Effect of Newtonian Heating/Cooling on Hydromagnetic Free Convection in Alternate Conducting Vertical Concentric Annuli

Abstract

This paper presents the effects of the Newtonian heating/cooling and the radial magnetic field on steady hydromagnetic free convective flow of a viscous and electrically conducting fluid between vertical concentric cylinders by neglecting compressibility effect. The derived governing equations of the model are first recast into the non-dimensional simultaneous ordinary differential equations using the suitable non-dimensional variables and parameters. By obtaining the exact solution of the simultaneous ordinary differential equations, the effects of the Hartmann number as well as the Biot number on the velocity, induced magnetic field, induced current density, Nusselt number, skin-friction and mass flux of the fluid are presented by the graphs and tables. The effect of the Biot number is to increase the velocity, induced magnetic field and induced current density in the case of the Newtonian heating and vice versa in the case of the Newtonian cooling, but the effect of Hartmann number is to decrease all above fields. Further, graphical representation shows that the velocity and induced magnetic field are rapidly decreasing, with increasing the Hartmann number, when one of the cylinders is conducting compared with when both the cylinders are non-conducting.

Appendix

$$\begin{array}{*{20}l} {{A}_{10} = \left\{ {\frac{{({1} + {R})}}{{({Bi}\,{\log}\,{\lambda } + {1})}}} \right\}{,} {A}_{11} = \left\{ {{A}_{10} {B}_{16} + {B}_{17} } \right\}{,} {A}_{21} = {A}_{31} = \left\{ {{A}_{10} {B}_{24} + {B}_{25} } \right\},} \hfill \\ {{A}_{12} = \left\{ {{A}_{10} {B}_{18} + {B}_{19} } \right\}, {A}_{23} = {A}_{33} = {0,} {A}_{22} = {A}_{32} = \left\{ {{A}_{10} {B}_{26} + {B}_{27} } \right\},} \hfill \\ {{A}_{13} = - \left\{ {{A}_{11} + {A}_{12} + {A}_{10} {B}_{12} + {B}_{13} {R}} \right\}, {B}_{11} = {B}_{21} = {B}_{31} = - \left\{ {\frac{Bi}{{({4} - {Ha}^{2} )}}} \right\},} \hfill \\ {{A}_{14} = \left\{ {\frac{1}{Ha}({A}_{11} - {A}_{12} ) - \frac{{{A}_{10} }}{4}({B}_{11} - {2B}_{12} ) + \frac{R}{2}{B}_{13} } \right\}{,B}_{16} = \left\{ {\frac{{({B}_{14} + {B}_{15} )}}{{{2(1} - {\lambda }^{Ha} {)}}}} \right\},} \hfill \\ {{A}_{24} = \left\{ {\frac{1}{Ha}({A}_{21} - {A}_{22} ) - \frac{{{A}_{10} }}{4}({B}_{11} - {2B}_{12} ) + \frac{R}{2}{B}_{13} } \right\},{D}_{27} = \left\{ {\frac{{{R}{\lambda }^{2} {\log}\,{\lambda }}}{{{4 (\lambda }^{{ - {2}}} - {\lambda }^{2} {)}}}} \right\},} \hfill \\ {{A}_{34} = \left[ {\frac{1}{Ha}({A}_{31} {\lambda }^{Ha} - {A}_{32} {\lambda }^{{ - {Ha}}} ) - \frac{{{A}_{10} {\lambda }^{2} }}{4}\left\{ {{B}_{11} ({1} - {2\log}\,{\lambda }) - {2B}_{12} } \right\} + {B}_{38} {R}{\lambda }^{2} } \right]{,}} \hfill \\ {{B}_{12} = {B}_{22} = {B}_{32} = \left\{ {\frac{4Bi}{{({4} - {Ha}^{2} )^{2} }} - \frac{1}{{({4} - {Ha}^{2} )}}} \right\},{B}_{13} = {B}_{23} = {B}_{33} = \left\{ {\frac{1}{{({4} - {Ha}^{2} )}}} \right\},} \hfill \\ {{B}_{14} = \left\{ {{B}_{11} {\lambda }^{2} {\log}\,{\lambda } + {B}_{12} ({\lambda }^{2} - {1})} \right\},{B}_{24} = {B}_{34} = \left\{ {\frac{{{B}_{11} {\lambda }^{2} {\log \lambda } + {B}_{12} ({\lambda }^{2} - {\lambda }^{{ - {Ha}}} )}}{{({\lambda }^{{ - {Ha}}} - {\lambda }^{Ha} )}}} \right\},} \hfill \\ {{B}_{15} = \frac{M}{4}\left\{ {({2B}_{12} - {B}_{11} )({\lambda }^{2} - {1}) + {2B}_{11} {\lambda }^{2} {\log}\,{\lambda }} \right\},{B}_{25} = {B}_{35} = \left\{ {\frac{{{B}_{13} {R(}{\lambda }^{2} - {\lambda }^{{ - {Ha}}} {)}}}{{({\lambda }^{{ - {Ha}}} - {\lambda }^{Ha} )}}} \right\},} \hfill \\ {{B}_{26} = {B}_{36} = \left\{ {\frac{{{B}_{11} {\lambda }^{2} {\log}\,{\lambda } + {B}_{12} ({\lambda }^{2} - {\lambda }^{Ha} )}}{{({\lambda }^{Ha} - {\lambda }^{{ - {Ha}}} )}}} \right\},{B}_{17} = \left\{ {\frac{{{B}_{13} {R(}{\lambda }^{2} - {1)}\left( {{1} + \frac{Ha}{2}} \right)}}{{{2(1} - {\lambda }^{Ha} {)}}}} \right\},} \hfill \\ {{B}_{27} = {B}_{37} = \left\{ {\frac{{{B}_{13} {R(}{\lambda }^{2} - {\lambda }^{Ha} {)}}}{{({\lambda }^{Ha} - {\lambda }^{{ - {Ha}}} )}}} \right\},{B}_{18} = \left\{ {\frac{{({B}_{14} - {B}_{15} )}}{{{2(1} - {\lambda }^{{ - {Ha}}} {)}}}} \right\},{B}_{38} = \left( {\frac{{{B}_{13} }}{2}} \right),} \hfill \\ {{B}_{19} = \left\{ {\frac{{{B}_{13} {R(}{\lambda }^{2} - {1)}\left( {{1} - \frac{Ha}{2}} \right)}}{{{2(1} - {\lambda }^{{ - {Ha}}} {)}}}} \right\},{C}_{11} = \left\{ {\frac{{{A}_{10} {D}_{17} }}{{{2(1} - {\lambda }^{2} {)}}} + \frac{{{R(1} - {\lambda }^{2} + {4}{\lambda }^{2} {\log}\,{\lambda }{)}}}{{{16(1} - {\lambda }^{2} {)}}}} \right\}{,}} \hfill \\ \end{array}$$

