Stability of density-stratified plane Poiseuille flow in a fluid layer adjacent to a porous layer

In this article, we examine the stability characteristics of plane Poiseuille flow (PPF) in a density-stratified fluid layer adjacent to a porous layer. The flow of a Newtonian fluid in a porous layer is governed by Darcy's law, with the Beavers–Joseph condition applied at the interface. The impact of stratification and flow penetration in the adjacent porous layer is examined through temporal linear stability analysis, primarily along the instability boundary. The mass diffusivity, density stratification, and fluid-porous thickness ratio are governed by the Schmidt number (Sc), the Froude number (Fr), and the thickness ratio (d̂), respectively. Our analysis indicates that for Sc≥200, the instability characteristics resemble a non-diffusive approximation, and are independent of Sc. In the case of density-stratified PPF in a fluid layer, the onset of instability may occur through either a two-dimensional Tollmien–Schlichting (TS) mode, a three-dimensional even-gravity fluid mode, or an odd-gravity fluid mode. However, upon replacing the channel's impermeable walls with the same fluid-saturated porous layers, two additional modes are identified: the gravity-porous mode (GPM) and the even-fluid mode. In the case of GPM, instability arises from density-stratified buoyant production in a porous layer and is counterbalanced by the dissipation of kinetic energy through surface drag. However, for all other modes of instability, viscous shear production causes instability and is balanced by other components in the energy spectrum. The TS mode prevails in both highly stratified and almost unstratified situations. The range of Fr, in which the above modes sustain, depends on the thickness ratio. It has also been found that instability may occur due to possible resonance between the TS wave and the gravity wave in a purely fluid channel and in a fluid adjacent to a porous layer at Sc=700. In general, increasing media permeability as well as the jump coefficient induces early instability. In contrast to a density-stratified, purely fluid channel, where instability of saline water occurs even for Re=297.40, the same phenomenon takes place when a porous layer is attached for at least 606.27 of Re. Overall, the inclusion of a homogeneous, isotropic porous layer delays the onset of instability.