = 45◦ .

Fig. 6. Neutral stability curve showing the evolution of ﬁrst upstream and ﬁrst downstream inertial LS mode with variation in values of imposed shear: G vs. k for Re = 0.1, H = 10, gl = 1, ls = 0, θ = 45◦ . ‘D’ and ‘U’ denote downstream and upstream inertial LS modes.

limit and require the presence of deformable LS interface for their existence. These upstream and downstream unstable modes are then modiﬁed by the presence of imposed shear stress at GL interface (Fig. 5). We show in Appendix B that these upstream/downstream unstable modes belong to a class of inertial liquid–solid (ILS) modes observed by Gaurav and Shankar (2009, 2010) for the case of ﬂuid ﬂow through neo-Hookean tubes and channels. Thus, we designate these modes as ILS-1d, ILS-2d, and so on for inertial LS ﬁrst downstream mode, second downstream mode, and so on, respectively. Similarly, ILS-1u, ILS-2u denote inertial LS ﬁrst upstream, second upstream modes, respectively. Since these ILS modes could become unstable at low Reynolds number and as will be demonstrated shortly that they dominate the stability behavior for a wide range of parameter sets in low-Re regime, therefore, creeping ﬂow limit (in which only the GL interfacial and short wave LS modes are present) cannot be used as an approximation to capture the true stability behavior in low-Re limit. Both the eigenspectrums in Fig. 5 also depict the effect of variation of τ s on the GL interfacial mode (refer inset). This ﬁgure shows that the GL mode becomes more stable with increase in τ s when the imposed stress is applied in positivex direction in agreement with the previously discussed low-k results. On the other hand, when the imposed stress is in negative-x direction, the increase in magnitude of τ s have a destabilizing effect on the GL mode and it becomes unstable when (2 + τs ) < 0. Fig. 5 suggests that the ﬁrst upstream (downstream) mode is the most unstable mode for positive (negative) values of imposed shear stress. Therefore, we constructed neutral curves for this ﬁrst upstream and downstream modes for positive and negative values of τ s , respectively (refer Fig. 6). Fig. 6 shows that the neutral curves corresponding to ﬁrst downstream

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and upstream modes for τs = 0 are indistinguishable from each other. For positive τ s , Fig. 6(a) shows that the neutral curve corresponding to downstream mode shifts upward while the upstream mode neutral curve shifts downward with increase in imposed shear stress. In contrast, when τ s < 0 in Fig. 6(b), it is the downstream mode neutral curve which moves down and upstream mode neutral curve which moves up on increasing the magnitude of imposed shear stress. While the neutral curves in Fig. 6(a) and (b) are shown for ﬁrst upstream (ILS-1u) and ﬁrst downstream modes (ILS-1d) for different values of τ s , we have veriﬁed that the other higher upstream and downstream modes also show similar behavior as observed in Fig. 6(a) and (b). Thus, the increase in positive (negative) imposed shear stress has a destabilizing (stabilizing) effect on upstream modes and stabilizing (destabilizing) effect on downstream modes. Both Fig. 6(a) and (b) show that a separate branch of neutral curve starts to appear as an island of instability in G vs. k plane with increase in magnitude of τ s . Fig. 6 shows that this island of instability starts appearing for |τ s | ≥ 0.2 and exists at much lower values of G than the upper branch of the neutral curve corresponding to the upstream/downstream (positive/negative τ s ) unstable mode observed above. With increase in magnitude of τ s , this island grows and ﬁnally merges with the upper branch of neutral curve. For example, at |τs | = 1, the neutral curve exists as a single branch starting from low wavenumbers, followed by a substantial dip in value of G at around k ~ 0.2, and then continuing to higher wavenumbers to ﬁnally merge with neutral curve corresponding to short wave LS mode instability. On the other hand, at |τs | = 0.5, a separate island of instability and an upper branch exist, and this upper branch merges with the short wave LS mode neutral curve at higher wavenumbers. Note that Fig. 6(b) also shows the neutral curve corresponding to the GL mode destabilization which occurs for (2 + τs ) < 0 when wall deformability parameter increases above a critical value.

