Instabilities of a free bilayer flowing on an inclined porous medium

The instabilities of a free bilayer flowing on an inclined Darcy-Brinkman porous layer have been explored. The bilayer is composed of a pair of immiscible liquid films with a deformable liquid-liquid interface and a liquid-air free surface. An Orr-Sommerfeld analysis of the governing equations and boundary conditions uncovers that this configuration can be unstable by a pair of long-wave interfacial modes at the free surface and at the interface together with a couple of finite wave-number shear modes originating from the inertial influences at the liquid layers. In particular, one of the shear modes originates beyond a threshold flow rate owing to the slippage at the porous-liquid interface and is found to be the dominant one even when the porous medium is moderately thin, porous, and permeable. The strength of the porous media mediated mode (a) grows with increase in porosity, (b) grows and then remains invariant with increase in thickness, and (c) initially grows and then decays with increase in the permeability of the porous layer. Further, the presence of a lower layer with smaller viscosity and a thicker upper layer is found to facilitate the growth of this newly identified porous media mode. Importantly, beyond a threshold upper to lower thickness and viscosity ratios and the angle of inclination the porous media mode dominates over all the other interfacial or shear modes, highlighting its importance in the bilayer flows down an inclined porous medium. The study showcases the importance of a porous layer in destabilizing a free bilayer flow down an inclined plane, which can be of importance to improve mixing, emulsification, and heat and mass transfer characteristics in the microscale devices.