A RESONANT GRAVITY-DRIVEN FLOW OF A POWER-LAW FLUID OVER A SLIPPERY TOPOGRAPHY SUBSTRATE

The gravity-driven flow of a power-law fluid over a slippery topography substrate is studied. The fluid flow, due to the gravity, is assumed to be steady and confined to the limit of small amplitude of the wall corrugation. As an analytic approach, we apply the Karman– Pohlhausen integral boundary-layer method and derive an asymptotic equation valid for rather thin films. Our results support the view that the resonance is associated with an interaction of the undulated film with capillary-gravity waves travelling against the mean flow direction in the linear case. The influence of the slip condition on the linear resonance phenomena for different power-law indices n is the main factor in this work.


Introduction
Gravity-driven fi lms falling on inclined planes are a fundamental problem in fl uid mechanics which has even an exact analytical solution. However, most applications are studied through topography form, related to heat exchangers and coating technologies [22]. Surface rollers, standing waves or hydraulic jumps are new phenomena resulting from the fl ow over a wavy bottom [23]. Small forces or periodic changes in parameters are the actual reasons for the presence of resonance [10]. Analytical and numerical studies [13,20] show that the presence of periodic undulation leads to a stable eff ect which increases with corrugation steepness. Th e fl ow and the instability of non-Newtonian fi lms along inclined surfaces were considered by many researchers. Th us, Gupta [6] and Berezin et al. [4] studied the stability of a second-order fl uid and a power-law fl uid, respectively. Argyriadi et al. [2] found that the extent of distortion depends on the height of corrugations, and so studying the liquid fi lm over a weakly undulating bottom is a simple theoretical technique. Th e eff ect of the slip length parameter was examined by Samanta [17], when the solid substrate has a slippery property. Beavers and Joseph [3] discussed the behavior of the interface between a fl uid and porous layers in a liquid fi lm which is governed by the Stokes and Darcy equations. Pascal [14] introduced a Navier-slip boundary condition ∂ = ∂ , s u u l y where κ = β s l is the eff ective slip length related to the permeability κ and to the empirical dimensionless parameter of Beavers and Joseph. In the framework of the integral boundary layer, the surface waves on a fi lm of a power-law fl uid were investigated by Dandapat and Mukhopadhyay [5]. Recently, Amaouche et al. [1] have developed an extension of the model equations derived by Ruyer-Quil and Manneville [16] which correctly predict the linear stability threshold. Th e porosity of the wavy bottom has an eff ect on the behavior of a thin liquid fi lm, where a destabilizing infl uence has been deduced by Th iele et al. [19], especially the existence of a jump boundary condition. Th e strong eff ect of resonance occurs at a dimensionless fi lm thickness of about unity [25]. Generation of higher harmonics under the eff ect of nonlinear resonance results from increasing the wall amplitude and is also found experimentally in a liquid fi lm fl owing over sinusoidal substrates [24]. Th e resonance has a diff erent case between thin and thick fi lms, where the quality of transportation of the bottom perturbation towards the free surface gradually deteriorates in thick fi lms. It remains sharp, but declines in amplitude. Other researchers focused on the infl uence of surfactants [15] and electric fi elds [21]. Th e resonant steady deformation was deduced experimentally by Argyriadi et al. [2], who showed that the dominant characteristic of the unstable free surface only slightly modulates in amplitude and phase during the passage of traveling waves. Saprykin et al. [18] studied inertial eff ects in the fl ow of a viscoelastic liquid over a step-down topography. Nevertheless, the infl uence of the slip length for a non-Newtonian fl uid fl owing over a wavy wall has not been studied. In this work, we model a non-Newtonian fl uid fi lm fl owing on an inclined corrugated slippery substrate in the fi eld of steady fl ow. Th e goal of the paper is to examine the linear resonance of the inclined slippery wall with weak periodic corrugations under the eff ect of the slip length parameter. Th e paper is organized as follows. Section 2 addresses the formulation of the problem. In Section 3 we derive the evolution equations of the free surface in the steady state. Section 4 examines the infl uence of the slippery property on the linear resonance phenomena for diff erent power-law indices n. Th e summary is given in Section 5.

