Effect of imposed shear on the dynamics of a contaminated two-layer film flow down a slippery incline
Muhammad Sani, Siluvai Antony Selvan, Sukhendu Ghosh, Harekrushna Behera
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p E FFEC T OF IMPOSED SHEAR ON THE DYNAMICS OF AC ONTAMINATED TWO - LAYER FILM FLOW DOWN A SLIPPERYINC LINE
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Muhammad Sani
Department of MathematicsSRM Institute of Science and TechnologyKattankulathur-603203, India [email protected]
Siluvai Antony Selvan
Department of MathematicsSRM Institute of Science and TechnologyKattankulathur-603203, India [email protected]
Sukhendu Ghosh
Department of MathematicsIndian Institute of TechnologyJodhpur, Rajasthan-342037, India [email protected]
Harekrushna Behera ∗ Department of MathematicsSRM Institute of Science and TechnologyKattankulathur-603203, India [email protected]
September 18, 2020 A BSTRACT
The linear instability of a surfactant-laden two-layer falling film over an inclined slippery wall isanalyzed under an influence of external shear which is imposed on the top surface of the flow. Thefree surface of the flow as well as the interface among the fluids are contaminated by insolublesurfactants. Dynamics of both the layers are governed by the Navier–Stokes equations, and the sur-factant transport equation regulates the motion of the insoluble surfactants at the interface and freesurface. Instability mechanisms are compared by imposing the external shear along and opposite tothe flow direction. A coupled Orr–Sommerfeld system of equations for the considered problem isderived using the perturbation technique and normal mode analysis. The eigenmodes correspond-ing to the Orr–Sommerfeld eigenvalue problem are obtained by employing the spectral collocationmethod. The numerical results imply that the stronger external shear destabilizes the interface modeinstability. However, a stabilizing impact of the external shear on the surface mode is noticed if theshear is imposed in the flow direction, which is in contrast to the role of imposed external shear onthe surface mode for a surfactant laden single layer falling film. Further, in the presence of strongimposed shear, the overall stabilization of the surface mode by wall velocity slip for the stratifiedtwo-fluid flow is also contrary to that of the single fluid case. The interface mode behaves differentlyin the two zones at the moderate Reynolds numbers and higher external shear magnifies the inter-facial instability in both the zones. An opposite trend is observed in the case of surface instability.Moreover, the impression of shear mode on the primary instability is analyzed in the high Reynoldsnumber regime with sufficiently low inclination angle. Under such configuration, dominance of theshear mode over the surface mode is observed due to the weaker impact of the gravitation force onthe surface instability. The shear mode can also be stabilized by applying the external shear in thecounter direction of the streamwise flow. Conclusively, the extra imposed shear on the stratifiedtwo-layer falling film plays an active role to control the attitude of the instabilities. K eywords Falling film · External shear · Multi-layer flow · Insoluble surfactant · Velocity slip · Orr–Sommerfeldequation ∗ Corresponding author
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The applications of falling film are often encountered in the coating process, food manufacturing, condenser andbiomedical engineering ([1], [2], [3], [4] and [5]). Several analysis were carried out to understand the hydrodynamicinstability of homogeneous falling film ([6], [7] and [8]). Initially, the low Reynolds number falling film was analyzedat the small inclination angle using the perturbation technique by [9]. It is observed that the small perturbations areamplified near the long-wave region while the disturbances are highly damped towards the short-wave region. Later,the asymptotic solution of falling film in both small and larger Reynolds number regions were developed by [10].Further, the analysis was carried out at a high Reynolds number regime to show the impact of shear and surface modeson the primary instability of the falling film ([6]). They observed that the surface mode stabilizes the fluid flow whilethe opposite trends are found in the case of shear mode.Similarly there are some applications like liquid-liquid extraction, bolus dispersion process, earth’s inner topographyand atmosphere, where the physical problems can be modeled as the two-layer/multi-layer flows ([11], [12], [13], [14]and [15]). In the two-layer/multi-layer fluid system, the overall instabilities can be controlled by the interface betweenthe fluids. The detailed analysis of interface-dominated viscosity and density stratified system for the free surface andthe Couette-Poiseuille flow were analyzed by [16] and [17], respectively. In the case of inertialess free surface flowwith an uniform densities, the presence of high viscous fluid in the upper layer destabilizes the falling film ([18]). Thisfurther results in the formation of wave at the surface and interface owing to the fluid-fluid and fluid-air interactions([19]). For the plane Poiseuille flow, the presence of viscosity jump at the interface, and interaction between the fluidand end walls stabilize the shear mode due to the lubrication effect ([20]). Such viscosity stratification can be usedin transporting the high viscous fluid through pipes, in which the less viscous fluid reduces the frictional losses at thesurface ([21]). This theory of linear stability analysis was later applied in the co-extrusion process in the polymerprocessing industry and validated with the experimental results by studying extensively the impact of density ratio,viscosity ratio, surface tension, and total flow rate on the instabilities of two-layer flows ([22]). Further, [14] studiedthe influence of viscosity stratification and miscibility on the instability of free falling film. A detailed overview onthe instabilities of miscible viscosity stratified flows is provided by [15].The primary instabilities play a crucial role in determining the qualities of many applications either by upgradingor degrading the end products ([23]). Farther, numerous passive mechanisms like influence of wall velocity slip([24, 25, 26, 27, 28]) and insoluble surfactant ([29, 30]) can be employed for controlling these instabilities in differentflow systems. In particular, the slippery wall is often realized physically as an interface between the fluid and porouslayer, which has profound environmental applications like the passage of water flow inside the crack and water-soilsystem ([31]). It was reported that the Darcian velocity in the porous layer rapidly changes to the interfacial velocity.