An exact solution of unsteady free convective
MHD flow past a hot vertical porous plate with variable temperature in slip
flow regime
K D Singh, Khem Chand and Sapna
Department
of Mathematics & Statistics, Himachal Pradesh University, Shimla -5
ABSTRACT:
An analysis of an unsteady free convective flow of a
viscous, incompressible, electrically conducting fluid past an infinite hot
vertical porous plate in slip flow
regime has been carried out under the following assumptions: (i)the suction velocity normal to the plate is constant,
(ii) the plate temperature is spanwise cosinusoidal fluctuating with time, (iii) the difference
between the temperature of the plate and the free stream is moderately large
causing free convection currents and (iv) the transversely applied magnetic
field and magnetic Reynold number are really small
and, hence, the induced magnetic field is negligible. The governing equations
have been solved by adopting complex variable notations and the expressions for
velocity and temperature fields obtained. The transient velocity, temperature,
and amplitude the shear stress and heat transfer at the plate have been
discussed.
KEYWORDS:
Unsteady, free convective, MHD, span wise fluctuating
temperature.
INTRODUCTION:
Free convection flow and heat transfer problems are
always important from the technological point of view because they have many
practical applications. The phenomenon of free convection arises when the
difference between the plate temperature and the free stream temperature is
appreciably large. This process of heat transfer has attracted the attention of
a number of worker owing to its numerous and wide ranging application, e.g., in
cooling of nuclear reactors, aeronautics, chemical engineering etc. These
problems of heat transfer turn out to be very complicated in fluid fuel nuclear
reactors as the wall temperature is not really constant. A number of studies
have appeared in the literature where the temperature of the plate is assumed
to be oscillating in time about a constant non zero mean1-4. The
other possibility of spanwise cosinusoidal
temperature of the vertical porous plate subjected to a uniform suction
velocity has been analyzed by Acharya and Padhy5.
The same problem has been extended further by Verma
et al.6 to non- Newtonian flow of second order fluid. Singh7 generalized
the problem by assuming the plate temperature to be spanwise
cosinusoidal fluctuating with time.
In view of the increasing technical applications using magnetohydrodynamic effect, it is desirable to extend many
of the available viscous hydrodynamics solutions to include the effect of
magnetic fields when the fluid is electrically conducting. Rassow8 has studied the effect of
transverse magnetic field on the two dimensional flow past a flat plate. Singh9-10
analyzed the hydromagnetic effects on the steady and
unsteady three dimensional flow past a flat porous plate. Soundalgekar11
investigated the free convection effects on the steady magnetohydrodynamic
flow past a vertical porous plate. We have extended Singhand
Chand12 work by including the radiative
heat effect along and slip flow regime.
Hence, the object of present paper is to study the
effects of the magnetic field on the free convection flow of an electrically
conducting viscous incompressible and radiating fluid past an infinite hot
vertical porous plate with constant suction when the plate temperature is span
wise cosinusoidal fluctuating with space and time in
slip flow regime.
Mathematical Analysis
We consider an infinite hot porous plate lying vertically
on the 
plane
such that the 
- axis is oriented in the direction of the
buoyancy force and 
- axis is normal to the plane of the plate.
The plate temperature is assumed to be spanwise cosinusoidal fluctuating with
time. All the fluid properties except the variation of density with
temperature, the influence of which is considered only in the body force term,
are considered to be constant. All the physical quantities will be independent
of because the plate is assumed to be infinite in
the - direction. Let (
) the components of velocity in the (
) directions respectively. Further, since the
plate is subjected to a constant suction velocity, i.e. 
