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. We could find only one study that examined the film flow past an anisotropic and inhomogeneous porous inclined plane; however, it focused on parameter regimes where the porous mode remains stable. Here, 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. 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 variation in anisotropic parameter. Our results clearly show that for a given Darcy number, the surface mode is the critical instability mode for isotropic and homogeneous porous media. However, when an anisotropic porous medium is considered 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 fluid mode (i.e., surface or shear mode) to porous mode with variations in inhomogeneity parameter. This switching of the most unstable mode from fluid mode to porous mode with variation in anisotropy and/or inhomogeneity in permeability has not been demonstrated in any of the earlier studies in the context of film flows. We also present an energy budget analysis to decipher the mechanism responsible for making the three modes unstable.