[24] studied the effect of wall slip on the linear stability of the homogeneous falling film and observed that the surfaceof film destabilizes while increasing the permeability of an inclined plane. Later, [25] investigated the homogeneousfalling film down the porous inclined plane using the Benny’s asymptotic expansion method, weakly-nonlinear andnonlinear analysis. Further, the depth-averaged model for the falling film down the slippery plane was developed by[26] and observed that the presence of slip amplifies the nonlinear travelling waves. Moreover, [27] and [28] haveshown the control of velocity slip on the instabilities due to continuous viscosity stratification.In both the homogeneous and heterogeneous fluids, the dynamics of surface and interfacial tension can be understoodby employing the insoluble surfactant. There are many studies in the literature analyzing an effect of insoluble surfac-tant on the homogeneous/heterogeneous falling films ([32, 33, 29, 1, 34] and [35]). On adding the insoluble surfactanton a falling film, the surface tension is altered by surface concentration and it gives rise to the extra Marangoni mode([29]). It was observed that the existence of Marangoni mode suppresses the surface instabilities of the falling film.[30] investigated the marangoni destabilization of shear imposed surfactant laden falling film due to the shear mode atthe small Reynolds number. A similar problem was also investigated by [36] to understand the influence of imposedshear on a free falling film. Other than this, the effect of insoluble surfactant on the falling film down the porousinclined plane is investigated by [1]. Besides, the presence of insoluble surfactant on the surface and interface of thetwo layer falling film is explored by [23]. Recently, the work of [30] is extended to an arbitrary wavenumber region by[37] and they observed that the surface mode stabilizes on applying the external shear opposite to the flow direction.Further, the effect of surfactant on the two-layer falling film down the inclined slipper plane has been investigated by[35].In the aforementioned studies, the passive mechanisms like effects of insoluble surfactant and wall-slip parameter areused to control the instabilities at surface as well as interface on the single/two-layer falling films. However, additionalinstability exists, and occurring due to the presence of insoluble surfactant at the top surface and interface for the two-layer falling film down a slippery wall. Hence, there is a need of some active mechanism for controlling unwantedinstabilities in such flows, which can be made possible by imposing the external shear stress in a constant rate on thetop surface of the flow. In the present study, the effects of imposed shear and wall velocity slip (hydrophobic kind2
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18, 2020wall) are analyzed on controlling the surface and interfacial instabilities in the presence of insoluble surfactant. TheOrr–Sommerfeld boundary-value problem for the proposed physical flow system is obtained using the perturbationtechnique and normal-mode analysis. The resultant eigenvalues are obtained by solving the system of equations usingthe spectral collocation method. The numerical analysis are widely carried out on the different flow parameters regimesfor varying external shear rate to understand overall control mechanism of imposed shear on multiphase falling film.The article is organized as follows: the mathematical formulation along with the base solution are discussed in Section2, the numerical results are discussed in Section 3 and the final outcomes of problem are highlighted in Section 4.
A two-dimensional flow of two-layer immiscible and incompressible Newtonian fluids down an inclined slippery planeof angle θ is considered as in Fig. 1. The upper layer of thickness H is known as fluid-1 having the density ρ andthe viscosity µ , whereas the lower layer of thickness H is denoted as fluid-2 having the density ρ and the viscosity µ . Further, the surface and interfacial elevations of each layers are represented as ζ (1) and ζ (2) , respectively. Theproblem is studied in the Cartesian coordinate system with an origin located at the flat interface, where the x and y -axes indicate the direction of streamwise and cross-streamwise flow for both the fluids. The motion of fluid-1 and Insoluble surfactant q , mr , mr Hy = = y Hy -= x y Figure 1: Schematic diagram for a two-layer thin film flow down in an inclined plane in the presence of insolublesurfactants.fluid-2 is governed by the two-dimensional Navier–Stokes equations, u ( j ) x + v ( j ) y = 0 , (1a) ρ j (cid:2) u ( j ) t + u ( j ) u ( j ) x + v ( j ) u ( j ) y (cid:3) = − p ( j ) x + µ j ( u ( j ) xx + u ( j ) yy ) + ρ j g sin θ, (1b) ρ j (cid:2) v ( j ) t + u ( j ) v ( j ) x + v ( j ) v ( j ) y (cid:3) = − p ( j ) y + µ j ( v ( j ) xx + v ( j ) yy ) − ρ j g cos θ, (1c)where j = 1 and j = 2 correspond the fluid-1 and fluid-2, respectively; u ( j ) and v ( j ) are known as the velocityof j th layer fluid in streamwise and cross-streamwise directions; further, the pressure of j th layer fluid and gravita-tional acceleration are represented as p ( j ) and g , respectively. In both the top surface and interface, the monolayeredinsoluble surfactant of concentration Γ ( j ) ( x, t ) diffuses into the each fluid resulting in the variations of local surface σ (1) and interfacial tension σ (2) of the fluid system. Thus, the relation between the surfactant concentration and thesurface/interfacial tensions is expressed as σ ( j ) = σ ( j )0 − E j (Γ ( j ) − Γ ( j )0 ) , (2)3 PREPRINT - S