Formulation of the problem
We consider a two-dimensional laminar fl ow of a thin layer of a power-law fl uid fl owing down an inclined slippery wavy substrate at inclination angle α. Th e contour of the topography is given by the periodic function where x is the coordinate in the mean fl ow direction, a is the amplitude of the undulation and λ is the wavelength. Th e fi lm is bounded above by a motionless gas at ambient pressure p gas . Th e liquid is considered to be non-Newtonian with constant density and surface tension σ 0 . Th e nonlinear shear-dependent viscosity may be written as where μ m is the constant for the particular liquid (μ m is a measure of the consistency of the fl uid; the higher the μ m , the more viscous the fl uid) and n is the positive power-law index. Th e case n = 1 represents a Newtonian fl uid with a constant dynamic coeffi cient of viscosity, while n < 1 and n > 1 correspond to the case of pseudo-plastic (shearthinning) and dilatant (shear-thickening) fl uids, respectively. We assume that the liquid fi lm is very thin and the induced gravity-driven fl ow is relatively slow so that the fl ow regime is close to that predicted by lubrication theory. Th e adopted (x, y) Cartesian coordinate system is oriented along the main fl ow direction, which is inclined at an angle α with respect to the horizontal plane, with (x, y) being the stream-wise and cross-stream directions of the fl ow, respectively. Th e continuity and momentum equations that govern the fl uid motion have the form where u and ν are the longitudinal and transverse velocity components, respectively, and p is the pressure. Here τ ij is the stress tensor defi ned by Th e above equations are subject to the following boundary conditions: On the wave substrate, the slip condition [11,12] has the form where the constant l 0 is the slip length, and b(x) is the wavy wall substrate.
On the free surface y = h(x, t), we have the following conditions: Th e above equations represent the normal, tangential stress balances and the kinematic boundary conditions, respectively, at the free surface interface.
Using the following dimensionless variables [9], we obtain a set of dimensionless parameters that describe the behavior of the model , where u is the mean velocity of the Nusselt fi lm and d is the mean fi lm thickness. Aft er using the above scales, we apply thin-fi lm approximation [8] and then truncate equations and the boundary conditions at order δ 2 , supposing that δcotα = O(1), δRe = O(1), and δBo -1 = O(1). Th us, the dimensionless form of the governing equations and boundary conditions are expressed as follow: Re cot , n p n y n (2.13) p is the non-dimensional pressure; and ( ) x is the non-dimensional fi lm thickness.

Free Surface Equation
In this Section, we derive the weakly evolution equation of the model and then study the case of a system which is based on limited values for the small parameter ζ = ≺ 1.
a d Since we have only retained the terms up to the fi rst order of δ, through the latter assumption we integrate Eq. (2.13) with the help of the dynamic normal boundary condition (2.15), where the denominator of the capillary term will be neglected. We obtain the following form of pressure: Substituting the right-hand part of equations (3.1) into the stream-wise momentum equation (2.12), we arrive at the expression Th e consistent second-order theory with the Karman-Pohlhausen approximation is supported by experiments in the parametric domain of interest, at least for small amplitude disturbances. Using the Leibniz rule to integrate the partial diff erential equation (3.2) from the bottom to the free surface, we obtain the evolution equation of the fi lm thickness  .3) is complemented with the dynamic boundary condition (2.16), the steady kinematic condition (2.17), the slip condition at the wavy substrate (2.14), and the continuity equation (2.11). It is to be noted that the momentum integral method has been used in connection with the boundary layer theory to obtain a single equation for the resonance capillary gravity waves with the bottom corrugation.
In the present application, we exploit the assumption of periodicity and define the domain over only one wall wavelength. Th is condition creates an additional degree of freedom, which is removed by fi xing either the fl ow rate or the mean liquid level. We select the former and enforce at the inlet the condition Our study concerns with the concept of steady state that was used by Heining et al. [7] and was confi ned to the limit of small amplitude of the wall corrugation [13]. To close the system, a specifi c velocity profile has to be introduced.
where the subscript represents the diff erentiation of the film thickness with respect to the variable x. Th is equation coincides with that in Ref. [9], when the slippery parameter is removed (γ = 0). Inertia, hydrostatic, capillary pressure, gravity-driven force and wall shear stress are physically responsible for the appearance of the nonlinear terms in Eq. (3.6) governing the film thickness. Let us now express the fl uid thickness f as a power series expansion in the form of asymptotic expansion of the parameter ζ 1, By substituting f into equation (3.6), where f 0 is the leading order term, which represents the steady film thickness for a flat incline. A system of linear ordinary diff erential equations will be degenerated from the nonlinear equation (3.6). At fi rst order we obtain: where the value of K 2 is formulated above, and the value of K 1 is found to be According to the periodicity of the inhomogeneity, we assume that the solution inherits the periodicity of the substrate and can be written as a periodic function with the same periodicity as the substrate contour; hence, = + where the constants A 1 and B 1 have the form Th e position of the free surface up to the fi rst order in the Cartesian coordinates is ( ) = +ζ +ζ = +ζ +Δϕ with the free-surface amplitude In order to classify the free surface response at leading order we defi ne the relative free surface amplitude and the relative fi lm thickness amplitude by Eq. (3.11) and by = +

