Benchmark solution for the hydrodynamic stability of plane porous-Couette flow

The hydrodynamic stability against small disturbances of plane Couette flow through an incompressible fluid-saturated fixed porous medium between two parallel rigid plates is investigated. The fluid flow occurs because of moving upper and lower plates with a constant speed in the opposite directions, and it is described by using the Brinkman-extended Darcy model with fluid viscosity different from the Brinkman viscosity. The resulting stability eigenvalue problem is solved numerically using the Chebyshev collocation method. The instigation of instability has been determined accurately by computing the critical Brinkman-Reynolds number as a function of the Darcy-Reynolds number. A comparative study between the plane porous-Poiseuille flow and the plane porous-Couette flow has been carried out, and the similarities and the differences are highlighted. For the Darcy and the inviscid fluid cases, the stability of fluid flow is analyzed analytically and found that the flow is always stable.