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18, 2020where E j , σ ( j )0 and Γ ( j )0 are the surface elasticity, reference tension and reference surfactant concentration, respectively.Moreover, j = 1 and j = 2 are respectively stand for the free surface and interface. For a static two-layer fluid, thekinematic conditions at the free surface and interface are expressed as v ( j ) = ζ ( j ) t + u ( j ) ζ ( j ) x at y = ζ ( j ) ( x, t ) . (3)Further, the dynamic boundary conditions corresponding to the tangential and normal shear stress balance at both thefree surface and interface are given as µ (cid:2) − u (1) x ζ (1) x + ( u (1) y + v (1) x ) (1 − ( ζ (1) x ) ) (cid:3) = (cid:2) σ (1) x ± τ (cid:3) q ζ (1) x ) at y = ζ (1) ( x, t ) , (4) p (1) = µ (cid:2) (cid:0) ζ (1) x (cid:1) (cid:3) n u (1) x ( ζ (1) x ) − ( u (1) y + v (1) x ) ζ (1) x + v (1) y o − σ ζ (1) xx (cid:2) ζ (1) x ) (cid:3) / at y = ζ (1) ( x, t ) , (5) µ (cid:2) − u (2) x ζ (2) x + ( u (2) y + v (2) x )(1 − ( ζ (2) x ) ) (cid:3) = µ (cid:2) − u (1) x ζ (2) x + ( u (1) y + v (1) x )(1 − ( ζ (2) x ) ) (cid:3) at y = ζ (2) ( x, t ) , (6) p (1) + µ (cid:2) ζ (1) x ) (cid:3) (cid:2) u (1) x (cid:0) − ( ζ (2) x ) (cid:1) + ( u (1) y + v (1) x ) ζ (2) x (cid:3) + σ (2) ζ (2) xx h (cid:2) ζ (2) x ) (cid:3) / = p (2) + µ (cid:2) ζ (2) x ) (cid:3)(cid:2) u (2) x (cid:0) − ( ζ (2) x ) (cid:1) + ( u (2) y + v (2) x ) ζ (2) x (cid:3) at y = ζ (2) ( x, t ) , (7)where τ represents the external shear applied to the two-layer fluid. Further, the external shear applied along thedirection of flow is denoted by + τ , whereas − τ denotes the external shear applied opposite to the flow direction.At the interface, the horizontal and vertical velocity components of both the layers are continuous and result in thefollowing boundary condition, u (1) = u (2) and v (1) = v (2) at y = ζ (2) ( x, t ) . (8)Along the bottom of the two-layer flow, there is a velocity slip and thus the Navier–slip condition with no penetrationof fluid resulting in, u (2) = βu (2) y and v (2) = 0 at y = − H , (9)where β is the slip parameter. The transport equation governing the evolution of surfactant concentration at both thesurface and interface are described as Γ ( j ) t + [( u ( j ) . t )Γ ( j ) ] x + Γ ( j ) κ ( u ( j ) . n ) = D j Γ ( j ) xx , (10)where u ( j ) . t and u ( j ) . n denote the velocities of the j th layer fluid along the streamwise and cross-streamwise directions,respectively, and D , D are the surfactant diffusivity at the top surface and the interface, respectively. The governing equations and its associated boundary conditions of a steady state two-layer fluid flow are obtainedby substituting ( u ( j ) , v ( j ) ) = ( U ( j ) ( y ) , and p ( j ) = P ( j ) ( y ) in the above equations (1)–(9). By applying the localparallel flow assumptions, the solutions of steady state flow are given by U (1) ( y ) = ρ g sin θH µ (cid:26) (1 ± ¯ τ ) yH − y H + δ ( rδ +2)2 m + βH ( δr +1) m ± ¯ τm (cid:18) βH + δ (cid:19)(cid:27) , (11) U (2) ( y ) = ρ g sin θH mµ (cid:26) (1 ± ¯ τ ) yH − ry H + δ ( rδ +2)2 + βH ( δr + 1) ± ¯ τ (cid:18) βH + δ (cid:19)(cid:27) , (12) P (1) ( y ) = ρ g cos θH (cid:18) − yH (cid:19) and P (2) ( y ) = ρ g cos θH (cid:18) − ryH (cid:19) , (13)where ¯ τ = τ /ρ H g sin θ is the non-dimensional form of the imposed shear stress. Further, the density ratio, depthratio and viscosity ratio are expressed as r = ρ /ρ , δ = H /H and m = µ /µ , respectively. The characteristicsvelocity of two-layer film flow down the inclined slippery bottom in the presence of external shear stress is obtainedby integrating the above base velocities over the film thicknesses of an each layer and averaging it with respect to thesum of thicknesses. This results in the following expression for the characteristic velocity U c = ρ g sin θH K µ , (14)4 PREPRINT - S
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18, 2020where K = 1 δ + 1 (cid:20)
13 + δ m + δm + δ r m + δ r m + ¯ τ (cid:18) − δ m (cid:19)(cid:21) + βH (cid:18) δrm + 1 m (cid:19) + ¯ τm (cid:18) βH + δ (cid:19) . It is worth to note that the above expression recovers the characteristics velocity of two-layer falling film in the absenceof slip bottom and external shear with no density stratification ([16]) by substituting τ = 0 , β = 0 and r = 1 . Inthe case of single layer fluid, the characteristics velocity of film flow having rigid bottom and no external shear ([9])is obtained by substituting r = 1 , β = 0 , m = 1 , δ = 0 and τ = 0 . Fig. 2 exhibits the non-dimensional base U/U c y / H τ = −0.5 τ = −0.3 τ = 0 τ = 0.3 τ = 0.5 (a) r = 0 . U/U c y / H τ = −0.5 τ = −0.3 τ = 0 (Bhat et al.(2020)) τ = 0 (Present theory) τ = 0.3 τ = 0.5 (b) r = 1 U/U c y / H τ = −0.5 τ = −0.3 τ = 0 τ = 0.3 τ = 0.5 (c) r = 5 Figure 2: Variation of non-dimensional velocity of base state flow (
U/U c ) with respect to the non-dimensional depth( y/H ) for different values of non-dimensional external shear ( τ ) showing the effect of density ratio r . The otherconstant flow parameters are m = 2 . , δ = 1 and β = 0 . .state velocity profile as a function of non-dimensional depth for difference values of dimensionless imposed shearstress. The velocity profile in the upper layer advances on increasing the external shear stress from negative to positivevalues. The positive and negative τ , restrictively, suggest the co and opposite direction of imposed shear with the flowdirection. The present result retrieve the result of [35] in the absence of external shear stress for m = 2 . , r = 1 and β = 0 . as in Fig. 2(b). When the shear stress is applied against the flow direction (i.e. for negative values of τ ), thenet force ( ρ g sin θ − τ H ) acting on the upper layer decreases and this weakens the base velocity along the surfacein the upper layer fluid. Contrarily, for the shear stress along the flow direction (i.e. for positive values of τ ), the netforce ( ρ g sin θ + τ H ) acting on the upper layer increases and enhances the base velocity of the upper layer fluid. Oncomparing the figures 2(a), (b) and (c), it is observed that the net force acting on the lower layer increases as comparedto the upper layer as a result of increase in the value of r .In this study, the reference length, velocity and time scales are chosen as H , U c and H /U c , respectively. Further,the reference scales for pressure, surfactant concentration and local surface tension for j th layer fluid are assumedas ρ j U c , Γ ( j )0 and σ ( j )0 , respectively. Using these scales, all the governing equations and its associated boundaryconditions (Eqs. (1)–(10)) are non-dimensionalized. 5 PREPRINT - S
