Thermosolutal convective instability of power-law fluid saturated porous layer with concentration based internal heat source and Soret effect

The onset of double-diffusive convective instability in a horizontal porous layer saturated by a power-law fluid that is subjected to concentration based internal heat source and Soret effect is investigated. Both these physical processes have a significant influence on the concentration field in porous media. Here the fluid flow is induced by constant but different temperature and concentration maintained across the boundary planes. The basic flow considered is a uniform throughflow which is inclined with respect to the horizontal x -axis. Considering the heat source (γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \gamma$\end{document}) and Soret effect (Sr) results in a non-linear basic temperature and concentration profiles. The resulting eigenvalue system of coupled ordinary differential equations along with the boundary conditions is solved to explore the combined effect of internal heat source and Soret parameter on the instability of the power-law fluid. The instability of the base flow appears in the form of oblique rolls, which can be reduced to longitudinal rolls or transverse rolls. Stationary instability is seen for the longitudinal rolls. For transverse rolls, the onset of stationary and oscillatory convective instability depends on Lewis number, Péclet number, Soret parameter and presence of the concentration based internal heat source. It is also evident that γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \gamma$\end{document} = 0 and Sr = 0 are singularities for the basic temperature and concentration profiles. The limiting behavior of these parameters is also investigated for their linear stability. Both stationary and oscillatory convective instability corresponding to the pseudoplastic and dilatant fluids is discussed for the case of γ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \gamma\rightarrow 0$\end{document}.