Infl uence of the slippery property on the resonance phenomenon
In the previous Section, we derived the weakly nonlinear evolution equations for the free surface in the presence of the slippery eff ect for diff erent power-law indices n. Th e linear resonance phenomena result from the amplitude amplifi cation of the free surface. Th e fi lm thickness, the bottom contour and the doubling with each other are the fundamental factors responsible for the presence of inhomogeneity.
In the following fi gures, the dependences are plotted for n = (solid curve) 0.5, (dotted curve) 1, and (dashed curve) 1.5. Figure 2(a) shows the amplitude of the free surface up to the fi rst order for diff erent power-law indices in the presence of the slippery property as a function of the Reynolds number. We conclude that the free surface amplitude strongly depends on the powerlaw index n. At a h = 1, three curves intersect with each other, since the free surface amplitude up to the fi rst order has the same magnitude as the topography amplitude. Th en each curve moves to a maximum value that corresponds to a fi nite value of the resonance Reynolds number. For n 1, the free surface amplitude is smaller than that in the Newtonian case whereas it is higher for n ≺ 1. Shear-thinning fl uid hence leads to an amplifi cation of the free surface. It is obvious that a less viscous liquid results in a stronger interaction of the free surface with the topography substrate. Th e phenomenon of the surface amplifi cation is called resonance in the literature (see, e.g., [25,8]), although it is not a dynamic process. It is clear that the behavior of free surface amplitude coincides with that reported in [9], when the wave substrate is free from the property eff ect.
On the other hand, we have found that the eff ect of the slip length parameter appears in the value of the resonance Reynolds number according to the power-law index n, as displayed in Fig. 2(b). For n 1, the resonance Reynolds number is close to the Newtonian case, whereas it is higher for n ≺ 1. In addition, the values of resonance Reynolds numbers for diff erent power-law indices n during the eff ect of the slip length parameter are less for γ = 0. Figure 3 depicts the amplitude of the fi lm thickness a f as a function of the Reynolds number. In both cases, it is found that the amplitudes show a maximum, depending on n. Th e power-law index n has, as already predicted by Fig. 2, a strong infl uence on both amplitudes. Shear-thickening fl uids lead to decreased amplitudes in both cases of the topography substrate, whereas shear-thinning fl uids lead to increased amplitudes.
Th e position of the maximum is referred to as the resonant Reynolds number. For slippery property γ = 0.15 the resonant Reynolds number decreases with increasing n, whereas the resonant Reynolds number for n on a slippery substrate slowly changes. Th e curves for non-slip and slip conditions are both related to each other and have the same qualitative shape, as shown in Fig. 3(a, b).

Conclusion
A linear resonance of the gravity-driven fi lm fl ow over a slippery topography substrate has been investigated, which extends the study of Heining et al. [9] by incorporating the slip condition at the substrate. We have derived the evolution equation of the free surface amplitude for diff erent power-law indices n under the eff ect of the slip length parameter. Th e obtained results describe the eff ects of slippery property in studying the linear resonance phenomena for small values of the resonance Reynolds number. Th ese results are validated with previous investigations of the resonance phenomenon in gravity-driven fi lms in the absence of slippery property (see, e.g., [25,7,9]). Th e dependence of relative amplitude during a gradual increase in the Reynolds number for both cases of the non-slip and slip length parameter at diff erent power-law indices n indicates that the maximum value for both non-slippery and slippery property are found for a shear-thinning fl uid. Moreover, the relative amplitude value at γ = 0.15 is less than the relative amplitude deduced at γ = 0 (see Fig. 2).