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The steady state flow is perturbed with infinitesimal disturbances such that the each dynamical variables are expressedas the sum of base state and perturbed state. These renewed variables are substituted in the dimensionless governingequations and boundary conditions, and linearized by removing the nonlinear terms with respect to the perturbations.The detailed procedure for obtaining the perturbed system of equations corresponding to a two-layer falling film isfound in [1] and [35]. Further, the perturbed variables are represented in the form of normal mode solution, whichis given by ˆ f ( x, y, t ) = f ( y ) e i k ( x − ct ) where ˆ f being the arbitrary perturbed function with amplitude f ( y ) . Here, k and c are the wavenumber and complex phase speed, respectively. Expressing the linearized governing equation andits associated boundary condition in the form of normal mode amplitudes yield the Orr–Sommerfeld (OS) system ofequation for the present physical model, which is given as follows: ( d yyyy − k d yy + k ) φ ( j ) − i kRe j (cid:2) ( U ( j ) − c ) ( d y − k ) φ ( j ) − d yy U ( j ) φ ( j ) (cid:3) = 0 , (15a) φ ( j ) + ( U ( j ) − c ) η ( j ) = 0 at y = 1 for j = 1 and y = 0 for j = 2 , (15b) d y φ ( j ) + d y U ( j ) η j + (cid:18) U ( j ) − c − i kP e j (cid:19) γ ( j ) = 0 at y = 1 for j = 1 and y = 0 for j = 2 , (15c) ( d yy + k ) φ (1) + d yy U (1) η + i kMa Ca γ (1) = 0 at y = 1 , (15d) d yyy φ (1) − k d y φ (1) − i kRe (cid:2) ( U (1) − c ) d y φ (1) − d y U (1) φ (cid:3) + i k (cid:20) i kU (1) − K cot θ − k Ca (cid:21) = 0 at y = 1 , (15e) d y φ (1) − d y φ (2) + ( m − d y U (2) η (2) = 0 at y = 0 , (15f) φ (1) − φ (2) = 0 at y = 0 , (15g) (cid:2) md yy U (2) − d yy U (1) (cid:3) η (2) + m ( d yy + k ) φ (2) − ( d yy + k ) φ (1) + i kmMa Ca γ (2) = 0 at y = 0 , (15h) d yyy φ (1) − k d y φ (1) − i kRe (cid:2) ( U (1) − c ) d y φ (1) − d y U (1) φ (cid:3) − m (cid:26) d yyy φ (2) − k d y φ (2) − i kRe (cid:2) ( U (2) − c ) d y φ (2) − d y U (2) φ (cid:3)(cid:27) + i k (cid:20) K ( r −
1) cot θ + mk Ca (cid:21) η (2) = 0 at y = 0 , (15i) d y φ (2) − βd yy φ (2) = 0 at y = − δ (15j) φ (2) = 0 at y = − δ, (15k)where M a j = E j Γ ( j ) /σ ( j )0 , Ca j = U c µ j /σ ( j )0 , P e j = U c H / D j , Re j = ρ j U c H /ν j are the Marangoni number,capillary number, Peclet number and Reynolds number of the j th layer fluid, respectively. Note that the two Reynoldsnumbers are co-related by Re = ( r/m ) Re . Further, d y represents the derivative with respect to y , and φ ( j ) , η ( j ) and γ ( j ) denote the amplitudes of stream function, free surface/interface elevation and surfactant concentrations, respec-tively. There is no closed-form analytical solution for the above system of equations (15)(a)–(f) while solving for thearbitrary wavenumber and Reynolds number. Hence, the above system is solved numerically to obtain the eigenvalue c corresponding to the OS system for the proposed physical problem. In this subsection, the numerical methods used for obtaining the eigenvalues for the derived OS boundary valueproblem associated to the physical configuration is explained in detail. To carry out the linear stability analysis forarbitrary wavenumber and Reynolds number, the spectral collocation method ([38] and [1]) based on the Chebyshevpolynomial is employed. The perturbation amplitudes of stream function in both the layers are written as the N -truncated series of Chebyshev polynomials, which is given as φ ( j ) = N X n =0 φ ( j ) n T n ( y ) for j = 1 , , (16)provided T n = cos( n cos − ( y )) with n = 0 , , , . . . N , where T n is the Chebyshev polynomial of first kind andChebyshev points are used to discretize each layer over the domain − ≤ y ≤ . Using the series expansion Eq. (16)6 PREPRINT - S
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18, 2020in the OS system of equations (15)(a)–(f) yields the matrix eigenvalue problem, which can be written in the generalizedform as AX = c BX , (17)where the A and B are the square block matrices. Further, c is the eigenvalue and its associated eigenvector is given by X = ( η (1) , η (2) , γ (1) , γ (2) , φ (1)1 , φ (1)2 , . . . , φ (1) N , φ (2)1 , φ (2)2 , . . . , φ (2) N ) . The eigenvalues ( c = c r + ic i ) are the complexphase speed for different perturbation waves, however, for studying the instability mechanism of the given physicalproblem, the primary dominant mode of disturbance is considered. In this Section the numerical results are discussed for a wide range of flow parameters. A Matlab 2014a subroutine isdeveloped for solving the above eigenvalue problem (Eq. (17)) for obtaining the eigenmodes for various combinationsof flow parameters. In general, in the case of two-layer flow in the presence of insoluble surfactant at the free surfaceand interface, there exist the two surfactant modes in addition to the unstable surface mode and interface mode ([11, 23]and [35]). Results of the current study aims to characterize the control mechanism of the imposed external shear onthese unstable modes. c r c i IM ISM SSM SM τ = −0.2 τ = 0 τ = 0.2 (a) c r c i IM ISM SSM SM τ = −0.2 τ = 0 τ = 0.2 (b) c r c i SHM IM ISMSSMSM τ = −0.2 τ = 0 τ = 0.2 (c) Figure 3: Distribution of eigenvalues showing the impact of external shear stress ( τ ) on the destabilized (a) surfacemode ( k = 0 . , θ = 0 . rad and Re = 5 ), (b) interface mode ( k = 1 . , θ = 0 . rad and Re = 25 ) and (c) shearmode ( k = 0 . , θ = 0 . ′ and Re = 8000 ). The other flow parameters are m = 0 . , r = 1 . , P e = P e = 10000 , Ca = Ca = 1 and β = 0 . .In Figs. 3(a), (b) and (c), the effect of external shear on the destabilized surface mode, interface mode and shear mode isanalyzed, respectively. Moreover, the modes corresponding to the free surface, interface, surface surfactant, interfacesurfactant and shear are denoted by SM, IM, SSM, ISM and SHM, respectively. It is evident from the Fig. 3 that the7 PREPRINT - S
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18, 2020eigenmodes satisfy the following condition based on the phase speed such as c r | SM < c r | SSM < c r | ISM < c r | IM
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18, 2020 −3 k ω i τ = −0.2 τ = −0.1 τ = 0 τ =0.1 (a) mr < −4 k ω i τ =−0.3 τ = −0.2 τ = 0 τ =0.1 (b) mr > Figure 5: Scaled growth rate ( ω i ) as a function of wavenumber (k) showing the effect of external shear for (a) mr < ( r = 1 . , Re = 25 ) and (b) mr > ( r = 2 . , Re = 100 ). The other flow parameters are m = 0 . , P e = P e =10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 and M a = M a = 0 . . Re k US β = 0 β = 0.04 β = 0.06 (a) mr < Re k US β = 0 β = 0.04 β = 0.06 (b) mr > Figure 6: Marginal stability curve corresponding to varying slip parameter for (a) mr < with r = 1 . and (b) mr > with r = 2 . . The other constant flow parameters are m = 0 . , P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 and M a = M a = 0 . .shows positive growth rate in the long-wave region. This occurs due to the presence of high dense fluid at lower layerwith mr > , which exposes to suppress the interfacial instability for larger wavenumber range. With increase in thevalues of external shear in Fig. 5(a), the instability enhances as a consequence of increasing net force acting on theupper layer. When the external shear is applied along the flow direction (i.e. for positive values of τ ), the increase in anet driving force near the interface is assisted by the external shear. While applying the external shear opposite to