Linear and weakly nonlinear stability analyses are carried out to understand the influence of anisotropic slip on the instability and transition characteristics of pressure-driven parallel flow in the fluid overlying a porous medium. The slip is induced on the upper plate dominating in the streamwise direction. The investigation is made by imposing Navier slip on the classical model considered by Aleria et al. (SIAM J. App. Math. , vol. 84, 2024, pp. 433–463). For finite-amplitude disturbances, a weakly nonlinear stability analysis based on the cubic-Landau theory is exploited. The bifurcation phenomena are investigated as a function of slip length at the critical instability point (CIP) and as a function of Reynolds number away from the CIP. The linear stability analysis shows that Squire’s theorem does not hold for anisotropic slip, and the mode of instability along the neutral curve is sensitive to slip length. Along the instability boundary, slip stabilises (destabilises) the porous mode (odd-fluid mode), whereas in the even-fluid mode, slip can have either a stabilising or destabilising effect. When the porous mode or odd-fluid mode dominates the flow instability, only the supercritical bifurcation exists at and away from the CIP. For each value of the depth ratio, there exists a finite interval of slip parameter in which the three-dimensional disturbances are least stable and the critical mode of instability is the even-fluid mode. Both the subcritical and supercritical bifurcations are possible for the even-fluid mode of instability and the supercritical bifurcation at the CIP always shifts to a subcritical bifurcation away from the CIP. The nonlinear kinetic energy analysis reveals that modifications in energy due to gradient production and viscous dissipation are mainly responsible for inducing the subcritical instability. The role of spanwise slip, Darcy number, porosity and Beavers–Joseph coefficient is also investigated. The results demonstrates a stabilising (destabilising) impact of spanwise slip (porosity and Beavers–Joseph coefficient), and instability as well as bifurcation characteristics is a function of {Da}/d, rather than individual Darcy number (Da) and depth ratio (d). Overall, this study finds a significant relationship among the critical modes of instability, dimension of the least stable disturbances, bifurcation phenomena and skin-friction coefficient. The present results also witness good experimental support for the stability of flow in the fluid overlying a porous medium and slippery flow in a single-fluid layer configuration.
Sharma et al. (Mon,) studied this question.