4.4. Identiﬁcation of critical mode of instability The results presented thus far demonstrate that in presence of positive (negative) imposed shear stress, the ﬁrst upstream (downstream) inertial LS mode becomes unstable at low Reynolds number and at signiﬁcantly low values of G as compared to τs = 0 case. For example, Fig. 6 shows that the critical value of G required to make ﬁrst upstream/downstream mode unstable for τs = 0 is Gcrit = 0.43. In contrast, Gcrit ≈ 3.5 × 10−3 for |τs | = 0.2 and Gcrit ∼ O(10−4 ) for |τ s | ~ O(1). Thus, the critical value of G required to make inertial LS modes unstable decreases by about two to three orders of magnitude in presence of imposed shear stress (positive/negative) as compared to the case when τs = 0. This signiﬁcant drop in critical value of G in presence of non-zero τ s is important in order to realize the above predicted inertial LS mode instability in experiments. As discussed in Section 4.1, the shear modulus corresponding to G ∼ O(10−1 ) is Es ~ 100 Pa which corresponds to highly soft solid layer which are diﬃcult to fabricate and use in such ﬂow instability experiments. Thus, even though an inertial LS mode instability is predicted numerically for τs = 0 when G ≥ 0.43, it will be unlikely to realize this instability in experiments. On the other hand, in presence of imposed shear stress |τs | = 0.2 or O(1), the wall deformability parameter is O(10−3 ) or O(10−4 ) which corresponds to Es ~ O(104 ) or O(105 ) Pa. Soft solid coatings or channels with soft walls with shear modulus of this range (103 –105 Pa) have been fabricated and used in experiments which investigate instabilities occurring in ﬂow past deformable solid surface (Neelamegam, Giribabu, & Shankar, 2014; Neelamegam & Shankar, 2015; Neelamegam et al., 2013; Verma & Kumaran, 2012; 2013; 2015). Thus, these inertial LS mode instabilities could be observed for ﬁlm ﬂows only in presence of an imposed shear stress. Fig. 6 also demonstrates that the ﬁrst upstream/downstream inertial LS mode remains the most unstable mode for positive/negative values of imposed shear stress. For negative values of τ s , GL mode could also become unstable when (2 + τs ) < 0 and G increases above a particular value. Fig. 6(b) shows the neutral curve corresponding to this destabilization of GL mode for τs = −3 and it shows that the GL mode becomes unstable for G ࣡ 0.035. However, this ﬁgure also shows that the inertial LS mode becomes unstable for much lower values of G(≈ 4 × 10−4 ) at τs = −3. Thus, even when the GL mode becomes unstable, it is the inertial LS mode that remains the most unstable mode. Fig. 7 shows the neutral stability curves for |τs | = 1 at different values of solid layer thickness. It demonstrates that the critical value of G required to trigger LS mode instability increases with decrease in solid thickness. For example, Gcrit ∼ O(10−3 ) for H = 10, Gcrit ∼ O(10−2 ) for H = 5, while for H = 1 and 2, Gcrit ~ O(1). Fig. 7(a) also shows that for τs = 1, the GL mode becomes unstable for a band of ﬁnite wavenumbers (except for H = 1) as depicted by a small island of instability (magniﬁed in the inset). The inertial LS mode remains the most unstable mode for thick solid layer (e.g. H = 10 and 5), however, the critical value of G is determined by small island of instability corresponding to GL mode for relatively thin solid (e.g. H = 2). This island of instability is not observed for negative values of τ s . Again, it is important to note that for H = 2, the critical G as determined by GL mode instability island is O(10−1 ) while the inertial LS mode becomes unstable for G > O(1). As mentioned above, these values of G yield very low shear modulus (Es ~ O(100) or 10 Pa) values which are non-realizable in experiments. On the other hand for thicker solid layer, G ∼ O(10−3 ) to trigger inertial LS mode instability which yields experimentally realistic values of shear modulus. Thus, the present composite ﬂow conﬁguration will become unstable because of inertial LS mode instabilities for relatively thick solid coatings. We have also plotted neutral curves at different values of solid thickness (data not shown) for the case when (2 + τs ) < 0 and the low-k GL mode perturbations could become unstable on increasing G along with inertial LS mode. The results show that decreasing solid layer thickness shifts the neutral curves upward belonging to both the inertial LS mode and low-k GL mode. A comparison of value of G for both modes show that the LS mode dominates for thick solid layer while the low-k GL mode becomes critical mode for relatively thin solid layers. However, the values of G required to trigger instabilities for

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Fig. 7. Neutral stability curves at different values of solid thickness: G vs. k data for Re = 0.1, gl = 1, ls = 0, θ = 45◦ .