theflow direction (i.e. for negative values of τ ), the net driving force acting at the interface decreases. Thus, increasingthe external shear value from negative to positive increases the interfacial instability and the same observation is madein the case of region mr > .It is evident from the Figs. 4 and 5 that the positive value of τ has destabilizing effect on the interfacial flow, whereasthe negative value shows the stabilizing effect on the interfacial mode. In Figs. 6(a) and (b), the effect of slip parameteron the marginal stability curve is plotted in the ( Re , k ) plane for mr < and mr > , respectively. The patterns ofneutral curve is same as observed in the Fig. 4. However, the presence of wall slip stabilizes the interfacial instability.On increasing the slip length, the velocity of the lower layer increases near the vivinity of the bottom wall. It resultsin a higher wall shear rate and redaction in frictional forces. Moreover, higher wall shear rate is well balanced by theinterfacial shear and this scenario is more significant for m < .The neutral stability curves show the effect of imposed shear for mr > provided that both m > and r > inFig. 7(a). On comparing with m < as in Fig. 4(b), it is noticed that, in addition to the unstable interfacial mode9 PREPRINT - S
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18, 2020 Re k U US τ = −0.2 τ = 0 τ = 0.1 (a) mr > k ω i τ = −0.2 τ = −0.1 τ = 0 τ = 0.10 0.2 0.40123 x 10 −3 k ω i (b) mr > Figure 7: Marginal stability curve corresponding to varying external shear for (a) mr > with r = 1 . and (b) itscorresponding scaled growth rate ( ω i ) as a function of wavenumber ( k ) with Re = 60 . The other constant flowparameters are m = 2 . , P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 , m = 2 . and M a = M a = 0 . . −3 −2 −1 0 1 2 3 40.40.50.60.70.80.91 k m mr=1 US τ = −0.2 τ = −0.1 τ = 0 τ =0.1 (a) mr < −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80.40.50.60.70.80.91 k m mr=1 US τ = −0.2 τ = −0.1 τ = 0 τ =0.1 (b) mr > Figure 8: Marginal stability curve of the interface mode in ( m, k ) plane for (a) mr < ( r = 1 . , Re = 25 ) and (b) mr > ( r = 2 . , Re = 100 ). The other flow parameters are P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 and M a = M a = 0 . .bandwidth at the long-wave region, there exist a weaker unstable bandwidth at shorter-wave region. Also, the externalshear plays dual role depending on the bandwidth of the unsatble waves. The critical Re for the instability increaseswith higher value of τ in the long-wave region, whereas a reverse pattern is observed in the shorter-wave region.Further, the variation of unstable mode bandwidth is opposite in the case of m > (Fig. 7(a)) as compared to that of m < (Fig. 4(b)). In Fig. 7(b), the growth rate corresponding to the dominant unstable mode is plotted as a functionof k for different values of τ in the case of m > . The growth rate decreases for increasing value of τ in the long-waveregion, however it follows reverse pattern for increasing value of external shear in the short-wave region. It is noticedthat if the external shear applied opposite to the flow direction in the free surface surface, triggers more long-waveinstability at the interface. On the other hand, the external shear increases the short-wave instability when it is appliedalong the direction of fluid flow.The marginal stability curve in the ( m, k ) plane is plotted for the configurations, mr < and mr > in Figs. 8(a)and (b), respectively when a more viscous fluid is flowing on a less viscous ( m < ). On the dashed line in both thefigures, m and r satisfies mr = 1 for the given value of the density ratio r . This line forms the critical bound forthe interfacial instability, namely the upper and lower bounds for the marginal stability region in Figs. 8(a) and (b),respectively. For mr < , the region of unstable mode bandwidth is more dominant near the long-wave region andless significant near the short-wave region. It is noticed that the present result validates the previous observation for10 PREPRINT - S
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18, 2020 −1 −0.5 0 0.5 10.511.522.533.544.5 k m mr=1 US τ = −0.2 τ = −0.1 τ = 0 τ =0.1 (a) Long-wave region −2 −1.5 −1 −0.5 0 0.5 1 1.5 20.511.522.533.544.55 k m mr=1 UU S τ = −0.2 τ = −0.1 τ = 0 τ =0.1 (b) Short-wave region Figure 9: Marginal stability curve of the interface mode in ( m, k ) plane for mr > with m > , r = 1 . , Re = 60 , P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 and M a = M a = 0 . . −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.8−0.6−0.4−0.200.20.40.60.81 φ r y mr<1, m<1mr>1, m<1mr>1, m>1 (a) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−1−0.8−0.6−0.4−0.200.20.40.60.81 φ i y mr<1, m<1mr>1, m<1mr>1, m>1 (b) Figure 10: Eigenfunction ( φ ) corresponding to the interface mode for various unstable zones with P e = P e =10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 and M a = M a = 0 . . m = 0 . as in Fig. 8(a), where there exist only short-wave instability. With increase in the values of τ , the unstablemode bandwidth increases. In the case of mr > from Fig. 8(b), the unstable mode bandwidth increases for increasein the values of m . Further, it is observed that the unstable mode bandwidth increases with stronger external shear τ .In both the figures, it is perceived that the critical condition of interfacial instability attained only one time for k = 0 .In Figs. 9(a) and (b), the bandwidth of unstable modes in both long- and short-wave regions are plotted respectivelyfor mr > provided that m > . In the case of long-wave region, the neutral conditions are attained two-times fora given Reynolds number. This depicts that there exist a sub-critical instability at long wave region while consideringthe high viscous fluid in the lower layer. With increase in the value of τ , the unstable mode bandwidth decreases whichis validated with the previous observation in Fig. 7(b). For the short-wave region as in Figs. 9(b), the unstable modebandwidth increases for increase in the external shear. Further, there is no occurrence of neutral condition in the caseof short-wave region. The observation are similar to the scales growth rate result for a fixed value of m greater thanunity (Fig. 7(b)).In Figs. 10(a) and (b), the real and imaginary part of eigenfunction, respectively, corresponding to the interface modeare plotted for three different unstable zones namely, (i) zone 1 ( mr < , m < ), (ii) zone 2 ( mr > , m < )and (iii) zone 3 ( mr > , m > ). The interfacial mode shape undergoes different variation in the three unstablezones under the influence of external shear. At the interface from Fig. 10(a), the velocity of fluid under zone 3 is moreas compared to other two zones resulting in the higher value of φ r for the zone 3. However, in the case of φ i as inFig. 10(b), the φ i corresponding to the zone 3 has the lowest value as compared to other zones at interface.11 PREPRINT - S
