Finite-amplitude Instability Analysis of Non-isothermal Annular Parallel Flow through Porous Medium

Weakly nonlinear stability analysis of stably stratified non-isothermal parallel water flow in a tall vertical annulus filled with a highly permeable porous medium is investigated in the current work. The flow stability is described by the volume averaged Navier Stokes equations [1, 2]. The effect of the curvature parameter on the bifurcation and instability of the considered flow has been investigated via finite-amplitude analysis centered around cubic Landau equation. A wide range of Reynolds number (Re), four different values of curvature parameter (C), two values of form drag coefficient (c_F), two values of media permeability ( in terms of Darcy number, Da) and a fixed value of Prandtl number (Pr = 7) are considered. The finite-amplitude analysis predicts supercritical as well as subcritical bifurcation in the vicinity of instability boundary. In most of the parametric space, the bifurcation is supercritical, whereas for Da = 10^-2 and cF = 6x10^-4 subcritical bifurcation exists in a very small range of Re that too for two values of C only. However, the non-isothermal parallel water flow through a vertical channel exhibits only supercritical bifurcation[3]. In the supercritical (subcritical) regime, the equilibrium (threshold) amplitude is investigated as a function of different controlling parameters. As the gap between the cylinders increases, the threshold amplitude below which the flow remains nonlinearly stable grows. Moreover, the influence of nonlinear interaction of different harmonics on the Nusselt number (Nu) and friction coefficient (C_f) on both the walls of the annulus has been also investigated. Nu for the distorted state is always less than the Nusselt number for the basic state. C_f at the inner wall is always greater than the C_f at the outer wall. The skin friction at the outer wall is either constant or a decreasing function of Ra.