Mixed Convection Boundary-layer Flow Past a Vertical Surface Embedded in a Porous Medium: Exponential Case

The steady, mixed convection boundary-layer flow on a vertical surface is considered when there is either an exponentially increasing/decreasing surface temperature with a constant outer flow or an exponentially increasing/decreasing outer flow with a constant surface temperature, characterized by the mixed convection parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}. A different behaviour is observed in aiding flow or in opposing flow, and whether the boundary condition is increasing or decreasing. An exponentially decreasing surface temperature sets up a flow at large distances driven by the outer flow with the temperature field decreasing at a rate proportional to x-3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{-3/2}$$\end{document}. An exponentially increasing surface temperature gives rise to a different behaviour for aiding and opposing flow. For aiding flow and temperature field increases as ex\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{e}^x$$\end{document}. For opposing flow the solution breaks down at a finite distance from the leading edge effectively where the wall velocity goes to zero at -logλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\log \lambda $$\end{document}. With an exponentially decreasing surface temperature for aiding flow the solution evolves to the free convection limit For opposing flow the solution breaks down at a finite distance from the leading edge, effectively where the wall velocity goes to zero. For an exponentially increasing outer flow the solution evolves to the forced convection limit.