Stability of a liquid film flowing down an inclined anisotropic and inhomogeneous porous layer: an analytical description

Abstract

We study the effect of anisotropy and inhomogeneity in the permeability of the porous layer on the stability of surface waves of an inclined fluid–porous double-layer system. The fluid is assumed to be Newtonian and the porous layer to be Darcian. The porous layer is saturated with the same fluid and the two layers are coupled at the interface via the Beavers–Joseph condition. Linear stability analysis is performed based on a long-wave approximation. The resulting eigenvalue problem is exactly solved up to third order in the wavenumber. The anisotropic behaviour of permeability, cross-stream component of permeability, surface tension and porosity are found to have only higher-order effects on the stability characteristics of the system. On the other hand, the inhomogeneous feature in the streamwise component of permeability play a dominant role in determining the stability of the gravity-driven surface waves; as do other system parameters such as the thickness of the fluid layer relative to that of the porous layer and the Beavers–Joseph coefficient.

Stability of non-isothermal Poiseuille flow in a fluid overlying an anisotropic and inhomogeneous porous domain

“…As widely done in the literature (Sun 1973;Chen & Chen 1988;Chang 2005Chang , 2006, a few parameters have been fixed as α BJ = 0.1, χ = 0.3, = 0.7 and G m = 10, which represents many porous materials' properties (Straughan 2002). In order to reduce complexities regarding the directional inhomogeneities, the inhomogeneity parameters are defined as η x = η z = e A(1+z m ) (Deepu et al 2015(Deepu et al , 2016, where the permeabilities in the x m as well as the z m direction increase and decrease vertically with positive and negative values of A, respectively. As discussed in the work of Chen & Hsu (1991), the inhomogeneity function in exponential form finds much more practical applicability than a linear form, which is due to the fact that the particle size distribution in a porous medium follows a lognormal distribution, i.e.…”

“…Note that when the value of all other parameters are kept constant, a change in ξ is due to a change in permeability in the z-direction only. Furthermore, the analysis in this section is based on two different values 10 −3 and 5 × 10 −4 of the Darcy number (Deepu et al 2016). In the following, we have analysed the stability characteristics of the considered flow through neutral stability curves.…”

“…Furthermore, the analysis in this section is based on two different values and of the Darcy number (Deepu et al. 2016). In the following, we have analysed the stability characteristics of the considered flow through neutral stability curves.…”

Falling film on an anisotropic porous medium

“…This motivated earlier studies on convection in an anisotropic porous medium (Castinel & Combarnous 1975;Epherre 1975). Deepu et al (2016) studied the hydrodynamic stability of a falling film over an almost horizontal anisotropic and inhomogeneous porous medium. They considered a generalised Darcy law to describe the flow in the porous medium coupled to the Beavers-Joseph boundary condition at the interface.…”

“…They discussed the surface instability mode as well as the shear mode of linear instability and concluded that anisotropy has no visible effect on the linear stability of the surface mode. Later, [26] provided a detailed analytical description of the flow over an anisotropic and inhomogeneous porous medium. They solved the linear eigenvalue problem constructed under the long-wave assumption up to first-order in the wavenumber and concluded that the anisotropy effect arises only at higher order.…”