Finite Amplitude Analysis of Poiseuille Flow in Fluid Overlying Porous Domain

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 433-463, April 2024.
Abstract. A weakly nonlinear stability analysis of isothermal Poiseuille flow in a fluid overlying porous domain is proposed and investigated in this article. The nonlinear interactions are studied by imposing finite amplitude disturbances to the classical model deliberated in Chang, Chen, and Straughan [J. Fluid Mech., 564 (2006), pp. 287–303]. The order parameter theory is used to ascertain the cubic Landau equation, and the regimes of instability for the bifurcations are determined henceforth. The well-established controlling parameters viz. the depth ratio [math] depth of fluid domain/depth of porous domain), the Beavers–Joseph constant [math], and the Darcy number [math] are inquired upon for the bifurcation phenomena. The imposed finite amplitude disturbances are viewed for bifurcations along the neutral stability curves and away from the critical point as a function of the wave number [math] and the Reynolds number [math]. The even-fluid-layer (porous) mode along the neutral stability curves correlates to the subcritical (supercritical) bifurcation phenomena. On perceiving the bifurcations as a function of [math] and [math] by moving away from the bifurcation/critical point, subcritical bifurcation is observed for increasing [math] and decreasing [math]. In contrast to only fluid flow through a channel, it is found that the inclusion of porous domain aids in the early appearance of subcritical bifurcation when [math]. A considerable difference between the computed skin friction coefficient for the base and the distorted state is observed for small (large) values of [math]. In addition, an intrinsic relation among the mode of instability, bifurcation phenomena, and secondary flow pattern is also observed.