Brinkman Flow of a Viscous Fluid Past a Reiner–Rivlin Liquid Sphere Immersed in a Saturated Porous Medium

This paper presents an analytical study of Stokes flow of an incompressible viscous fluid past an immiscible Reiner–Rivlin liquid sphere embedded in porous medium using the validity of Brinkman’s model. The stream function solution of Brinkman equation is obtained for the flow in porous region, while for the inner flow field, the solution is obtained by expanding the stream function in a power series of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document}. The flow fields are determined explicitly by matching the boundary conditions at the interface of porous region and the liquid sphere. Relevant quantities such as shearing stresses and velocities on the surface of the liquid sphere are obtained and presented graphically. It is found that dimensionless shearing stress on the surface is of periodic nature and its absolute value decreases with permeability 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} and almost constant for all the representative values of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document}; on the other hand, the permeability parameter increases the velocity in the vicinity of the liquid sphere. The mathematical expression of separation parameter SEP is also calculated which shows that no flow separation occurs for the considered flow configuration and also validated by its pictorial depiction. The drag coefficient experienced by a liquid sphere embedded in a porous medium is evaluated. The dependence of the drag coefficient on permeability parameter, viscosity ratio and dimensionless parameter S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S$$\end{document} is presented graphically and discussed. Some previous well-known results are then also deduced from the present analysis.