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Similarity analysis in magnetohydrodynamics: effects of hall and ion-slip currents on free convection flow and mass transfer of a gas past a semi-infinite vertical plate

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Summary

The effects of Hall and ion-slip currents on a steady free convection flow and mass transfer past a semi-infinite vertical plate in a viscous incompressible electrically conducting fluid are investigated when the fluid is subjected to a strong non-uniform magnetic field. The similarity solutions are obtained using a scaling group of transformations. These are the only symmetry transformations admitted by the field equations. The derived fundamental equations under the assumption of a small magnetic Reynolds number are solved numerically by employing the shooting method. The axial and transverse components of the velocity function profiles are computed for various values of magnetic parameters, Hall and ionslip current parameters.

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Abbreviations

B o :

imposed magnetic field

\(\mathop B\limits^ \to \) :

magnetic induction vector

\(\mathop g\limits^ \to \) :

acceleration due to gravity

\(\mathop j\limits^ \to \) :

electric current density vector

\(\mathop E\limits^ \to \) :

electric field vector

x, y, z :

Cartesian coordinates

u, v, w :

velocity components along (x, y, z)-axes

T :

temperature

C :

concentration

C p :

specific heat

k :

thermal conductivity

D :

diffusion coefficient

p :

fluid pressure

U ∞ :

free stream velocity

θ:

dimensionless temperature

φ:

dimensionless concentration

M :

magnetic parameter\(\left( {\frac{{\sigma B_0^2 v}}{{\rho U_\infty ^2 }}} \right)\)

Pr:

Prandtl number\(\left( {\frac{{\varrho vC_p }}{k}} \right)\)

Sc:

Schmidt number\(\left( {\frac{v}{D}} \right)\)

Gr:

Grashof number\(\left( {\frac{{vg\beta (T_W - T_\infty )}}{{U_\infty ^3 }}} \right)\)

Gc :

modified Grashof number\(\left( {\frac{{vg\beta *(C_W - C_\infty )}}{{U_\infty ^3 }}} \right)\)

η:

similarity variable

F 1(η),F 2(η),F 3(η),F 4(η),F 5(η):

similarity functions

c 1;c 2;c 3;c 4;c 5,c 6,c 7 :

constants

ν:

kinematic viscosity

ρ:

density

β :

coefficient of thermal expansion

β*:

coefficient of expansion with concentration

β e :

Hall parameter

β i :

ion-slip parameter

σ:

electric conductivity

τ e :

electron collision time

τ i :

ion collision time

ω w :

cyclotron frequency of an electron

ω i :

cyclotron frequency of an ion

μ e :

magnetic permeability

λ:

parameter of transformations

x, y, z :

quantities along (x, y, z)-axes

w :

wall condition

∞:

free stream condition

o :

constant condition

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Author notes

  1. A. A. Megahed

    Present address: Department of Mathematical Engineering, Faculty of Engineering, Cairo University, Gamaa Street, 12613, Giza, Egypt

  2. S. R. Komy & A. A. Afify

    Present address: Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt

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    1. A. A. Megahed
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    3. A. A. Afify
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    Megahed, A.A., Komy, S.R. & Afify, A.A. Similarity analysis in magnetohydrodynamics: effects of hall and ion-slip currents on free convection flow and mass transfer of a gas past a semi-infinite vertical plate. Acta Mechanica 151, 185–194 (2001). https://doi.org/10.1007/BF01246917

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