Effect of Newtonian Cooling/Heating on MHD Free Convective Flow Between Vertical Walls with Induced Magnetic Field

Abstract

An analysis is performed for the steady MHD free convective flow between two vertical walls assuming that the fluid is viscous, incompressible, and electrically conducting. The impacts of the Newtonian cooling/heating and induced magnetic field have been considered in the mathematical formulation of the problem. The nondimensionalized simultaneous differential equations, governing the problem, have been solved analytically for the temperature, the velocity, and the induced magnetic field. The manifestations have been made for the induced current density, the skin-friction, and the mass flux. The impact of the Hartmann number, the Biot number, and the magnetic Prandtl number on the velocity, the induced magnetic field, and the induced current density diagrams have been presented by considering a temperature-dependent source/sink. It is inspected that the velocity, the induced magnetic field, and the induced current density diagrams have decreasing tendency with rise in the value of the Hartmann number. Further, it is also noticed that with enhancement in the magnetic Prandtl number the velocity diagram decreases, but the induced magnetic field and the induced current density diagrams have increasing nature. It is beheld that the impression of Newtonian cooling/heating is to reduce/raise the velocity as well as the induced magnetic field and the induced current density. The impacts of the governing parameters on the skin-friction and mass flux have also been concluded dealing with their numerical values given in the tables.

Appendix

$$\begin{array}{*{20}l} {E_{1} = \frac{\sqrt S }{Bi \, \sin \sqrt S + \sqrt S \, \cos \sqrt S },\quad E_{3} = Ha\sqrt {Pm} ,\;\;E_{15} = \frac{1}{{2E_{7} }},} \hfill \\ {E_{2} = \frac{Bi}{Bi \, \sin \sqrt S + \sqrt S \, \cos \sqrt S },\quad E_{4} = \frac{{E_{1} }}{{S + E_{3}^{2} }}, \;\; E_{5} = \frac{{E_{2} }}{{S + E_{3}^{2} }},} \hfill \\ {E_{16} = \frac{{E_{6} }}{2},\quad E_{11} = \frac{{E_{6} }}{{\exp \left( {E_{3} } \right)}},\;\; E_{12} = - \frac{{E_{4} \cos \left( {\sqrt S } \right) + E_{5} \sin \left( {\sqrt S } \right)}}{{\exp \left( {E_{3} } \right)}},} \hfill \\ {E_{13} = - \frac{1}{{E_{7} \exp \left( {E_{3} } \right)}}, \quad E_{17} = - \frac{{E_{4} E_{7} + E_{8} }}{{2E_{7} }},\;\;E_{18} = \frac{{E_{6} }}{2} - \frac{{E_{6} - E_{11} }}{{1 - E_{10} }},} \hfill \\ {E_{14} = \frac{{E_{8} \cos \left( {\sqrt S } \right) + E_{9} \sin \left( {\sqrt S } \right)}}{{E_{7} \exp \left( {E_{3} } \right)}}, \quad E_{19} = \frac{{E_{4} + E_{12} }}{{1 - E_{10} }} - \frac{{E_{4} E_{7} + E_{8} }}{{2E_{7} }},} \hfill \\ {E_{20} = E_{15} + \frac{{E_{13} }}{{1 + E_{10} }},\quad E_{21} = \frac{{E_{6} }}{2} - \frac{{E_{6} }}{{1 + E_{10} }},\;\;E_{22} = \frac{{E_{4} + E_{14} }}{{1 + E_{10} }} - \frac{{E_{4} E_{7} + E_{8} }}{{2E_{7} }},} \hfill \\ {E_{23} = \frac{{E_{19} E_{20} - E_{15} E_{22} }}{{E_{18} E_{20} - E_{15} E_{21} }},\quad E_{24} = \frac{{E_{19} - E_{18} E_{23} }}{{E_{15} }},\;\;E_{25} = E_{17} - E_{15} E_{24} - E_{16} E_{23} ,} \hfill \\ {E_{26} = - E_{4} - E_{25} - E_{6} E_{23} ,\quad E_{27} = E_{6} E_{23} ,\;\;E_{28} = - E_{7} E_{26} ,\;\;E_{29} = E_{7} E_{25} } \hfill \\ {F = - S,\quad F_{3} = Ha\sqrt {Pm} ,\;\;F_{1} = \frac{\sqrt F }{Bi \, \sinh \sqrt F + \sqrt F \, \cosh \sqrt F },} \hfill \\ {F_{2} = \frac{Bi}{Bi \, \sinh \sqrt F + \sqrt F \, \cosh \sqrt F },\quad F_{4} = - \frac{{F_{1} }}{{F - F_{3}^{2} }},\;\;F_{5} = - \frac{{F_{2} }}{{F - F_{3}^{2} }},} \hfill \\ {F_{6} = - \frac{Pm}{{F_{3} }},\quad F_{7} = - \frac{{F_{5} Pm}}{\sqrt F },\;\;F_{8} = - \frac{{F_{4} Pm}}{\sqrt F },\;\;F_{9} = \exp \left( { - 2F_{3} } \right),} \hfill \\ {F_{10} = \frac{1}{{Pm \, \exp \left( {F_{3} } \right)}},\quad F_{14} = - \frac{1}{{2F_{6} }},\;\;F_{11} = - \frac{{F_{4} \cosh \sqrt F + F_{5} \sinh \sqrt F }}{{\exp \left( {F_{3} } \right)}},} \hfill \\ {F_{12} = \frac{1}{{F_{6} \exp \left( {F_{3} } \right)}},\quad F_{17} = \frac{{2F_{12} F_{6} - F_{9} - 1}}{{2F_{6} (1 + F_{9} )}},\;\;F_{18} = \frac{{F_{9} - 1}}{{2Pm(1 + F_{9} )}},} \hfill \\ {F_{13} = - \frac{{F_{7} \cosh \sqrt F + F_{8} \sinh \sqrt F }}{{F_{6} \exp \left( {F_{3} } \right)}},\quad F_{25} = \frac{{F_{21} }}{Pm},\;\;F_{15} = \frac{{2F_{10} Pm - F_{9} - 1}}{{2Pm(1 - F_{9} )}},} \hfill \\ {F_{16} = \frac{{2F_{11} F_{6} + F_{4} F_{6} F_{9} + F_{4} F_{6} + F_{7} - F_{7} F_{9} }}{{2F_{6} (1 - F_{9} )}},\quad F_{24} = - \frac{{F_{21} }}{Pm} - F_{23} - F_{4} ,} \hfill \\ {F_{19} = \frac{{2F_{13} F_{6} + F_{4} F_{6} + F_{7} F_{9} + F_{7} - F_{4} F_{6} F_{9} }}{{2F_{6} (1 + F_{9} )}},\quad F_{21} = \frac{{F_{16} F_{17} - F_{14} F_{19} }}{{F_{15} F_{17} - F_{14} F_{18} }},} \hfill \\ {F_{22} = \frac{{F_{16} - F_{15} F_{21} }}{{F_{14} }},\quad F_{20} = \frac{{F_{7} - F_{4} F_{6} }}{{2F_{6} }},\;\;F_{23} = F_{20} - F_{14} F_{19} - \frac{{F_{21} }}{2Pm},} \hfill \\ {F_{26} = F_{24} F_{6} ,\quad F_{27} = - F_{23} F_{6} .} \hfill \\ \end{array}$$