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MHD Mass Transfer Flow Past an Impulsively Started Semi-Infinite Vertical Plate with Soret Effect and Ramped Wall Temperature

  • N. AhmedEmail author
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

Abstract

An exact solution to the problem of a hydromagnetic natural convective mass transfer flow of an incompressible viscous electrically conducting non-Gray optically thin fluid past an impulsively started semi-infinite vertical plate with ramped wall temperature in presence of appreciable radiation, thermal diffusion and uniform transverse magnetic field is presented. The magnetic Reynolds number is assumed to be small enough to neglect the induced hydromagnetic effects. Closed form Laplace Technique is adopted to get the exact solutions of the resultant non-dimensional governing equations. The influences of thermal radiation, ramped parameter, magnetic field, thermal diffusion and time on the flow and transport characteristics are studied graphically.

Keywords

Optically thin Thermal radiation Thermal diffusion Natural convection 

Nomenclature

\( {\textbf{B}} \)

Magnetic flux density;

\( B_{0} \)

Strength of the applied magnetic field, \( {\text{Tesla or}}\, \frac{\text{Weber}}{{m^{2} }} \);

\( C \)

Molar species concentration, \( \frac{\text{kmol}}{{{\text{m}}^{ 3} }} \);

\( C_{p} \)

Specific heat at constant pressure, \( \frac{\text{J}}{{{\text{kg}}\,{\text{K}}}} \);

\( C_{\infty } \)

Concentration far away from the plate, \( \frac{\text{kmol}}{{{\text{m}}^{ 3} }} \);

\( C_{w} \)

Species concentration at the plate, \( \frac{\text{kmol}}{{{\text{m}}^{ 3} }} \);

\( D_{M} \)

Mass diffusivity, \( \frac{{{\text{m}}^{ 2} }}{\text{s}} \);

\( D_{T} \)

Molar thermal diffusivity, \( \frac{{{\text{m}}^{ 2} {\text{kmol}}}}{{{\text{K}}\,{\text{s}}}} \);

\( e_{b\lambda } \)

Planck function;

\( {\textbf{g}} \)

Gravitational acceleration vector;

\( g \)

Acceleration due to gravity, \( \frac{\text{m}}{{{\text{s}}^{ 2} }} \);

\( {\text{Gr}} , \)

Thermal Grashof number;

\( {\text{Gm}} \)

Solutal Grashof number;

\( {\mathbf{J}} \)

Current density vector;

\( K_{\lambda } , \)

Absorption coefficient; \( \frac{ 1}{\text{m}} \)

M

Magnetic parameter;

p

Pressure, \( \frac{\text{N}}{{{\text{m}}^{ 2} }} \);

\( \Pr \)

Prandtl number;

\( {\mathbf{q}} \)

Fluid velocity vector;

\( {\mathbf{q}}_{{\mathbf{r}}} \)

Radiative heat flux vector;

\( q_{r} \)

Radiation heat flux, \( \frac{\text{W}}{{{\text{m}}^{ 2} }} \);

\( Q \)

Radiation parameter;

\( Ra \)

Ramped parameter;

\( {\text{Sc}} \)

Schmidt number;

\( {\text{Sr}} \)

Soret number;

\( t^{\prime} \)

Time, s;

\( t_{0} \)

Characteristic time, \( {\text{s}} \);

\( T \)

Temperature, \( {\text{K}} \);

\( T_{w} \)

Isothermal temperature, \( {\text{K}} \);

\( T_{\infty } \)

Temperature far away from the plate, \( {\text{K}} \);

\( u^{\prime} \), \( x^{\prime} \)

Component of fluid velocity, \( \frac{{\text{m}}}{{\text{s}}} \);

\( U_{0} \)

Plate velocity, \( \frac{{\text{m}}}{{\text{s}}} \);

\( \left( {x^{\prime},y^{\prime},z^{\prime}} \right) \)

Cartesian coordinate system, \( \left( {{\text{m}},{\text{m}},{\text{m}}} \right) \);

Greek symbols

\( \sigma \)

Electrical conductivity, \( \frac{ 1}{{\left( {{\text{Ohm}} \times {\text{m}}} \right)}} \);

\( \uprho \)

Fluid density, \( \frac{\text{kg}}{{{\text{m}}^{ 3} }} \);

\( \uprho_{\infty } \)

Fluid density far away from the plate, \( \frac{\text{kg}}{{{\text{m}}^{ 3} }} \);

\( \upmu \)

Coefficient of viscosity, \( \frac{{\text{kg}}}{{\text{ms}}}\;{\text{or}}\;\frac{{{\text{Ns}}}}{{{\text{m}}^{2} }} \);

\( \upkappa \)

Thermal conductivity, \( \frac{\text{W}}{{{\text{m}}\,{\text{K}}}} \);

\( \upbeta \)

Coefficient of thermal expansion, \( \frac{1}{\text{K}} \);

\( {\overline{\upbeta }} \)

Coefficient of solutal expansion, \( \frac{ 1}{\text{kmol}} \);

\( \upupsilon \)

Kinematic viscosity, \( \frac{{{\text{m}}^{ 2} }}{\text{s}} \);

\( {\upvarphi } , \)

Viscous dissipation of energy per unit volume, \( \frac{\text{J}}{{{\text{m}}^{3} }} \);

Subscripts

\( w \)

Refers to physical quantities at the plate;

\( \infty \)

Refers to physical quantities far away from the plate;

AMS Subject Classification

76W05 

PACS

44.27. +g 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsGauhati UniversityGuwahatiIndia

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