, thus, following Acharya and Padhy5,
Verma et.al,6 and Singh7, 
is
independent of 
and so
we assume 
throughout. Hence, the free convective flow of
an electrically conducting, viscous incompressible fluid in the presence of a
magnetic field of uniform strength 
applied perpendicular to the plate along - axis is governed by the following
equations:
Fig.1. Physical configuration of the problem
RESULT AND DISCUSSION:
In order to study the influence of Reynolds number (
), the
Prandtl number(
) , magnetic field parameter (
), the free convection parameter i.e. the
thermal Grashof number (
), the Radiation parameter (
),frequency of oscillation (
) and the slip parameter (
) on the flow field in the boundary layer
region, the numerical values of the fluid velocity, fluid temperature and the
coefficient of the skin friction and the
coefficient of the heat transfer computed from the analytical solutions
reported in the previous section are displaced graphically in figs 2 to 11 for
various values of the Reynolds number, the Prandtl
number, the Grashof number, the Radiation parameter
and the slip parameter. We have taken
the value of 
and 
through our calculation. The effects of
Reynold number on flow velocity have been
shown in Fig.2. it is observed from this figure that velocity profile
increases with increasing Reynold number
). Fig.3 depits the influence of the magnetic
field parameter () on the velocity field. It
is observed from the figure that with the increase in magnetic field parameter
velocity decreases. In Fig.4. the variations in the Prandtl number () has been presented. Here
we analyse that with the increase in Prandtl number 
the velocity decreases. The variation in
velocity profile with radiation parameter () is presented in Fig.5. The
study of this figure shows that the velocity decreases with the enhancement in
the radiation parameter(). The variation of velocity
profile with frequency of oscillation is expressed in Fig.6. Clearly the velocity profile decreases as 
increases. From Fig.7. we observed that with
the increase in Grashof number(), velocity of fluid also
increases. From Fig.8. we observed that with the increase in slip parameter(), velocity profile
increases. The variation in temperature with various parameters is expressed in
Fig.9. Here we analyzed that with the increase in Reynold number (),Prandtl number () and frequency of
oscillation () the temperature profle
increases. Whereas with the increase in radiation parameter (), temperature profile
decreases. Fig.10.shows the variation of
skin friction with requency of oscillation for various parameters.
Fig.11. shows the variation of Nusselt number (Nu) with frequency of
oscillation for various parameters. Table.1. shows the variation of phase of
skin friction with various parameters. In this table we observed that with the increase in Reynold
number 
,Prandtl number () and slip parameter (h) the
phase of skin friction increases and with the increase in magnetic field
parameter (M) and radiation parameter (N),the phase of skin friction decreases.
The phase does not change with Grashof number 
). Table.2. show the
variation of phase of Nusselt number with various parameters. Here we analyzed
that with the increase in Reynold number , Prandtl number (), the phase of Nusselt
number decreases whereas with the increase in radiation parameter(N) phase of
Nusselt number increases.
Fig.2. Variation in velocity profile with various value of 
number and 
, 
, 
, 
, 
and 
.
Fig.3. Variation in velocity profile
with various value of Magnetic field parameter (M) and 
, , , 
, 
and 
.
Fig.4. Variation of velocity profile
with various value of Prandtl number (
)and, 
, , and .
Fig.5. Variation of velocity profile
with various value of radiation parameter 
and 
,
,
,
,
and 
.
Fig.6. Variation of velocity with
various value of frequency of oscillation (ω) and, , , , and .
Fig.7. Variation of velocity with
various value of Grashof number(Gr)
and , , , , and .
Fig.8. Variation of velocity with
various value of slip parameter (h) and , Gr=1, M=1,
N=0.5, Pr=0.7 and ω=1.
Fig.9. Variation of temperature profile with Re, Pr, N
and ω for h=0.5
Fig.10. Variation of skin friction with frequency of
oscillation.
Fig.11. Variation of Nusselt
Number (Nu) with frequency of oscillation.