Fig. 8. Neutral stability curves demonstrating that the low-k mode instability dominates for ﬁlm ﬂow past a vertical plane at low Reynolds numbers: Data for different H and Re = 0.01, τs = −3, gl = 1, ls = 0, θ = 90◦ .

thin solid correspond to very low and non-practical values of shear moduli, and for thick solid coatings (H = 5, 10 etc.), the (ﬁrst downstream) inertial LS mode dominates and yields practically feasible values of shear moduli. The angle of inclination was set at θ = 45◦ in all the results presented thus far. These results suggest that shear-imposed ﬁlm ﬂow past a ﬂexible inclined surface becomes unstable for thick solid coating due to excitation of inertial LS mode instability. The low-k GL mode instability (whenever) present due to interaction between imposed shear and wall deformability becomes the most unstable mode for suﬃciently thin solid layer. However, it is unlikely to be realizable in practical systems because the excitation of this GL mode instability requires very soft solid layer which is diﬃcult to fabricate and use in practice. Thus, it seems that the low-k GL mode instability predicted in Section 3 will not be important for shear-imposed ﬁlm ﬂow past a deformable solid surface. The expression for c (1 ) suggests that for low Reynolds number and moderate angl gle of inclination, the term containing cot θ provides a signiﬁcant stabilizing contribution which makes G suﬃciently higher such that this instability becomes unrealizable for practical systems. This stabilizing contribution due to cot θ becomes small at higher inclination angles and vanishes for vertical inclined plane. Fig. 8 presents neutral stability curves for θ = 90◦ at different values of solid thickness. This ﬁgure clearly shows that the threshold value of G required to trigger low-k GL mode instability and inertial LS mode instability is almost similar for H = 10. The low-k GL mode instability becomes the dominant mode of instability at H ≤ 5 and the wall deformability parameter G ∼ 10−4 –10−3 for H = 1–5 to trigger the GL interfacial mode. This implies that this low-k GL mode instability will be important for shear-imposed ﬁlm ﬂow past a deformable vertical surface at low Reynolds numbers. 4.5. Instability suppression using deformable solid layer All the numerical results presented thus far are related to activation of LS/GL interfacial instabilities at low Reynolds numbers. The low-k GL interfacial mode discussed in Section 3 does not become unstable at low Reynolds number (Re ≤ O(10−1 ))

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Fig. 9. Neutral stability curves demonstrating instability suppression: G vs. k data for Re = 1.25, gl = 1, ls = 0, θ = 45◦ .

for ﬁlm ﬂow past a rigid inclined plane (i.e. for H and/or G = 0). This long wave GL interfacial mode is commonly referred as free surface mode in literature. This free-surface instability becomes active when Re increases above a critical value, and this critical value of Reynolds number is an O(1) quantity for moderate angle of inclinations (for example, refer Eq. (15) for H and/or G = 0) Yih (1963). Several previous studies related to ﬁlm ﬂow past ﬂexible inclined plane have demonstrated a complete suppression of this GL interfacial mode instability without excitation of any of the LS interfacial mode (Baingne & Sharma, 2019; Gaurav & Shankar, 2007; Jain & Shankar, 2007; Shankar & Sahu, 2006). However, all of these studies demonstrated instability suppression in absence of any imposed shear at GL interface. The results in preceding sections show that the presence of imposed shear stress at GL interface substantially enhances the inertial LS mode instabilities. In view of the presence of this enhancement of inertial LS mode instability, we evaluate whether it is still possible to use a deformable solid coating in presence of non-zero τ s to obtain a stable ﬁlm ﬂow conﬁguration when the ﬂow otherwise remains unstable in the limit of rigid inclined plane. Further note that the expression of c (1 ) in Eq. (15) shows that the soft solid layer gl has a stabilizing effect and inertia has a destabilizing effect on GL mode for (2 + τs ) > 0. Thus, we use only positive values of τ s in presenting the results related to instability suppression. Fig. 9(a) shows the neutral stability curves for two different values of solid thickness for Re = 1.25 and τs = 0.25. We ﬁrst discuss the neutral curve data set for H = 5 in Fig. 9(a). The expression of ﬁrst correction to wave speed in Section 3 (Eq. (15)) shows that the GL interface is unstable in the rigid limit (H and/or G → 0) and increasing H and/or G has a stabilizing effect on the GL interfacial mode. Accordingly, Fig. 9(a) shows that the GL interface remains unstable for G → 0 and at lower values of G. As G is increased above the lower neutral curve, there is a transition from unstable to stable GL mode perturbations. Note that this lower neutral curve shows that the threshold value of G above which GL mode is stabilized can be determined using long wave asymptotic results (i.e. Eq. (15)). With further increase in G for a given H (= 5 in Fig. 9(a)), we encounter two neutral curves: one corresponding to destabilization of inertial LS mode, and other corresponding to destabilization of GL interface (Baingne & Sharma, 2019; Gaurav & Shankar, 2007; Tomar & Sharma, 2018) at suﬃciently higher values of G. Importantly, there exists a wide gap in between the lower and upper neutral curves where the GL interfacial instability is suppressed due to the presence of deformable solid layer without triggering any other instabilities. The values of G is in between O(10−3 ) − O(10−1 ) where the present ﬁlm ﬂow conﬁguration remains stable. When the solid thickness is increased (e.g. H = 10 in Fig. 9(a)), the neutral curve corresponding to inertial LS mode instability begins to appear at suﬃciently low values of G which leads to closing of stability window in terms of parameter G. On the other hand, a stable gap is obtained for smaller solid thickness for the parameters corresponding to Fig. 9(a). For example, wall deformability parameter G is in between O(10−2 ) − O(1 ) for H = 1 or 2 (data not shown) for both GL and LS interfaces to remain stable. These results illustrate the potential of using a soft solid layer in obtaining stable ﬁlm ﬂow conﬁguration when the ﬁlm otherwise remains unstable for ﬂow past a rigid inclined plane. Fig. 9(a) demonstrated that the stability window begins to appear below an optimum value of solid thickness (e.g. H ࣠ 5) and vanishes for suﬃciently thick solid layer. Thus, we constructed neutral curves at H = 5 and at higher values of τ s while keeping all other parameters same as in Fig. 9(a). Recall that increasing (positive) τ s have a destabilizing effect on GL mode in rigid limit (refer Eq. (15)). Fig. 9(b) shows the neutral curves when τ s is increased to 0.5 and 1. We again observe similar set of neutral curves as depicted in Fig. 9(a) i.e. lower neutral curve corresponding to GL instability suppression and upper neutral curves corresponding to GL and LS interfacial modes destabilization. The inertial LS mode neutral curve is present as an island of instability for τs = 0.5, and the stability window is determined by this inertial LS mode and lower GL mode neutral curves. The data in Fig. 9(b) for τs = 0.5 shows that there still exists an adequate gap between the two neutral curves and hence, it is possible to obtain stable ﬁlm ﬂow for τs = 0.5. On the other hand, the stable gap ceases to exist for

