Effect of anisotropy and inhomogeneity on the stability of liquid film flowing down a porous inclined plane

We examined the linear stability of a Newtonian liquid film flow past a porous inclined plane. Falling film on inclined permeable planes displays three instability modes: surface, shear, and porous mode. Most of the previous studies have examined the film flow past an isotropic and homogeneous porous medium. To the best of our knowledge, only one study examined the film flow past an anisotropic and inhomogeneous porous inclined plane, however, it focused on parameter regimes where the porous mode remains stable. We explored the parameter regimes where all three modes become unstable and investigated the effect of the anisotropic and inhomogeneous variations in the permeability of the porous medium on all three unstable modes. The generalized Darcy model is used to describe the flow through the porous medium along with the Beavers-Joseph condition at the fluid-porous interface. We show that there is a switching of dominant instability mode from surface mode to porous mode with variations in anisotropic parameter. The surface mode is critical instability mode for isotropic and homogeneous porous media. However, for an anisotropic porous medium with high wall normal permeability than the wall parallel permeability, the porous mode becomes the most unstable mode. We show a similar exchange of dominant instability mode from surface or shear mode to porous mode with variations in inhomogeneity parameter. This switching of the most unstable mode mode to porous mode with variations in anisotropy and/or inhomogeneity in permeability has not been demonstrated in any earlier work in context of film flows.