Effects of Rotation and Heat Source on MHD Free Convective Flow on Vertically Upwards Heated Plate with Gravity Modulation in Slip Flow Region

Abstract

In the present paper we study free convective unsteady flow of a viscous incompressible and rotating fluid in a porous medium past an infinite vertical plate with constant heat source and gravity modulation to the flow region. Expressions for velocity, temperature and concentration distribution are obtained by using perturbation technique. Also the expressions for the skin-friction coefficient, rate of heat transfer and Sherwood number are derived. The effects of Prandtl Number ($P_{r}$), Grashof number ($G_{r}$), modified Grashof number ($G_{c}$), frequency of oscillation, rotation parameter (E), heat source parameter ($S^{*}$), gravity modulation parameter (α) and permeability parameter (K₀) are analyzed and studied through graphs. It is found that primary velocity of fluid decreases with increase of rotation parameter whereas secondary velocity is increased near to the wall, and rate of heat transfer decreases with increase of heat source parameter.

Appendix

$$ A_{1} = \frac{{R_{1} S_{c} }}{{R_{1}^{2} - S_{c} R_{1} - S_{c} \left( {K_{c} + \frac{i\omega }{4}} \right)}} $$

$$ A_{2} = \frac{{R_{5} P_{r} }}{{R_{5}^{2} - P_{r} R_{5} + P_{r} \left( {S_{H} - \frac{i\omega }{4}} \right)}} $$

$$ A_{3} = \frac{{ - G_{r} }}{{R_{5}^{2} - R_{5} - \left( {L + \frac{1}{{k_{0} }}} \right)}} $$

$$ A_{4} = \frac{{ - G_{c} }}{{R_{1}^{2} - R_{1} - \left( {L + \frac{1}{{k_{0} }}} \right)}} $$

$$ A_{5} = \frac{{R_{9} C_{9} }}{{R_{9}^{2} - R_{9} - \left( {L + \frac{1}{{k_{0} }} + \frac{i\omega }{4}} \right)}} $$

$$ A_{6} = \frac{{ - \left( {1 - A_{2} } \right)G_{r} }}{{R_{7}^{2} - R_{7} - \left( {L + \frac{1}{{k_{0} }} + \frac{i\omega }{4}} \right)}} $$

$$ A_{7} = \frac{{R_{5} A_{3} - A_{2} G_{r} - \alpha G_{r} }}{{R_{5}^{2} - R_{5} - \left( {L + \frac{1}{{k_{0} }} + \frac{i\omega }{4}} \right)}} $$

$$ A_{8} = \frac{{ - \left( {1 - A_{1} } \right)G_{c} }}{{R_{3}^{2} - R_{3} - \left( {L + \frac{1}{{k_{0} }} + \frac{i\omega }{4}} \right)}} $$

$$ A_{9} = \frac{{R_{1} A_{4} - A_{1} G_{c} - \alpha G_{c} }}{{R_{1}^{2} - R_{1} - \left( {L + \frac{1}{{k_{0} }} + \frac{i\omega }{4}} \right)}} $$

$$ R_{1} = \frac{{S_{c} + \sqrt {S_{c}^{2} + 4K_{c} S_{c} } }}{2} $$

$$ R_{3} = \frac{{S_{c} + \sqrt {S_{c}^{2} + 4\left( {K_{c} + \frac{i\omega }{4}} \right)S_{c} } }}{2} $$

$$ R_{5} = \frac{{P_{r} + \sqrt {P_{r}^{2} - 4S_{H} P_{r} } }}{2} $$

$$ R_{7} = \frac{{P_{r} + \sqrt {P_{r}^{2} + 4\left( {S_{H} - \frac{i\omega }{4}} \right)P_{r} } }}{2} $$

$$ R_{9} = \frac{{1 + \sqrt {1 + 4\left( {L + \frac{1}{{k_{0} }}} \right)} }}{2} $$

$$ R_{11} = \frac{{1 + \sqrt {1 + 4\left( {L + \frac{1}{{k_{0} }} + \frac{i\omega }{4}} \right)} }}{2} $$

$$ K_{12} = \frac{{R_{3} S_{c} }}{{R_{3}^{2} - S_{c} R_{3} - \left( {K_{c} + \frac{i\omega }{4}} \right)S_{c} }} $$

$$ C_{9} = \frac{{ - \left( {1 + R_{5} \varphi_{1} } \right)A_{3} - \left( {1 + R_{1} \varphi_{1} } \right)A_{4} }}{{1 + R_{9} \varphi_{1} }} $$

$$ C_{11} = \frac{{ - \left( {1 + R_{9} \varphi_{1} } \right)A_{5} - \left( {1 + R_{7} \varphi_{1} } \right)A_{6} - \left( {1 + R_{5} \varphi_{1} } \right)A_{7} - \left( {1 + R_{3} \varphi_{1} } \right)A_{8} - \left( {1 + R_{1} \varphi_{1} } \right)A_{9} }}{{1 + R_{11} \varphi_{1} }} $$