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τs = 1 due to presence of inertial LS mode instability at suﬃciently low value of G. To summarize, our results show that a soft solid layer could be used to obtain stable ﬁlm ﬂow conﬁguration for several cases when the ﬁlm ﬂow past a rigid inclined plane is unstable. 5. Conclusion This work investigated the effect of wall deformability on the linear stability of shear imposed liquid ﬁlm ﬂowing past an inclined plane. For this ﬁlm ﬂow past an inclined plane which is coated with a deformable solid layer, there exists two interfaces which could possibly become unstable: (i) the gas–liquid (GL) interface, and (ii) the deformable liquid–solid (LS) interface. The two tunable (dimensionless) parameters for the soft solid layer are solid thickness (H) and deformability parameter G (or equivalently, shear modulus). The GL interface exhibits a Yih type long wave instability in the limit of rigid inclined plane (Smith, 1990; Yih, 1963). However, this GL mode instability is activated only in presence of inertia. We show that for shear-imposed ﬁlm ﬂow past an inclined plane with deformable wall, this GL mode becomes unstable even in creeping ﬂow limit when the shear stress is negative and when G increases above a critical value for a given solid thickness. There also exists an unstable short wave LS mode in zero Reynolds number limit, however, the GL mode becomes unstable much early as G is increased. Thus, the GL mode dominates the stability behavior of the composite ﬂuid–solid system in creeping ﬂow limit. We further examined the stability of this ﬁlm ﬂow conﬁguration for low Reynolds numbers (Re O(10−1 )). We show that multiple unstable eigenmodes with positive (downstream modes) and negative (upstream modes) real part of wave speed emerge due to the presence of deformable LS interface for low Re for ﬁlm ﬂow in absence of any applied shear stress at GL interface. However, unrealistically high value of G corresponding to an ultra-soft solid layer is required to excite these unstable LS modes. The presence of shear stress at GL interface dramatically reduces the critical value of G required to trigger these LS mode instabilities, thus, making these instabilities realizable in an experiment. We further show that the most unstable upstream/downstream LS mode (among the multiple LS eigenmodes) remains the critical mode of instability for positive/negative shear stress in low-Re regime. An important implication of this result is that the creeping ﬂow limit cannot be used as an approximation for low Reynolds number regime to predict the most dangerous instability mode for shear-imposed ﬁlm ﬂow past a deformable inclined plane. For negative imposed shear stress when the GL mode instability is possible on increasing the magnitude of shear; we also show that for vertical inclined plane, either of the GL mode or downstream LS mode will dictate the stability behavior depending on thickness of the soft solid layer. We ﬁnally show that for parameter regimes where the shear-imposed ﬁlm ﬂow remains unstable for rigid incline, a soft solid coating can be used to suppress the GL interface instability without exciting any other (LS) modes by appropriately selecting the values of solid thickness and deformability parameter (or equivalently, shear modulus). To conclude, the present work demonstrates the potential of a deformable wall in control and manipulation of instabilities for shear-imposed ﬁlm ﬂow past an inclined plane. Declaration of Competing Interest The authors declare that they do not have any ﬁnancial or nonﬁnancial conﬂict of interests. Acknowledgment The authors thank the Science and Engineering Research Board (SERB), Government of India for funding the work through project grant CRG/2018/003532. Appendix A. Low wavenumber equations A1. Leading order The governing equations for liquid layer at leading order are:

dv˜ z(0) + iv˜ x(0) = 0, dz

d v˜ x(0) − i p˜ (0) = 0, dz2 2

d p˜ (0) = 0. dz

(A.1)