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In this subsection, the influence of external shear on the surface mode instability is investigated for different set offlow parameters. In Fig. 11(a), the marginal stability curves are plotted in ( Re , k ) plane for various imposed shearrate. It is noticed that the present theory is in well-agreement with the existing result of [35] in the case of two-layerfilm flow down the slippery plane in the absence of imposed external shear at the free surface (for τ = 0 ). Further,our results confirms that the external shear applied in the direction of film flow reduces the bandwidth of the unstablesurface mode bandwidth and on applying the external shear in the opposite direction to the film flow strengthens thebandwidth of surface mode instability. Thus, in the case of considered two-layer flow, the influence of imposed shearon the surface mode is exactly opposite to that for the shear imposed surfactant laden single layer falling film ([37]),where the surface unstable mode bandwidth increases/decreases for positive/negative values of external shear rate (seeFig. 11(b). It is really interesting to see that presence of two different fluid layer with density/viscosity variationcompletely changing the stability mechanism of imposed shear. Note that, on applying the external shear in the flowdirection to the stratified flow system, the velocity of the upper layer fluid increases near the surface and consequently,there is a decrease in the velocity of lower layer as well as interfacial velocity. However, the change in interfacialshear overcomes the shear rate at free surface region, which cases an hindrance to the inertia force on the convectiveflow and yields a stabilizing effect on the surface mode. By contrast, if the external shear stress acts opposite to thestreamwise flow direction, a strong back-flow phenomena occurs at the free surface region, which enhances the surfacemode instability. Re k US τ = −0.2 τ = 0 [Bhat and Samanta (2020)] τ =0 [Present theory] τ = 0.2 (a) Re k US τ = −0.2 τ = 0 τ = 0.2 (b) Figure 11: (a) Neutral stability curve corresponding to varying external shear ( τ ) when the fixed flow parametersare P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = r = 1 , m = 0 . , r = 1 . and M a = M a = 0 . , and (b) Stability boundaries for limiting single fluid falling film with imposed shear ([37]).In Fig. 12(a), the effect of slip length on the marginal stability curve is illustrated in the ( Re , k ) plane in the absenceof external shear. The neutral curve behaves differently for very smaller values of k as shown in inset plot of Fig. 12(a)as compared to the larger values of k . A dual role of wall velocity is observed similar to the case of single fluid fallingfilm. For the onset of instability k ∈ [0 , . , there is a mild destabilization by the reduction in the critical Reynoldsnumber with the increasing value of slip parameter, whereas the surface mode bandwidth decreases with larger β forshorter-waves. The wall shear rate changes with the variation of the slip length, which in turn affects the interfacialand surface velocity of the base flow. Further, when inertial effect is strong enough the surface instability stabilizesgradually for increase in the value of β . In order to validate the destabilizing effect of slip parameter on the long wavesas seen in Fig. 12(a), the corresponding growth rate is plotted for the region < k < . in Fig. 12(b) and it isobserved that the growth rate of surface instability accelerates for larger values of slip length.The role of wall slip in the presence of imposed shear is captured in Fig. 13. The neutral stability curves in the ( Re , k ) plane and its corresponding growth rate as a function of k are analyzed for the same flow parameters as consideredin Fig. 12 after including a strong external shear effect ( τ = − ). Here, the external shear is applied opposite to thedirection of two-layer film flow (i.e. for negative values of τ ). It is observed from Fig. 13(a) that the unstable modebandwidth decreases for increase in the value of β for all values of Re . There is no longer any destabilizing effectof the wall velocity slip close to the criticality (in particular for k → ). It fact ensures that slip parameter suppliesonly stabilizing effect on the surface mode instability, when external shear is included. This is possibly because ofthe energy balancing between the extra imposed shear at the free surface to the interfacial tension. The corresponding12 PREPRINT - S
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18, 2020 Re k US β = 0 β = 0.04 β = 0.1 β = 0.141.1 1.2 1.3510 x 10 −3 Re k (a) −4 k ω i β = 0 β = 0.04 β = 0.1 β = 0.14 (b) Figure 12: (a) Marginal stability curve corresponding to varying slip length ( β ) and (b) its corresponding scaled growthrate ( ω i ) as a function of wavenumber ( k ) with Re = 1 . . The other constant flow parameters are P e = P e =10000 , Ca = Ca = 1 , τ = 0 , θ = 0 . rad, δ = r = 1 , m = 0 . , r = 1 . and M a = M a = 0 . . Re k US β = 0 β = 0.04 β = 0.1 β = 0.140.6 0.7 0.8510 x 10 −3 Re k (a) k ω i β = 0 β = 0.04 β = 0.1 β = 0.14 (b) Figure 13: (a) Neutral stability curves corresponding to varying slip length ( β ) and (b) its corresponding scaled growthrate ( ω i ) as a function of wavenumber ( k ) with Re = 5 . The other constant flow parameters are P e = P e = 10000 , Ca = Ca = 1 , τ = − , θ = 0 . rad, δ = r = 1 , m = 0 . , r = 1 . and M a = M a = 0 . .growth rate as a function of wavenumber for different values of β is plotted in Fig. 13(b). It is noticed that the temporalgrowth rate of the surface instability decreases for stronger velocity slip which advances the wall shear rate.The Marginal stability curves are drown in Fig. 14(a) and (b), respectively, in the (1 /m, k ) and (1 /r, k ) plane forexploring the substantial effect of external shear on surface mode. It is observed that, the unstable mode bandwidth isprevalent near the small wavenumber range, which gradually increases for lower values of /m . When the viscosityin the upper layer fluid is much smaller as compared to the lower layer fluid, the long-wave instabilities are moresignificant. Further, with increase in the viscosity of upper layer, the long-wave instability corresponding to the surfacemode diminishes initially and remains