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MHD Convective Flow of Stratified Viscous Fluid Past of Flat Plate in Slip Flow Regime: Analysis with Variable Thermal Conductivity and Heat Source

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Abstract

Convective flow of a unsteady, incompressible, viscous stratified fluid through porous medium past a moving porous plate under slip boundary conditions for velocity field and jump in temperature in the presence of an absorption type heat source under the influence of magnetic field applied normal to the flow is studied. Using regular perturbation technique, the solutions for velocity and temperature distribution are obtained. The expressions for skin-friction and rate of heat transfer are also derived. The effects of various parameters on velocity field and temperature field are depicted graphically, whereas skin-friction and rate of heat transfer are presented in tabular form and discussed.

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Abbreviations

B :

Variable magnetic field

B 0 :

Magnetic field component along y*-axis

K T :

Thermal conductivity of the fluid

\( K_{{T_{0} }} \) :

Constant thermal conductivity

Q :

Dimensional heat source

Q 0 :

Constant heat source

K′:

Permeability of the medium

T :

Non-dimensional temperature of the fluid

t :

Non-dimensional time

u :

Mean velocity

U 0 :

Scale of free stream velocity

y :

Dimensional coordinate

T 0 :

Constant temperature

T w :

Temperature of the wall

n :

Frequency

m 1 :

Maxwell reflection coefficient

L :

Mean free path

a :

Thermal accommodation

β :

Small positive number

ε:

Small parameter

γ :

Spin gradient viscosity

μ :

Variable viscosity of the fluid

μ 0 :

Constant viscosity of the fluid at y = 0

ρ :

Variable density of the fluid

ρ 0 :

Constant density of the fluid at y = 0

σ :

Variable electrical conductivity of the fluid

σ 0 :

Constant electrical conductivity of the fluid at y = 0

υ 0 :

Constant dynamatic viscosity of the fluid at y = 0

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Authors and Affiliations

  1. Department of Applied Science and Humanities, Rama Institute of Engineering & Technology, Mandhana, Kanpur, 209217, UP, India

    N. P. Singh

  2. Department of Mathematics, V. S. S. D. College, Kanpur, 208002, India

    Atul Kumar Singh

  3. Department of Mathematics, C. L. Jain (PG) College, Firozabad, 283203, India

    Ajay Kumar Singh, Usha Singh & Rajiv Kumar

Authors
  1. N. P. Singh
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  2. Atul Kumar Singh
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  3. Ajay Kumar Singh
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  4. Usha Singh
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  5. Rajiv Kumar
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Corresponding author

Correspondence to N. P. Singh.

Appendix

Appendix

$$ R_{1} = \frac{1}{2}\left( {\sqrt {S^{2} + 4\alpha_{0} } - S} \right),\quad R_{2} = \frac{1}{2}\left( {\sqrt {S^{2} + 4M_{1} } - S} \right) $$
$$ K_{1} = \frac{1}{{1 + R_{1} H_{2} }},\quad K_{2} = \frac{1}{{1 + R_{2} H_{1} }}, $$
$$ K_{3} = \frac{{\left( {1 + A_{1} H_{2} } \right)}}{{\left( {1 + A_{1} H_{2} } \right)^{2} + B_{1}^{2} H_{2}^{2} }}, $$
$$ K_{4} = \frac{{B_{1} H_{2} }}{{\left( {1 + A_{1} H_{2} } \right)^{2} + B_{1}^{2} H_{2}^{2} }}, $$
$$ K_{5} = \frac{{1 + A_{2} H_{1} }}{{\left( {1 + A_{2} H_{1} } \right)^{2} + B_{2}^{2} H_{1}^{2} }}, $$
$$ K_{6} = \frac{{B_{2} H_{1} }}{{\left( {1 + A_{2} H_{1} } \right)^{2} + B_{2}^{2} H_{1}^{2} }}, $$
$$ A_{1} = - \frac{S}{2} - \frac{{X_{1} }}{2},\quad A_{2} = - \frac{S}{2} - \frac{{X_{3} }}{2}, $$
$$ B_{1} = - \frac{{X_{2} }}{2},\quad B_{2} = - \frac{{X_{4} }}{2}, $$
$$ G_{1} = S^{2} + 4\alpha_{0} ,\quad G_{2} = 4n\Pr , $$
$$ G_{3} = S^{2} + 4M_{1} ,\quad G_{4} = 4n, $$
$$ X_{1} = \sqrt {\frac{{G_{1} + \sqrt {G_{1}^{2} + G_{2}^{2} } }}{2}} ,\quad X_{2} = \frac{{G_{2} }}{{2X_{1} }}, $$
$$ X_{3} = \sqrt {\frac{{G_{3} + \sqrt {G_{3}^{2} + G_{4}^{2} } }}{2}} ,\quad X_{4} = \frac{{G_{4} }}{{2X_{3} }}, $$

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Singh, N.P., Singh, A.K., Singh, A.K. et al. MHD Convective Flow of Stratified Viscous Fluid Past of Flat Plate in Slip Flow Regime: Analysis with Variable Thermal Conductivity and Heat Source. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 82, 269–274 (2012). https://doi.org/10.1007/s40010-012-0041-9

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