The conditions at GL interface (evaluated at z = 0) are:

˜ (0) = v˜ z(0) , i[vx − c(0) ]h

dv˜ x(0) 2 ˜ (0) + dz vx h = 0, dz

p˜ (0) = 0,

(A.2)

where Eq. (A.2) refer to kinematic condition, tangential and normal stress balances, respectively. The leading order conditions at LS interface (linearized about the mean position z = 1) are:

v˜ z(0) = 0,

v˜ x(0) = 0,

dv˜ x(0) 1 du˜x(0) = , dz G dz

p˜ (0) = p˜ (s0) .

(A.3)

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First two Eq. (A.3) represent velocity continuity conditions, which clearly show that the leading order ﬂuid velocities satisfy no slip condition at LS interface. Last two Equations in (A.3) represent stress continuity at LS interface. The governing equations in liquid layer at leading order (Eq. (A.1)) along with conditions at gas–liquid interface (last two equations of (A.2)), and velocity continuity at LS interface (ﬁrst two equations of (A.3)) are used to solve for leading order pressure and velocity ﬁeld in liquid layer:

v˜ x(0) = 2(z − 1 )h˜ (0) ,

p˜ (0) = 0,

v˜ z(0) = −i(z − 1 )2 h˜ (0) .

(A.4)

The leading order wave speed corresponding to GL mode is calculated from leading order kinematic condition (ﬁrst equation of (A.2)) and is given as:

c ( 0 ) = 2 + τs . gl

(A.5)

Note that the solid deformation ﬁeld is not required for the calculation of leading order wave speed. This implies that the ﬂexible solid layer do not play any role in determining c (0 ) . Therefore, the stability of the system at this order remains gl independent of the presence of deformable wall and is identical to that of the shear-imposed falling ﬁlm down a rigid inclined plane (Smith, 1990; Wei, 2005). This is attributed to the velocity continuity at LS interface (ﬁrst two equations of (A.3)) showing the existence of no slip condition at leading order. However, the stress continuity conditions at LS interface (last two equations of (A.3)) suggest that the leading order ﬂuid velocity ﬁeld exerts stresses at the liquid–solid interface, resulting in the development of a leading order deformation ﬁeld in the solid layer. Consequently, these deformations in turn, can alter the original ﬂuid velocity ﬁelds at O(k) and ﬁnally, the effect of wall deformability can be witnessed in ﬁrst correction to the wave speed. The governing equations for solid layer at leading order are:

du˜z(0) + iu˜x(0) = 0, dz

d u˜x(0) 2

dz

2

d p˜ (s0) = 0. dz

− iG p˜ (s0) = 0,

(A.6)

Eq. (A.6) subjected to continuity of stresses at LS interface (last two equations in (A.3)) and no deformation conditions at z = 1 + H : u˜z(0 ) = 0, u˜x(0 ) = 0, are solved for the leading order deformation ﬁeld:

˜ (0 ) . u˜z(0) = −iG(z − 1 − H )2 h

(A.7)

A2. First correction As c (0 ) is real, it is required to carry out calculations at O(k). At O(k), the governing equations for liquid are: gl

dv˜ z(1) + iv˜ x(1) = 0, dz

d p˜ (1) = 0, dz

(A.8a)

d v˜ x(1) − i p˜ (1) = Re i(vx − c(0) )v˜ x(0) + (dz vx )v˜ z(0) . gl dz 2 2

(A.8b)

The kinematic condition and stress balances at z = 0 are:

˜ ( 1 ) − ic ( 1 ) h ˜ (0) = v˜ z(1) , i(vx − c(0) )h gl gl

(A.9a)

dv˜ x(1) 2 ˜ (1) + dz v x h = 0 , dz

(A.9b)

˜ ( 0 ) = 0, p˜ (1) + (2 cot θ )h

and the velocity continuity conditions at z = 1 are:

v˜ z(1) = −ic(0) u˜z(0) ,

v˜ x(1) + (dz vx )u˜z(0) = −ic(0) u˜x(0) .