constant for larger values of µ . On increasing the external shear, the unstablemode bandwidth decreases owing to the balancing of the shear rate at the free surface and the shear layer near theinterface. This happens as a consequence of momentum conservation in the two-layer fluid system. From Fig. 15(a),it is found that the bandwidth of unstable modes is more significant near the small wavenumber region and it graduallydecreases, then remains invariant for increasing the value of /r . The effect of density ratio on the marginal stabilitycurve is more significant for the values of /r ∈ [0 , . , which implies that the net driving force in the upper layerfluid increases due to the presence highly dense lower layer fluid, constituting more surface instabilities. With increasein the values of /r , the density of upper layer fluid increases, which decreases the surface instabilities to some extent.However, there is no effect of upper layer density on the surface instability of two layer fluid for larger values of /r .13 PREPRINT - S
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18, 2020 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.800.20.40.60.811.21.41.61.82 k / m U SS τ = −0.2 τ = 0 τ = 0.2 τ = 0.5 (a) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.800.050.10.150.20.250.30.350.4 k / r U SS τ = −0.2 τ = 0 τ = 0.2 τ = 0.5 (b) Figure 14: Marginal stability curves: (a) in the (1 /m, k ) plane with r = 1 . and (b) in the (1 /r, k ) plane with m = 0 . for different values of τ when Re = 5 , P e = P e = 10000 , Ca = Ca = 1 , θ = 0 . rad, δ = 1 and M a = M a = 0 . .This occurs due to the balanced net driving force between the upper and lower fluids as the value of r is approachingtowards the unity along with the negligible effect of inertia on the unstable surface modes. Re k US m= 0.5m= 1m= 1.5 (a) Re k US r= 1.1r= 1.5r= 2 (b) Figure 15: Effects of viscosity and density stratification on the neutral stability curves on the surface mode: (a) forvaring viscosity ratio ( m ) with r = 1 . and (b) for varing density ratio ( r ) with m = 0 . . The other flow parametersare P e = P e = 10000 , M a = M a = 0 . , Ca = Ca = 1 , δ = 1 , τ = 0 . and β = 0 . .In Figs. 15(a) and (b), the neutral stability boundaries are plotted to describe the influence of viscosity and densityratio on the surface mode. The numerical results unveil that both the viscosity and density ratio play a dual role inthe surface instability. It is observed that the unstable region corresponding to the surface mode deceases close to thethreshold of instability, while it increases away from the threshold of instability after a critical value of wave numberas soon as the viscosity ratio magnifies. In fact, the viscous shear stress of the upper layer enhances with the highervalue of viscosity ratio to balance the same of the lower layer at the interface that causes a slower flow of the upperfluid layer and yields a stabilizing effect when inertial effects are not strong enough. Whereas, at the moderate wavenumbers when the inertial effects dominants over the viscous effects a destabilizing impact of viscosity stratification isseen. However, the effect of density ratio on the surface mode instability is completely opposite to that of the viscosityratio (Fig. 15(b)). In particular, for r > the lower layer fluid becomes more dense than the upper layer and it makesslower the impact of the driving force of the heavier lower fluid layer because of the larger magnitude of the depthwisegravitational force.Figs. 16(a) and (b) exhibit the real and imaginary part of φ , respectively, for different values of m . The mode shapevariation for φ r corresponding to the surface mode is negligible as compared to φ i . The velocity of fluid increases14 PREPRINT - S
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18, 2020gradually towards the surface, however the velocity of fluid at interface dampens in the case of m = 2 . as comparedto m = 0 . owing to the high viscous friction near the interface. This further reduces the velocity of fluid near theinterface. φ r y m=0.5m=2.5 (a) −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02−1−0.8−0.6−0.4−0.200.20.40.60.81 φ i y m=0.5m=2.5 (b) Figure 16: Eigenfunction ( φ ) corresponding to the surface mode for various m with r = 1 . , Re = 5 , τ = 0 . , P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = r = 1 and M a = M a = 0 . . Re k US τ = −0.1 τ = 0 τ = 0.1 τ = 0.2 (a) −3 k ω i τ = −0.1 τ = 0 τ = 0.1 τ = 0.2 (b) Figure 17: (a) Neutral stability curve corresponding to the varying external shear ( τ ) and (b) its corresponding scaledgrowth rate ( ω i ) as a function of wavenumber ( k ) with Re = 8000 . The other constant flow parameters are P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 , m = 0 . , r = 1 . and M a = M a = 0 . . In this subsection, the effect of external shear on the primary instability corresponding to the shear mode appearing athigh Reynolds number is analyzed for the various flow parameters. Further, the analysis is made for sufficiently smallinclination angle to compare the primary instability due to the surface and shear modes.Fig. 17(a) depicts the neutral stability curves in the ( Re , k ) plane showing the impact of the external shear force ( τ ).It is noticed that the onset of instability occurs early at lower critical Re for higher value of τ . The correspondingscaled growth rate as a function of wavenumber for various external shear is plotted in Fig. 17(a) at the Reynoldsnumber Re = 8000 . It is worth to note that the external shear rate plays a dual role on the growth rate curves for agiven value of Re . The temporal growth of the dominant mode raises for stronger imposed shear till a critical valueof wave number ( k = 0 . ) and a bifurcation is noticed thereafter ( k > . ). This is validated with the result ofmarginal stability curves in Fig. 17(a) for the value of Re = 8000 . This dual variations of growth rate with respect tothe wavenumber is due to the energy interaction between the surface/interfacial instabilities and the shear mode.15 PREPRINT - S
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18, 2020 k m US τ = −0.1 τ = 0 τ = 0.1 τ = 0.2 Figure 18: Marginal stability curve corresponding to the varying external shear ( τ ) in the ( m, k ) plane with Re =8000 . The other constant flow parameters are P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , θ = 0 . rad, δ = 1 , r = 1 . and M a = M a = 0 . . Re k τ = −0.1 τ = 0 τ = 0.1 τ = 0.2 UUS U
