Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous layer are impermeable, isothermal and isosolutal. The generalized Darcy’s equation for power-law fluid is employed in the momentum balance, along with Oberbeck–Boussinesq approximation for buoyancy effect in the system. It is considered that the viscous dissipation is non-negligible. The basic temperature and concentration profiles for the considered horizontal throughflow are determined analytically, and the linear stability analysis is performed on this basic solution. The eigenvalue problem is solved numerically using bvp4c in MATLAB. The conditions for the onset of instability of base flow in terms of critical Rayleigh number and the corresponding wave number are obtained. It is found that critical values are affected by the flow governing parameters such as the Péclet number (Pe), power-law index (n), viscous dissipation (Ge), angle of inclination (α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha )$$\end{document}, buoyancy ratio (N) and Lewis number (Le). For the considered range of these parameter values, the instability of the basic flow is discussed for the convective rolls. It is observed that instability of any general convective roll lies between that of the longitudinal and transverse rolls. Also, the effect of viscous dissipation on the onset of instability for both the longitudinal and transverse rolls is discussed in detail. This source of heating inside the porous layer increases the thermal instability of base flow. Further, the asymptotic analysis for vanishingly small Péclet number is presented as Pe=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Pe = 0$$\end{document} is a singular point. The boundary between the stationary and the oscillatory instabilities is given using the buoyancy ratio parameter and the Lewis number in this limiting case of Pe→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Pe\rightarrow 0$$\end{document}.