Magnetohydrodynamic creeping flow around a weakly permeable spherical particle in cell models

Abstract

The present paper studies the impact of applied uniform transverse magnetic field on the flow of incompressible conducting fluid around a weakly permeable spherical particle bounded by a spherical container. Analytical solution of the problem is obtained using Happel and Kuwabara cell models. The concerned flow is parted in two regions, bounded fluid region and internal porous region, to be governed by Stokes and Darcy’s law respectively. At the interface between the fluid and the permeable region, the boundary conditions used are continuity of normal component of velocity, Saffman’s boundary condition and continuity of pressure. For the cell surface, Happel and Kuwabara models together with continuity in radial component of the velocity has been used. Expressions for drag force, hydrodynamic permeability and Kozeny constant acting on the spherical particle under magnetic effect are presented. Representation of hydrodynamic permeability for varying permeability parameters, particle volume fraction, slip parameter and Hartmann numbers are represented graphically. Also, the magnitude of Kozeny constant for weakly permeable and semipermeable sphere under a magnetic effect has been presented. In limiting cases many important results are obtained.

Appendix

$$\begin{aligned}&S_1=K_{1/2}(\alpha ), \quad S_2= K_{3/2}(\alpha ), \nonumber \\&S_3=K_{1/2}(\alpha /\eta ), \quad S_4= K_{3/2}(\alpha /\eta ), \nonumber \\&U_1=I_{1/2}(\alpha ), \quad U_2= I_{3/2}(\alpha ), \nonumber \\&U_3=I_{1/2}(\alpha /\eta ),\quad U_4= I_{3/2}(\alpha /\eta ), \nonumber \\&W_1= S_1U_4+S_4U_1,\quad W_2=S_2U_4-S_4U_2, \nonumber \\&W_3=W_1\alpha +W_2,\quad W_4= W_1\alpha +3W_2,\nonumber \\&W_5= S_1U_2+S_2U_1,\quad W_6=S_3U_1-S_1U_3,\nonumber \\&W_7= S_3U_2+S_2U_3,\quad W_8= S_3U_4+S_4U_3,\nonumber \\&W_9= W_6\alpha -3W_7,\quad W_{10}=W_5\eta +W_8, \nonumber \\&W_{11}=W_6\alpha - W_7,\quad W_{12}= W_9\beta ^2-W_{11}\alpha ^2,\nonumber \\&W_{13}=W_3\alpha ^2- W_4\beta ^2,\quad W_{14}=2 W_1\beta ^{2}+W_3 \,\alpha ,\nonumber \\&W_{15}=2W_{6}\,\beta ^2+ W_{11}\alpha ,\quad W_{16}=-2W_{12}\,\eta +W_{13}\alpha , \nonumber \\&W_{17}=2W_{16}\,\eta ^3{+}6W_{14}\,\eta ^2,\quad W_{18}=2\alpha \eta W_{15}+\alpha ^2 W_{14}, \nonumber \\&W_{19}=W_2\alpha ^2+W_4,\quad W_{20}={W_9}-{W_7} {{\alpha }^{2}}, \nonumber \\&W_{21}={W_2}\alpha +{W_4}, \quad W_{22}={W_6}-{W_7} \alpha , \nonumber \\&W_{23}=2 W_{21}{{\beta }^{2}}+{W_{19}}\, \alpha , \quad W_{24}=2 W _{22}\, {{\beta }^{2}}+{W_{20}} \,\alpha , \nonumber \\&Z=6\eta ^2+\alpha ^2, \nonumber \\&\Delta _{1}=W_{19}{\beta }^{2}-2{W_2}\, {\alpha }^{2},\nonumber \\&\Delta _{2}=W_{19}( 2( \beta ^2-\alpha ^2) {\eta }^{3}-{{\alpha }^{2}}) -2 \alpha \left( {W_2} \alpha +{W_1}\right) {{\beta }^{2}}, \nonumber \\&\Delta _{3}=W_{17}+W_{18},\nonumber \\&\Delta _{4}=W_{4}Z+2W_{9}\alpha \eta ,\nonumber \\&\Delta _{5}=2 \alpha ( {W_{20}} {{\beta }^{2}}+2 {W_7} {{\alpha }^{2}}) \eta +Z {\Delta _1}, \nonumber \\&\Delta _6=-4 {W_{20}} ( {{\beta }^{2}}{-}{{\alpha }^{2}}) {{\eta }^{4}}{+}6 {W_{23}} {{\eta }^{2}}{+}2 {W_{24}} \alpha \eta -{\Delta _2} \alpha , \nonumber \\&\Delta _7=-12 ( { W }_5\,{{\beta }^{2}} {{\eta }^{4}}+{W_8}( {{\beta }^{2}}-{{\alpha }^{2}}){{\eta }^{3}}),\nonumber \\&\Delta _8= {W_1}{Z}-2 {W_9}{{\eta }^{4}}{-}\eta ( {W_4} \alpha {{\eta }^{2}}{+}6 {W_{10}}\, {{\eta }^{3/2}}-2 {W_6} \alpha ). \end{aligned}$$