Conjugate natural convection flow over a vertical surface with radiation

Numerical study of conjugate natural convection flow over a finite vertical surface with radiation is reported in this article. Rosseland diffusion approximation is used to express the radiative heat flux term. The governing boundary-layer equations are made dimensionless by means of a suitable form of non-similarity transformation. These equations are obtained in three regimes: (1) upstream (when ξ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \rightarrow 0$$\end{document}), (2) downstream (when ξ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \rightarrow \infty $$\end{document}) and (3) entire regime and are solved numerically. The solutions in the upstream and downstream regimes are obtained via shooting method whereas two-point implicit finite difference method is used to get the solutions for the entire regime. It is seen that asymptotic solutions give accurate results when compared with the numerical solution of the entire regime. The results indicate that the flow field and the temperature distributions are greatly influenced by thermal radiation parameter , Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_d$$\end{document}, surface temperature parameter, θw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _w$$\end{document} and Prandtl number Pr. It is established from the analysis that recirculation occurs in the flow specifically for Rd=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_d=1.5$$\end{document}.