MHD Flow Through a Perturbed Channel Filled with a Porous Medium

The main aim of this paper is to investigate the effects of a slightly perturbed boundary on the MHD flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and an arbitrary smooth function h. Employing asymptotic analysis with respect to ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, we derive the first-order effective model. We can clearly observe the nonlocal effects of the small boundary perturbation with respect to the Hartmann number since the asymptotic approximation is derived in explicit form. Theoretical error analysis is also provided, rigorously justifying our formally derived model.