ShearmodebandwidthSurfacemodebandwidth (a) Re k τ = −0.1 τ = 0 τ = 0.1 τ = 0.23000 3500 40000.30.350.4 Re k S Shear modebandwidthSurface modebandwidth UU (b) Figure 19: Marginal stability curve for the surface and shear modes in the ( Re , k ) plane for (a) θ = 0 . rad and (b) θ = 0 . rad. The other constant flow parameters are P e = P e = 10000 , Ca = Ca = 1 , β = 0 . , δ = r = 1 , m = 0 . , r = 1 . and M a = M a = 0 . .In Fig. 18, the marginal stability curves are portrayed including direct and opposite external shear effects in the ( m, k ) plane. It is observed that the bandwidths are more significant for the smaller m and the values of m promotes moreinstability in the perturbed flow. On the other hand, for the larger values m (i.e. m > . ), there is a stable modebandwidths enhancing the primary stability of the system. Further, the unstable shear mode bandwidth increases forincrease in the value of τ and the observations in the ( m, k ) are quit similar to that in the ( Re , k ) plane (Fig. 17(a)).The behavior of neutral stability curves with showing the role of external shear are plotted in the ( Re , k ) plane fortwo different inclination angle θ = 0 . ′ and θ = 0 . ′ in Fig. 19(a) and (b), respectively. Further, the comparison studyis made for both the shear and surface mode at very high Reynolds number range. It is observed from Fig. 19(a) thatfor θ = 0 . ′ the unstable shear mode bandwidth increases with the superior imposed shear rate, whereas the unstablesurface mode bandwidth sinks for increase in the external shear. At θ = 0 . ′ , the critical Reynolds number of surfacemode is smaller than the shear mode and it implies that the surface mode dominates the shear mode at this situation. InFig. 19(b) with θ = 0 . ′ , the unstable mode bandwidth fall short for higher range of external shear and the equivalentobservation are noticed in the case of surface mode (shown in the inset of Fig. 19(b)). Further, the critical Reynoldsnumber of shear mode is greater than the surface mode, which shows that the shear mode dominates over the surfacemode at θ = 0 . ′ . Therefore, at the lower inclination angle shear mode becomes dominant over the surface mode asthe role of the gravitational force to derive the surface mode is weaker for this configuration.16 PREPRINT - S
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The present study deals with the role of imposed external shear on the surfactant laden two-layer flow falling down aslippery pane, using the linear stability analysis. The governing equations and its associated boundary conditions arelinearized applying the method of perturbation. Further, by employing the normal mode solutions for the perturbedvariables, the Orr–Sommerfeld equation corresponding to each layer is derived, and solved the coupled system utilizingthe spectral collocation method. There exist different unstable eigenmodes, namely, surface mode, interface mode, andtwo surfactant modes assisting the surfactant transportation at the free surface and interface. In addition, at significantlyhigh Reynolds numbers the shear modes are also recognized. The competitive effect of external shear rate on theunstable modes is analyzed by imposing the additional shear in the direction ( τ > ) and counter direction ( τ < ) ofthe flow. Moreover, based on the analysis and numerical results, the present study summarizes that: • At the moderate Reynolds numbers, both the surface and interfacial instabilities exist for certain range ofwave numbers. The external shear has significant impact on the surface mode as well as interface mode. Thebehaviour of the interfacial instability varies depending on the viscosity ratio of the fluids. On consideringthe less viscous fluid in the lower layer ( m < ), the shorter-wave and long-wave instabilities occur for theconfigurations mr < and mr > , respectively, provided that the fluid in the lower layer is more dens( r > ). Irrespective of both configurations, the interfacial instabilities is enhanced for stronger externalshear in the flow direction, whereas the extra shear in the opposite direction is favorable to stabilize theinterface mode. Contrariwise, in the case of high viscous fluid in the lower layer with r > , there exist aweaker unstable mode bandwidth in the shorter wave number region, in addition to the dominant unstablelong-wave mode bandwidth. Quit interestingly, the higher imposed shear (with τ > ) shows a dual role, andtriggers more shorter-wave instability and weaker long-wave instability. Further, the wall velocity slip has astabilizing effect on the interface mode. The marginal stability curves confirm that the neutral condition forthe interfacial instability is attained twice, resulting in the existence of the sub-critical Reynolds number. • A reverse impact of stronger external shear is noticed on surface mode instability. It is observed that thesurface instabilities stabilizes with the increase in the external shear owing to the amplification of interfacialinstabilities as a consequence of the momentum conservation. This is a fundamental finding and contraryto the role of imposed external shear on the surface mode for a surfactant laden single layer falling film.In the absence of external shear, analogous to the single fluid case, the wall velocity slip destabilizes theonset of surface mode instability but stabilizes it away from the critical Reynolds number. However, inthe presence of strong external shear against the flow direction, the surface velocity reduces and resulting aoverall stabilization of the surface mode by the slip parameter, which is again a new finding for the stratifiedtwo-layer falling film. • It is recognized that the unstable shear mode contributes to the primary instability of the flow for higherReynolds number region. At a very small inclination angle the tangential component of the gravitationalforce is weak enough and consequently, the surface mode becomes less potent and the shear mode dominants.The results suggest for higher values of external shear rate, the shear mode bandwidth increases, however, theopposite trends are observed in the case of surface mode. Further, on reducing the inclination angle, the shearmode befalls unstable early at lower critical Reynolds number than the surface mode. Moreover, to stabilizethe shear mode one need to impose the superfluous shear along the reverse direction of the streamwise flow.
Acknowledgment
HB and SG gratefully acknowledge the financial support from SERB, Department of Science and Technology,Government of India through CRG project, Award No. CRG/2018/004521.
Data Availability:
The data that supports the findings of this study are available within the article, highlighted in therelated figure captions and corresponding discussions.
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