Table.1. The phase 
of skin friction.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0.5 |
1 |
1 |
0.7 |
0.5 |
0.5 |
0 |
0.071176 |
0.14139 |
0.20977 |
0.27557 |
0.33825 |
|
1 |
1 |
1 |
0.7 |
0.5 |
0.5 |
0 |
0.14109 |
0.27500 |
0.39663 |
0.50373 |
|
|
0.5 |
5 |
1 |
0.7 |
0.5 |
0.5 |
0 |
0.071176 |
0.14319 |
0.20977 |
0.27557 |
0.33825 |
|
0.5 |
1 |
4 |
0.7 |
0.5 |
0.5 |
0 |
0.037284 |
0.074419 |
0.11126 |
0.14767 |
0.18352 |
|
0.5 |
1 |
1 |
7 |
0.5 |
0.5 |
0 |
0.15493 |
0.28088 |
0.38074 |
0.46275 |
0.53221 |
|
0.5 |
1 |
1 |
0.7 |
5 |
0.5 |
0 |
0.068823 |
0.13677 |
0.20303 |
0.26692 |
0.32793 |
|
0.5 |
1 |
1 |
0.7 |
0.5 |
0 |
0 |
0.04011 |
0.087369 |
0.12948 |
0.16984 |
0.20809 |
|
0.5 |
1 |
1 |
0.7 |
0.5 |
1 |
0 |
0.077129 |
0.15318 |
0.22717 |
0.29828 |
0.36589 |
Table. 2.
The phase 
of Nusselt Number.
|
|
|
|
|
|
|
|
|
|
|
0.5 |
0.7 |
0.5 |
0 |
-0.034375 |
-0.068418 |
-0.10181 |
-0.13429 |
-0.16561 |
|
1 |
0.7 |
0.5 |
0 |
-0.066386 |
-0.13037 |
-0.19003 |
-0.24422 |
-0.29252 |
|
0.5 |
7 |
0.5 |
0 |
-0.23102 |
-0.37847 |
-0.46515 |
-0.51981 |
-0.55704 |
|
0.5 |
0.7 |
2 |
0 |
-0.033879 |
-0.067435 |
-0.10036 |
-0.13239 |
-0.16330 |
PHYSICAL INTERPRETATION OF
RESULTS:
With an increase in Reynold number,
velocity increases. As we know that Reynold number is the “ratio of inertial
forces to the viscous forces” as the Reynold number increases viscous forces
decreases and the decrease in viscosity give ease to the flow of fluid. With an
increase in magnetic field parameter, velocity decreases. The presence of
magnetic field in an electrically conducting fluid introduces a force called
Lorentz force which acts against the flow, if magnetic field is applied in the
normal direction as cosider in the present problem. This type of resistive
force tends to slow down the flow field. With the rise in Prandtl number i.e.
flow is decelerated. Higher Prandtl number posses higher viscosities and lower
thermal coductivities, therefore fluid will flow slower. As a result the
velocity will decrease substaintially with increasing Prandtl number. With the increase of radiation parameter,
there is decrease in velocity i.e. flow is decelerated. This may be attributed
to the fact that the increase in radiation parameter implies more interaction
of radiation with momentum boundary layer. As we increase the Grashof number,
velocity of fluid also increases. Because the Grashof number is the “ratio of
buoyancy forces to the viscous forces” and higher Grashof number posses higher
buoyancy forces and lower lower viscosities, which give rise to the flow of
fluid and velocity get accerlated. As we
increase the slip parameter(h) velocity profile goes on increasing. In
the temperature profile it is observed that with the increase in Prandtl numbr 
temperature decreases, as it is the ratio of
kinematic viscosity to thermal diffusivity. As Pr increases thermal diffusivity
decreases and hence there is decrease in temperature. With the increase in
Reynold number skin friction increases,
as we know that Reynold number is the “ratio of inertial forces to the
viscous forces”. With an increase in magnetic field parameter, skin
friction decreases. The presence of
magnetic field in an ellectrically conducting fluid introduces a force called
Lorentz force which acts against the flow, if magnetic field is applied in the
normal direction. This type of resistive force tends to slow down the flow.
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Appendix:
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Received on
15.12.2014 Accepted on 02.01.2015 ©A&V
Publications all right reserved Research J. Engineering and Tech. 6(1):
Jan.-Mar. 2015 page102-109 DOI: 10.5958/2321-581X.2015.00015.X |
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