Can isothermal plane Couette flow in fluid overlying porous layer be linearly unstable?

Abstract

The present study aims to examine the temporal linear stability analysis of isothermal plane Couette flow over a porous layer using the two-domain approach. The flow in the porous layer is described by the unsteady Darcy–Brinkman equations, whereas it is characterised by the Navier–Stokes equations in the fluid layer. In contrast to the Darcy model, it is observed that the isothermal plane Couette flow becomes unstable for such a superposed system on the inclusion of the Brinkman term. From the stability analysis, the two-dimensional mode is found to be least stable, and two modes of instability, namely porous mode and mixed mode are obtained under the consideration of the Darcy–Brinkman model along with advection term (DBA model). For Darcy number $(\delta )=0.01$ , depending on the value of the stress-jump coefficient, mixed mode controls the instability of the system at small values of depth ratio $(\hat {d})$ , and it disappears for relatively high values of $\hat {d}$ , where the porous mode dominates. In addition, it has been observed that when $\hat {d}=0.1$ , the critical mode of instability is found to be mixed for $\delta >0.02$ and porous for $\delta \le 0.02$ . The stress-jump coefficient destabilises the flow in terms of energy production through perturbed stresses at the interface. As observed in the case of isothermal plane Poiseuille flow studied by Chang, Chen & Straughan (J. Fluid Mech., vol. 564, 2006, pp. 287–303), here also depth ratio (Darcy number) stabilises (destabilises) the flow. However, this characteristic does not remain valid when the advection term is eliminated from the considered momentum equation. For a certain range of $\hat {d} (\delta )$ , the destabilising (stabilising) characteristic of the respective parameters are encountered when the fluid mode of instability prevails.