Variable Thermal Conductivity and Heat Source Effect on Hydromagnetic Flow of Viscous Stratified Fluid Past a Vertical Porous Plate Through Porous Medium
Khem Chand1, Bharti Sharma²
1Department of Mathematics and Statistics, H.P. University-Shimla, India.
2Research Scholar, Department of Mathematics and Statistics, H.P. University-Shimla, India.
*Corresponding Author E-mail: khemthakur99@gmail.com, bharti926sharma@gmail.com
ABSTRACT:
An analysis of unsteady convective flow of viscous stratified fluid through porous medium past an infinite vertical porous plate in presence of heat source under the influence of magnetic field applied normal to the plane of the plate has been presented. The dimensionless governing equations are solved by using Laplace transform technique. The solution are expressed in terms of exponential and complimentary error function. The effects of various parameters on velocity and temperature field are depicted graphically, whereas skin friction and rate of heat transfer are presented in tabular form and discussed
KEYWORD:
INTRODUCTION:
The study of MHD free convective flow with heat transfer has attracted the attention of a number of scholars due to its diverse applications. In astrophysics and geophysics it is applied to study the stellar and solar structures, radio propagation through the ionosphere etc. In engineering we find its applications in MHD pumps, bearings and converters etc. From technological point of view, MHD convection flow problem are also very significant in the fields of stellar and planetary magnetosphere, aeronautics, chemical engineering and electronics. Convective heat transfer in porous media has been the subject of many recent studies due to practical applications. The attention stems from the wide range of engineering applications such as transpiration cooling, porous bearing, solar collectors, nuclear waste repository, energy storage units, electronic cooling, thermal insulation, packed bed heat exchangers, heat pipes, drying technology, catalytic reactors, petroleum industry and geothermal systems. Channabaspa and Ranganna (1976) studied the flow of viscous stratified fluid with variable viscosity past a porous bed and observed that stratification may provide a technique for studying the pore size in porous medium. Viscosity variations play an important role in petroleum industry, geothermal system and thermal insulation. Choudhury and Das (2000) presented magneto-hydrodynamic boundary layer flows of non-Newtonian fluid past a flat plate. Das et al. (2009) studied mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source. Gupta and Sharma (1978) discussed stratified fluid considering a porous bed and a moving impermeable plate. The effects of radiation on MHD flow due to moving vertical porous plate with variable temperature was investigated by Garg (2013). Kim (2000) presented unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction. Mukhopadhyay and Mandal (2015) investigated magneto hydrodynamic mixed convection slip flow and heat transfer over a vertical porous plate. Nayak et al. (2013) studied unsteady MHD flow of a visco-elastic fluid along vertical porous surface with chemical reaction. Park and Hayun (1998) studied unsteady flow of stratified viscous fluid considering a vertical buoyant. Singh (1986) use Laplace transform technique to study unsteady stratified Couette flow. Shaprio and Fedorovich (2006) examined natural convective stratified flow along a vertical plate for two different physical configurations. Non-Darcy free convection from vertical surfaces in thermally stratified porous media has been investigated by Singh and Tiwari (1993). Singh and Singh (2003) studied heat and mass transfer in MHD flow of viscous fluid past a vertical plate under oscillatory suction velocity. Das et al. (2011) studied the effect of heat source on MHD free convection flow past an oscillating porous plate in the slip flow regime. The objective of the present section is to analyze the effect of variable thermal conductivity and heat source on MHD unsteady convective flow of incompressible viscous stratified fluid through porous medium past an infinite vertical porous plate.
MATHEMATICAL FORMULATION:
Consider an
unsteady hydro magnetic free convective flow of a incompressible, electrically
conducting viscous stratified fluid past an infinite vertical porous plate
embedded in porous medium with constant suction velocity 
.
Coordinate system is chosen in such a way that 
is
taken along the plate in the upward direction, 
is
normal to it and 
perpendicular
to 
. A
uniform magnetic field of strength 
is
applied perpendicular to the plate. Initially at time 
both
the plate and fluid are at rest and assumed to be same temperature as at the
edge of the boundary layer i.e. 
respectively.
At time 
the
plate at 
starts
moving with a velocity 
in its
own plane and is heated with temperature 
. Since
the plate is infinitely long in 
and 
directions,
therefore
all the physical quantities except pressure depends upon 
and 
only.
Figure 1 Geometrical configuration of the problem.
RESULTS AND DISCUSSION:
MHD free
convective flow of viscous incompressible stratified fluid under the effect of
variable viscosity and heat source through porous medium past a vertical moving
plate in the presence of variable magnetic field is analysed. The closed form
solutions for the velocity, temperature, skin friction and rate of heat
transfer are obtained analytically and evaluated numerically for different
value of governing parameters. To have better insight of physical problem the
variation of physical quantities with flow parameters are shown graphically. To
be realistic the value of Prandtl number 
are
chosen to be 0.71 and 7 which correspond to air and water respectively. It is
clear from figure 2 that the permeability parameter 
enhance
the fluid velocity. The figure also reveal that fluid velocity reduces with the
increasing Hartmann number 
. From
figure 3 we observed that fluid velocity increases with stratification
parameter 
. In
this figure curve with 
present
the absence of stratification, which clearly show that velocity profile
decreases in the absence of stratification parameter. Singh et al. (2012)
observed that the velocity profile increases with increase of stratification
(
). Hence
our result agrees with Singh et al. (2012) in the absence of convection and
suction. Figure 4 illustrate that fluid temperature decreases with the
increasing Prandtl number 
. The
figure also reveal that fluid temperature increases with the increase of heat
source parameter. From figure 5 we observed that temperature increases with the
increase of stratification parameter whereas it decreases in the absence of
stratification 
.Variation
in velocity profile with heat source parameter shown in table 1. We observed
from this table that with the increase in heat source parameter velocity
profile increases. Table 2 present the variation in skin friction coefficient
(τ). It observed from this table that an increase in permeability
parameter and stratification parameter lead to increase in coefficient of skin
friction, while an increase in Hartmann number lead to decrease in coefficient
of skin friction. The variation in Nusselt number is listed in Table 3 It is
noticed from this table that the increase in heat source parameter and
stratification parameter the Nusselt number increases. The values in the table
clearly show that rate of heat transfer decreases with the increases in Prandtl
number.
CONCLUSIONS:
The main conclusion of this study are:
1. The stratification increases the velocity and temperature profile.
2. The skin friction and rate of heat transfer increases with the increase of stratification parameter.
3. The permeability parameter accelerates and the magnetic field parameter diminishes the velocity profile.
4. The heat source parameter enhances and the Prandtl number diminishes the temperature profile.