gl

(A.10)

gl

The above equations show that the solid deformation ﬁeld enters the low wavenumber calculations via conditions (A.10). Furthermore, Eq. (A.10) infer that the leading order deformations are responsible for creating O(k) ﬂuid velocities. Therefore, the solid contribution is now expected to appear in the ﬂuid velocity ﬁeld at ﬁrst correction. The solution to Eqs. (A.8a) and (A.8b) can be obtained by using Eqs. (A.9b) and (A.10):

˜ (0) cot θ , p˜ (1) = −2h (1 )

v˜ z where

v˜ x(1) = i f1 h˜ (0) + 2(z − 1 )h˜ (1) ,

(A.11a)

˜ ( 0 ) − i ( z − 1 )2 h ˜ (1 ) , = f2 h

(A.11b)

1 (z − 1 ) z3 − 3z2 + 3z + 3 Re + GH 2 + 2c(0) GH − (z − 1 ) z2 − 2z + 2 Re − (z2 − 1 ) cot θ , gl 12 6

3 z+2 (0 ) 1 2 2 + τs 2 f 2 = − (z − 1 ) z − 3z + 3z + 9 Re + cot θ − c (z − 1 )2 z2 − 2z − 5 Re + (2 + H − 2z )GH f 1 = − ( 2 + τs )

1

60

−(2 + τs )(z − 1 )GH 2 .

3

gl 12

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The ﬁrst correction to the wave speed for the gas–liquid mode c (1 ) can be obtained from O(k) kinematic condition Eq. (A.9a): gl

2 4 c (1 ) = i (2 + τs )Re − cot θ − 2(2 + τs )GH . gl 15 3

(A.12)

Appendix B. Idenitiﬁcation of unstable LS modes in low Reynolds number regime Fig. 6(a) and (b) depicted the evolution of the inertial LS modes neutral curves with respect to the variation of τ s . It was also mentioned that multiple unstable modes emerge because of presence of deformable LS interface for non-zero Reynolds number. Thus, in order to gain more insight on the nature of origin of the unstable upstream and downstream modes observed in Fig. 6, we now examine the behavior of neutral curves for these unstable modes with variation in Reynolds number. Fig. B.10 gas–liqu the neutral curves for ﬁrst upstream and ﬁrst downstream modes at τs = 1 and −1 for different values of Reynolds number. These ﬁgures clearly show that the neutral curves divide into two branches when Re ≤ 0.01: an upper branch which continues to low wavenumbers, and a lower branch which exists as an island of instability for a narrow l −k band of wavenumbers. The overall critical value of G (Gal ) above which we expect ﬂow (LS interface) to become unstable c is determined by this lower branch of neutral curve (which merges with upper branch for suﬃciently high Reynolds number, for example, Re ࣡ 0.1). The upper branch which continues to low-k regime, also shows a minimum in terms of G which we denote as Glow c . Fig. B.10 shows that this upper branch of neutral curve keeps shifting towards low wavenumbers with decrease in Reynolds number. The value of Glow is not inﬂuenced by variation in Re while the critical wavenumber klow keeps c c decreasing with decrease in Reynolds number for this upper low-k branch of the neutral curve. Similar trends of shifting of neutral curves, critical G remaining independent of Re, and critical wavenumber decreasing with decrease in Reynolds number were also observed in low-k low-Re regime by Gaurav and Shankar (2009, 2010) for ﬂuid ﬂow in deformable tubes and channels. They plotted the variation of critical conditions with Reynolds number and demonstrated that Glow ∼ O ( 1 ), c klow ∼ Re1/2 , and crlow ∼ Re−1 . We also observed a similar scaling behavior for critical conditions in low-Re low-k regime c (Re and k 1) for this upper branch of neutral curves. These scalings for critical conditions with respect to Re in k 1 regime for the present case are shown in Fig. B.11(a) and (b) for τs = 1 and τs = −1, respectively. While the neutral curves in Fig. B.10 and scalings of critical conditions in Fig. B.11 are shown for the ﬁrst upstream or downstream unstable mode at τs = ±1, we have veriﬁed that similar features are observed at other values of τ s and for higher downstream and upstream unstable modes. Similar to Gaurav and Shankar (2009, 2010), we have performed an asymptotic analysis in low-Re low-k regime by assuming the above observed scalings in order to capture the instability of upstream and downstream modes. The results from our asymptotic analysis show an excellent match with our numerical results. An outline of asymptotic analysis is given in Appendix C. These low-Re low-k modes were identiﬁed as a type of shear waves in elastic solid which exist at ﬁnite inertia in solid (Achenbach, 1973) and gets destabilized by ﬂuid ﬂow (Gaurav & Shankar, 2009; 2010). Based on the nature of neutral curves and the scalings shown by critical conditions as a function of Re in low-k regime, we infer that the unstable eigenmodes observed above belong to the same class of unstable shear waves in elastic solid which were identiﬁed by Gaurav and Shankar for ﬂuid ﬂow in neo-Hookean deformable tubes or channels (Gaurav & Shankar, 2009; 2010). Since, these modes exist only when Re is non-zero and the liquid–solid interface is deformable, we have referred these unstable modes as inertial liquid–solid (upstream/downstream) modes (ILS-u/d modes).

