Role of slip on the linear stability of a liquid flow through a porous channel

The linear stability of a liquid flow bounded by slippery and porous walls is studied for infinitesimal disturbances of arbitrary wavenumbers. The Orr-Sommerfeld type eigenvalue problem is formulated by using the normal mode decomposition and resolved based on the Chebyshev spectral collocation method along with the QZ algorithm. The results are computed numerically in detail for various values of the flow parameters. The presence of an upper wall slip shows a destabilizing effect on the fluid layer mode, but it shows a stabilizing effect on the porous layer mode. On the other hand, the decreasing value of the depth ratio has a stabilizing effect on the fluid layer mode but it has a destabilizing effect on the porous layer mode. In fact, there occurs a competition between the most unstable porous layer mode and the most unstable fluid layer mode to control the primary instability. The most unstable porous layer mode triggers the primary instability unless the upper wall slip dominates the effect of the porous layer otherwise the most unstable fluid layer mode triggers the primary instability. A new phase boundary is detected in the plane of the depth ratio and slip length, which separates the domain of the most unstable porous layer mode from the domain of the most unstable fluid layer mode.