Instabilities of a confined two-layer flow on a porous medium: An Orr–Sommerfeld analysis

• We model the instabilities of a two-layer Poiseuille flow on a porous layer.
• Presence of porous media acts as destabilizing influence due to the added inertia.
• An inertia driven shear mode with an interfacial mode destabilizes the system.
• Interesting transitions from dominant shear to interfacial mode are shown.
• The findings are of importance in improving heat/mass transfer and mixing.

Instabilities of a pressure driven two-layer Poiseuille flow confined between a rigid wall and a Darcy–Brinkman porous layer are explored. A linear stability analysis of the conservation laws leads to an Orr–Sommerfeld system, which is solved numerically with appropriate boundary conditions to identify the time and length scales of the instabilities.fde The study uncovers the coexistence of twin instability modes, (i) long-wave interfacial mode—engendered by the viscosity stratification across the interface and (ii) finite wave number shear mode—originating from the inertial stresses, for almost all combinations of the viscosity (μrμr) and thickness (hrhr) ratios of the liquid layers. The presence of the porous layer reduces the frictional influence on the films, which significantly alters the length and time scales of the shear mode while the interfacial mode remains dormant to this effect. This is in stark contrast to the two-layer flow confined between non-porous plates where an unstable interfacial (shear) mode is observed when μr>hr2(μr