Instability in Poiseuille flow in a porous medium with slip boundary conditions and uniform vertical throughflow effects

In this study, the effect of uniform vertical throughflow/crossflow on the instability of Poiseuille flow in a porous medium was investigated using the Brinkman model. The effect of slip boundary conditions on instability, in particular, was studied. The Chebyshev collocation method was utilized to approximate the eigenvalue system of this problem. It has been found that the throughflow Reynolds number RT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{T}$$\end{document} has both stabilizing and destabilizing effects, and it has the same stabilizing/destabilizing effect on fluid flow in both positive and negative directions. The results for the flow of a Navier–Stokes fluid through channel case and clear fluid case, are likewise obtained as a particular result of this work. Moreover, flow in a planar channel is found to be stable to small disturbances for all values of the Reynolds number, but for a plane–parallel channel, the flow is unstable if the Reynolds number exceeds a critical value, dependent on a Darcy number, throughflow Reynolds number RT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{T}$$\end{document} and slip length.