Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation

The present research work is dedicated to examine the heat and mass transport phenomenon due to unsteady MHD flow of Williamson nanofluid between the permeable channel with heat source/sink. The influence of buoyancy and thermal radiation effects are also considered for the present model. The flow equations are reduced to an equivalent nonlinear coupled partial differential equations (PDE) through suitable transformation. The numerical simulation is performed to attain the solution of the nonlinear system via the Crank-Nicolson finite difference scheme. The influence of various emerging parameters on velocity, temperature and concentration profiles are developed. The magnetic parameter plays the significant role to enhance the heat transfer rate but reduces the velocity profile. The velocity of the fluid increases gradually with respect to time and this influence is dominant at the center of the channel. Additionally, the velocity profile exhibits the increasing behavior by varying the Reynolds number for a small time. In the entire study it is analyzed that the temperature profile increases for increasing values of Reynolds, thermophoresis, Brownian motion and heat source numbers, while a low temperature profile is attained for Biot numbers. The thermophoresis and Brownian motion parameters provide the decreasing concentration profile however, an increasing result is noticed for Biot numbers. An increase in the values of Williamson 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$\end{document}, illustrates the increase effects on the skin friction coefficient when η=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \eta=1$\end{document} while it shows the decrease behavior at η=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \eta=0$\end{document}. The effects of the Reynolds number on streamlines pattern is presented. It is noticed that the higher values of Re affects the stream line pattern.