$$\begin{array}{*{20}l} {{C}_{12} = \left\{ {\frac{{{A}_{10} {D}_{16} }}{{{2(1} - {\lambda }^{{ - {2}}} {)}}} + \frac{{{R(}{\lambda }^{2} - {1)}}}{{{16(1} - {\lambda }^{{ - {2}}} {)}}}} \right\},{C}_{21} = ({A}_{10} {D}_{26} + {D}_{27} ){,C}_{13} = - ({C}_{11} + {C}_{12} ),} \hfill \\ {{C}_{14} = {C}_{24} = \left\{ {\frac{1}{2}({C}_{11} - {C}_{12} ) + {A}_{10} {D}_{15} - \frac{R}{16}} \right\}{,C}_{22} = ({A}_{10} {D}_{28} + {D}_{29} ){,}} \hfill \\ {{C}_{23} = {C}_{33} = {0,C}_{31} = ({A}_{10} {D}_{36} + {D}_{37} ){,C}_{32} = ({A}_{10} {D}_{38} + {D}_{39} ){,}} \hfill \\ {{C}_{34} = \left[ {\frac{1}{2}({C}_{11} {\lambda }^{2} - {C}_{12} {\lambda }^{{ - {2}}} ) + {A}_{10} {\lambda }^{2} \left\{ {{(D}_{33} {\log}\,{\lambda } + {D}_{34} {)\log}\,{\lambda } + {D}_{35} } \right\} - \frac{R}{16}{\lambda }^{2} ({1} - {\log}\,{\lambda })} \right]{,}} \hfill \\ {{D}_{11} = {D}_{21} = {D}_{31} = - \left( {\frac{Bi}{8}} \right),{D}_{29} = \left\{ {\frac{{{R}{\lambda }^{2} {\log}\,{\lambda }}}{{{4(}{\lambda }^{2} - {\lambda }^{{ - {2}}} {)}}}} \right\},{D}_{12} = {D}_{22} = {D}_{32} = \left( {\frac{Bi}{16} - \frac{1}{4}} \right),} \hfill \\ {{D}_{13} = {D}_{23} = {D}_{33} = \left( {\frac{{{D}_{11} }}{2}} \right),{D}_{14} = {D}_{24} = {D}_{34} = \left\{ {\frac{{({D}_{12} - {D}_{11} )}}{2}} \right\},} \hfill \\ {{D}_{15} = {D}_{25} = {D}_{35} = \left\{ {\frac{{({D}_{11} - {D}_{12} )}}{4}} \right\},{D}_{28} = \left\{ {\frac{{{\lambda }^{2} {\log}\,{\lambda }{(D}_{21} {\log}\,{\lambda } + {D}_{22} {)}}}{{({\lambda }^{2} - {\lambda }^{{ - {2}}} )}}} \right\}{,}} \hfill \\ {{D}_{16} = \left\{ {{D}_{11} {\lambda }^{2} ({\log}\,{\lambda })^{2} + {D}_{12} {\lambda }^{2} {\log}\,{\lambda } - {2}{\lambda }^{2} {\log}\,{\lambda }{(D}_{13} {\log}\,{\lambda } + {D}_{14} {)} - {2D}_{15} ({\lambda }^{2} - {1})} \right\},} \hfill \\ {{D}_{17} = \left\{ {{D}_{11} {\lambda }^{2} ({\log}\,{\lambda })^{2} + {D}_{12} {\lambda }^{2} {\log}\,{\lambda } + {2}{\lambda }^{2} {\log}\,{\lambda }{(D}_{13} {\log}\,{\lambda } + {D}_{14} {)} + {2D}_{15} ({\lambda }^{2} - {1})} \right\},} \hfill \\ {{D}_{26} = \left\{ {\frac{{{\lambda }^{2} {\log}\,{\lambda }{(D}_{21} {\log}\,{\lambda } + {D}_{22} {)}}}{{({\lambda }^{{ - {2}}} - {\lambda }^{2} )}}} \right\}{,D}_{36} = \left\{ {\frac{{{\lambda }^{2} {\log}\,{\lambda }}}{{({\lambda }^{{ - {2}}} - {\lambda }^{2} )}}({D}_{31} {\log}\,{\lambda } + {D}_{32} )} \right\},} \hfill \\ {{D}_{37} = \left\{ {\frac{{{R}{\lambda }^{2} {\log}\,{\lambda }}}{{{4(}{\lambda }^{{ - {2}}} - {\lambda }^{2} {)}}}} \right\},{D}_{38} = \left\{ {\frac{{{\lambda }^{2} {\log}\,{\lambda }}}{{({\lambda }^{2} - {\lambda }^{{ - {2}}} )}}({D}_{31} {\log}\,{\lambda } + {D}_{32} )} \right\}{,D}_{39} = \left\{ {\frac{{{R}{\lambda }^{2} {\log}\,{\lambda }}}{{{4(}{\lambda }^{2} - {\lambda }^{{ - {2}}} {)}}}} \right\}.} \hfill \\ \end{array}$$