Linear Instability of a Horizontal Thermal Boundary Layer Formed by Vertical Throughflow in a Porous Medium: The Effect of Local Thermal Nonequilibrium

In this paper we investigate the onset of convection in a saturated porous medium where uniform suction into a horizontal and uniformly hot bounding surface induces a stationary thermal boundary layer. Particular attention is paid to how the well-known linear stability characteristics of this boundary layer are modified by the presence of local thermal nonequilibrium effects. The basic conduction state is determined and it is found that the boundary layer forms two distinct regions when the porosity is small or when the conductivity of the fluid is small compared with that of the solid. A linearised stability analysis is performed which results in an ordinary differential eigenvalue problem for the critical Darcy–Rayleigh number as a function of the wave number and the two nondimensional parameters, H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document} and 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}, which are associated with local thermal nonequilibrium. This eigenvalue problem is solved numerically by first approximating the equations by fourth order compact finite differences, and then the critical Rayleigh number is computed iteratively using the inverse power method and minimised over the wavenumber. The variation of the critical Rayleigh number and wavenumber with H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H$$\end{document} and 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} is presented. One of the unusual effects of local thermal nonequilibrium is that there exists a parameter regime within which the neutral curve is bimodal.