Fig. B.10. First upstream (a) and ﬁrst downstream (b) modes neutral curves as a function of decreasing Reynolds number: G vs. k data for H = 10, gl =

1, ls = 0, θ = 45◦ .

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Fig. B.11. Scalings of critical conditions with Re: Data for H = 10, gl = 1, ls = 0, θ = 90◦ .

Appendix C. Asymptotic analysis for low-Re low-k unstable LS modes In this appendix, we provide a brief outline of the asymptotic analysis used to capture the unstable upstream or downstream LS modes observed in Section 4.3 and also discussed in Appendix B. The details of the asymptotic procedure are given in Gaurav and Shankar (2009) who carried out a similar analysis for pressure-driven ﬂow of a Newtonian ﬂuid in a neo-Hookean deformable tube. Based on the scalings shown in Fig. B.11, = (Re/G )1/2 is chosen as the small parameter and k = k0 with k0 ~ O(1) as the scaled dimensionless wavenumber. The limit 1 implies k 1, and Re 1 because G ~ O(1) as shown in Fig. B.11. The complex wave speed c is expanded as:

c = −2 (c(0) + c(1) + · · · ).

(C.1) O( −1 )

O( −2 )

Assuming v˜ z ∼ O(1 ), the relative scalings of dynamical variables in ﬂuid ﬁlm are v˜ x ∼ and p˜ ∼ from ﬂuid continuity Eq. (9a) and x-momentum balance Eq. (9b), respectively. The scalings of variables in deformable solid layer are u˜z ∼ O( ) from continuity of normal velocity at z = 1, u˜x ∼ O(1 ) from solid continuity Eq. (11a), and p˜ s ∼ O( −1 ) from xmomentum balance (Eq. (11b)) in solid layer. The variables in ﬂuid and solid layer are thus expanded as:

1 1 1 1 1 1 {v˜ z , v˜ x , p˜, u˜z , u˜x , p˜ s } = v˜ z(0) , v˜ x(0) , 2 p˜(0) , u˜z(0) , u˜x(0) , p˜(s0) + v˜ z(1) , v˜ x(1) , 2 p˜(1) , u˜z(1) , u˜x(1) , p˜(s1) 1 1 1 + 2 v˜ z(2) , v˜ x(2) , 2 p˜ (2) , u˜z(2) , u˜x(2) , p˜ (s2) + · · ·

These expansion along with k = k0 are substituted in linearized governing equations for liquid layer (Eqs. (9a)–(9c)), solid layer (Eqs. (11a)–(11c)), and boundary conditions (Eqs. (12a)–(14)). The resulting governing equations after making these substitutions are solved at each order in to evaluate the eigenfunctions in liquid and soft solid layer at different orders in . The analysis reveals that it is suﬃcient to evaluate liquid eigenfunctions correct up to order O( 2 ) and solid eigenfunctions correct to O( ) to determine the stability of the present ﬂow conﬁguration. The liquid eigenfunctions at O( 0 ), O( 1 ), and O( 2 ) are given as:

p˜ (0) = 0,

v˜ x(0) = A1 ,

p˜ (1) = c(0) A1 +

6 A4 , k20

p˜ (2) = (c(1) + ik0 )A1 ,

v˜ z(0) = A2 − ik0 A1 z, v˜ x(1) =

i (2A3 + 3A4 z )z, k0

v˜ x(2) =

(C.2a)

v˜ z(1) = (A3 + A4 z )z2 ,

1 ( 4A3 + 3A4 z )c ( 0 ) z 3 , 12

v˜ z(2) = −

i k0 c ( 0 ) ( 5A3 + 3A4 z )z 4 . 60

(C.2b) (C.2c)

The solid eigenfunctions correct up to O( ) are given as:

p˜ (s0) = G4 (c(0) )2 , iG1 iG4 iG2 , √ sin B1 z − √ cos B1 z + 2 ( 0 ) k0 G k0 c G G1 G2 u˜z(0) = − cos B1 z − sin B1 z + G4 z + G3 , ( k0 c ( 0 ) )2 G ( k0 c ( 0 ) )2 G u˜x(0) =

k20 c(0)

(C.3a) (C.3b) (C.3c)

