MHD mixed convection Poiseuille flow in a porous medium: New trends of Caputo time fractional derivatives in heat transfer problems⋆

This paper discusses new trends of fractional derivatives in heat transfer problems. More exactly, in this work magnetohydrodynamic (MHD) mixed convection Poiseuille flow of electrically conducting, an incompressible viscous fluid with memory, in a vertical channel filled with porous medium is studied under the influence of an oscillating pressure gradient. The vertical channel is taken in stationary state with non-uniform walls temperature. The problem is formulated in terms of fractional differential equations with Caputo time fractional derivatives. The closed forms of the non-dimensional temperature, velocity, Nusselt numbers and skin friction coefficients on the walls are determined by employing the Laplace transform method. The solutions are presented in terms of the time-fractional derivative of the Wright function, Robotnov and Hartley F -function and Lorenzo-Hartley R -function. Similar solutions for ordinary fluid, corresponding to the fractional parameter equal to one, are obtained as a particular case of the fractional problem. The influences of the fractional parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \alpha$\end{document} , Péclet number Pe and Reynolds number Re on the heat and momentum transfer are studied. It is found that the heat transfer can be enhanced in the fluid with memory. Fluids described with a fractional model flow faster/slower than the ordinary fluid, depending on the Reynolds number/Péclet number.