# 1. GOVERNING EQUATIONS: Equation of continuity Modified

March 25, 2019 Law

1. INTRODUCTION:
This chapter initially explains the basic equations governed by the principle of magnetohydrodynamics. The theory of continuum mechanics is highly essential to study this concept . Navier-stokes equation is obligatory in this field. Michael Faraday in the year 1832 was the first to document the experiment on MHD. It has various modelling jobs i.e., dynamo action, nuclear fusion etc.
Heat and mass transfer are applied numerously in the industrial uses.
This chapter highlights the heat and mass transfer due to various effects such as Grashof number, Sherwood number, Prandtl number, velocity field, skin friction, Nusselt number and induced magnetic field.

1.1 GOVERNING EQUATIONS:
Equation of continuity

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Modified Navier-Stokes equation

EnergyEquation

Species Continuity Equation

Ampere’s law

Gauss’s law of magnetism

Ohm’s law

Magnetic induction equation

This paper considers a flow that is 2D of an electrically conducting fluid (viscous as well as incompressible) under the influence of a uniform magnetic field and induced magnetic field. These are the following assumptions:
(i) The density in the buoyancy force term are non-constant and the rest of the fluid properties are constant.
(ii) The plates are put through constant suction.
(iii) The Eckert number E is assumed to be small.
(iv) The plate is presumed to be electrically non- conducting.
Using the above assumptions with usual boundary layer approximations the equations from (1) to (10) reduces to:
Equation of continuity
which is satisfied with , a constant. …………………………………..(11)
Momentum equation

Energy Equation

Magnetic induction equation

Species Continuity Equation

The boundary conditions are-

It introduces the following non-dimensional quantities:

Its governing equations are:

subject to the boundary conditions

1.3 METHOD OF SOLUTIONS
The solution of the equation (18) subject to the boundary conditions given in the equation (21) is

Under the boundary conditions given in equation(21), to solve the equations (17),(19)and (20),the following form of the solutions are taken:
Putting equations (1.3.2),(1.3.3),(1.3.4) in the equations (17),(19) and (20) and equating the coefficient of like degree terms and neglecting terms of higher order , following differential equations have been derived:

The boundary conditions given by equation (21) reduces to:

The solutions of the equations from (1.3.5) to (1.3.10) subject to the boundary conditions given in the equation (1.3.11) are:

1.4. SKIN FRICTION:
The non-dimensional skin friction at the plate y=0 is given by:

where
and

1.5. RATE OF HEAT TRANSFER:
The non-dimensional heat flux at the plate y=0 in terms of Nusselt number Nu is given by-

where
and
=

1.6. RATE OF MASS TRANSFER:
The non-dimensional mass flux at the plate y=0 in terms of Sherwood number is given by-

=
=

where the constants are –
,
, ,

, ,
, ,

Pm=1,Pm=1.5,Pm=2

1.8.: CONCLUSIONS:
(i) Fluid motion is accelerated by magnetic field which is transverse.
(ii) Increase in M and S increases the velocity asymptotically to the value 1.
(iii) Increase in Schmidt number decreases the skin friction.
(iv)Increase in magnetic Prandtl number increases the induced magnetic field asymptotically to the value 1.

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