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(1 )

p˜ s

where

i =2 − c ( 0 ) c ( 1 ) G4 , k0

(C.4a)

u˜x(1) =

1 √ {(G1 f1 + G2 f2 ) cos B1 z + (G2 f1 + G1 f2 ) sin B1 z}, 2B2 G

(C.4b)

u˜z(1) =

i {(G2 f3 − G1 f4 ) cos B1 z − (G1 f3 + G2 f4 ) sin B1 z}, 2B2 c ( 0 ) G

(C.4c)

√ B 1 = k 0 c ( 0 ) G,

f1 = B1 2ik0 c

B2 = ( k0 c ( 0 ) )3 ,

(0 ) (1 )

c

z − 3 τs ,

f2 = k0 c(0) 3ic(1) + 2k0 c(0) G(z + τs )z + 2,

f3 = B1 2ik0 c(0) c(1) z − 4z − 5τs , f4 = 2 G(k0 c(0) )2 (z + τs )z + 5ik0 c(0) c(1) − 2. Note that the imposed shear stress term appears in solid deformation ﬁeld at O( ), and hence, the imposed shear stress is expected to have appreciable effect on stability of liquid–solid modes. These liquid and solid eigenfunctions are then substituted in boundary conditions (Eqs. (12a)–(14)) to obtain a set of linear homogeneous equations which can be represented as M · CT = 0 where C = {A1 , A2 , A3 , A4 , G1 , G2 , G3 , G4 }. The determinant of matrix M is expanded as a series in and setting this determinant equal to zero gives the characteristic equation as follows:

f ( 0 ) ( c ( 0 ) ) + f ( 1 ) ( c ( 0 ) , c ( 1 ) ) + · · · = 0, f(0)

(C.5)

f(1)

where and are the leading order and ﬁrst correction terms of the determinant, respectively. Setting the leading order term of Eq. (C.5) equal to zero yields the leading order wave speed, and this leading order characteristic equation admits multiple roots for c(0) all of which are real, and could be positive or negative. Thus, the perturbations are neutrally stable to leading order, and c(0) positive/negative corresponds to downstream/upstream traveling neutrally stable waves. Out of all the multiple roots for c(0) , the root having the least magnitude (i.e. the perturbation wave traveling with slowest speed) is termed as ﬁrst downstream (for c(0) positive) or ﬁrst upstream (for c(0) negative) mode. The leading order solutions (of equation f (0 ) (c (0 ) ) = 0) in increasing order of magnitude of values of c(0) form higher downstream (for c(0) positive) and upstream (for c(0) negative) modes. Further the scaling analysis reveals that inertial effects in solid remain important even in low Reynolds number limit. This can be inferred as follows: The inertial terms in solid are O(k2 c2 Re), and based on the numerically obtained scalings for k and c (refer Fig. B.11 and as mentioned above), it can be seen that the inertial and elastic terms in governing equations for solid (Eqs. (11b) and (11c)) are comparable to each other. Also, Eq. (C.2a) shows that the leading order perturbation velocity in liquid layer admits a plug ﬂow proﬁle which is possible only when tangential motion is allowed at the boundaries. This is true for the boundary at z = 0 which is a gas–liquid interface. The other boundary at z = 1 is liquid–solid interface and tangential velocity is allowed only if the LS interface is deformable. Thus, the above discussion shows that these modes will exist in presence of inertia and with deformable LS interface. Therefore, these modes are referred as inertial liquid–solid (ILS) modes in the present work.

Fig. C.12. Comparison of asymptotic solution with eigenspectrum using spectral code for ﬁrst, and second upstream (IL S-1u, IL S-2u) and downstream (ILS1d, IL S-2d) L S modes. Symbols circle, square, diamond, upward triangle, downward triangle, and right-side triangle represent eigenvalues obtained from spectral code while “+” and “×” data points obtained from asymptotic solution: data for Re = 0.001, k = 0.001, H = 10, G = 1, gl = 1, ls = 0, θ = 45◦ .

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Since, c(0) is real, it is necessary to evaluate the ﬁrst correction to wave speed which is determined by setting the ﬁrst correction to determinant equal to zero, i.e. f (1 ) (c (0 ) , c (1 ) ) = 0. The calculations for the asymptotic analysis are performed using the symbolic software package MATHEMATICA. Recall that there exist multiple solutions to c(0) which could correspond to downstream (positive c(0) ) or upstream traveling (negative c(0) ) modes. For each of these values of leading order wave speed (c(0) ), there exists a unique solution for c(1) which is purely imaginary, and hence, determines the stability of the system. The value of c(1) from our asymptotic analysis demonstrates that for positive τ s and a given G, the increase in magnitude of imposed shear stress has a destabilizing effect on upstream traveling modes and stabilizing effect on downstream modes. 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Effect of wall deformability on the stability of shear-imposed film flow past an inclined plane

Effect of wall deformability on the stability of shear-imposed film flow past an inclined plane

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