EFFECTS OF THERMAL
CONDUCTIVITY ON UNSTEADY MHD
FREE CONVECTIVE FLOW OVER A
SEMI INFINITE VERTICAL PLATE
P. LOGANATHAN
Department of Mathematics, Anna University Chennai, Chennai, India logu@annauniv.edu
P. GANESAN
Department of Mathematics, Anna University Chennai, Chennai, India ganesan@annauniv.edu
D. IRANIAN Research Scholar
Department of Mathematics, Anna University Chennai, Chennai, India call_iranian2004@yahoo.com
Abstract
The numerical study of effects of thermal conductivity on unsteady MHD free convective flow over an isothermal semi infinite vertical plate is presented. It is assumed that the thermal conductivity of the fluid as a linear function of temperature. A magnetic field is applied transversely to the direction of the flow. The boundary layer equations of continuity, momentum and energy equations are transformed into non-linear coupled equations and then solved using implicit finite-difference method of Crank-Nicholson type. A parametric study is performed to illustrate the influence of thermal conductivity, magnetic parameter and Prandtl number on the velocity and temperature profiles. In addition, the local and average skin friction, Nusselt number at the plate are shown graphically for both air and water. An analysis of the results obtained shows that the flow field is influenced appreciably by the strength of magnetic field, thermal conductivity at the wall of the plate.
Keywords: unsteady, free convection, MHD, thermal conductivity, vertical plate, finite difference.
1. Introduction
The most common type of body force, which acts on a fluid, is due to gravity so that the body force can be defined as in magnitude and direction by the acceleration due to gravity. Free convection flow occurs frequently in nature. Heat losses from hot pipes, ovens etc surrounded by cooler air, are at least in part, due to free convection. Sometimes, electromagnetic effects are important. Magneto convection plays an important role in various industrial applications such as magnetic control of molten iron flow in the steel industry, liquid metal cooling in nuclear reactors, salt water, collision less plasmas and magnetic suppression of molten semi-conducting materials. The problem of transient laminar – free convection flow past a semi infinite vertical plate under different plate conditions was studied by many researchers.
conductivity. Hossain [13] studied the viscous and Joule heating effects on MHD free convection flow with variable plate temperature. Soundelgekar et al. [14] presented transient free convection of dissipative fluid past an infinite vertical porous plate. Takhar et al. [15] presented radiation effects on MHD free convection flow of a radiating gas past a semi infinite vertical plate. Soundalgekar and Mohammed [16] presented free convection effects on MHD flow past an impulsively started infinite vertical isothermal plate. Gokhale [17] studied magneto hydrodynamic transient-free convection past a semi infinite vertical plate with constant heat flux. Sattar and Maleque [18] studied unsteady MHD natural convection flow along an accelerated porous plate with hall current and mass transfer in a rotating porous medium. Sattar et al. [19] studied free convection flow and heat transfer through a porous vertical flat plate immersed in a Porous Medium. Abdus Samad and Mansur Rahman [20] presented thermal radiation interaction with unsteady MHD flow past a vertical porous plate immersed in a porous medium.
The spite of all these studies, unsteady MHD free convective flow over a semi infinite vertical plate with thermal conductivity has not received the attention of any researcher. Hence, the objective of the present investigation is to study the effects of thermal conductivity on unsteady MHD free convective flow over a semi infinite isothermal vertical plate.
2. Mathematical Analysis
A two dimensional laminar unsteady flow of a viscous, incompressible fluid past a semi infinite vertical plate is considered. Assume that x axis is to be directed upward along the plate and the y axis is taken normal to the plate. Initially, it is assumed that the plate and the fluid are at the same temperature. Then at the time
t
0
, the temperature of the plate is suddenly raised toT
and is maintained at the same value with time and a magnetic field of uniform strength B0 is applied normal to the plate. The magnetic Reynolds number on the flow is taken to be small so that the induced magnetic field is negligible. It is assumed that thermal conductivity is presented in the fluid flow. Under these assumptions and incorporating the Boussinesq approximation within the boundary layer, the governing equations of continuity, momentum and energy respectively are given by (Schlichting [6], Eckert and Drake [7])) ( ) ( ) ( ) ( 3 2 1 0 2 2 2 0 2 2 y T c k y T v x T u t T u B T T g y u y u v x u u t u y v x u p
The initial and boundary conditions are
) ( , , , , , : , , : 4 0 0 0 0 0 0 0 0 0 0 0 y as T T u x at T T v u y at T T v u t y and x all for T T v u t w
Now introduce the following non dimensional quantities
) ( ) ( , 5 2 3 νρ Gr L B σ M , T T L g Gr , k c νρ Pr , L νGr t t T T T T T , ν vLGr V , ν uLGr U , L yGr Y L x X 2 1 2 2 0 w p 2 2 1 w 4 1 2 1 4 1
) ( )
( )
(T k 1 T 6
K
Where
denotes the thermal conductivity parameter,k
is the fluid stream thermal conductivity respectively. By introducing the above non dimensional quantities the equations (1), (2) and (3) are reduced to the non dimensional form. ) (9 (8) (7) 2 2 2 2 2 Y T γ Y T T) γ (1 Pr 1 Y T V X T U t T MU T Y U Y U V X U U t U 0 Y V X UThe corresponding boundary conditions are
) ( , , , , , : , , : 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 Y at T U X at T V U Y at T V U t Y and X all for T V U t
3. Numerical Solution of the problem
The governing equations (7-9) are unsteady, coupled and non-linear with initial and boundary conditions. An implicit finite - difference scheme of Crank-Nicholson type has been employed to solve the nonlinear coupled equations, as described (Thomas algorithm) in Carnahan et al [2]. The finite difference equations corresponding to equations (7 – 9) are as follows
) ( , , , , ) ( , . , . , , , , , . , , , , . , , , , , 13 2 1 1 2 1 1 1 1 2 2 1 2 1 1 1 1 2 1 1 2 1 1 1 4 1 1 1 1 1 1 2 1 1 1 1 1 Y n j i T n j i T Y n j i T n j i T Pr Y n j i T n j i T n j i T n j i T n j i T n j i T n j i T n j i T Pr Y n j i T n j i T n j i T n j i T n j i V X n j i T n j i T n j i T n j i T n j i U t n j i T n j i T
The region of integration is considered as a rectangle with sides Xm ax(1) andYm ax(14), where corresponding to
y
which lies far from the momentum and energy boundary layers. Computations are carried out till the steady state is reached. The steady state solution is assumed to have been reached when the absolute difference between values ofU
andT
. The subscripti
designates the grid point alongX
direction,j
designates alongY
direction and the subscript n along the t direction. An appropriate mesh sizes considered for the calculation areX 0.02, Y 0.25 and the time step is taken ast 0.10. During any one-time step, the coefficientsU
in,jandV
in,jappearing in the difference equations are treated as constants. The valuesU
,
V
,
T
are known at all grid points att
0
from all the initial conditions. The finite difference solution procedure described by Soundalgekar and Ganesan [1] is employed to solve the governing equations (1-3). The local truncation error is O(t2 Y2 X)and it tends to zero as
t
,Y and Xtend to zero. Hence the scheme is compatible. Stability and compatibility ensures convergence.4. Discussion of Results
In order to determine the evolution of the boundary layer, the governing partial differential equations (7-9) subject to the boundary conditions (10) has been solved numerically using Crank-Nicholson method (Thomas algorithm) as described in Carnahan et al. [2]. Numerical computations have been carried out for different values of the parameters entering into the problem. The transient velocity, temperature profiles for different physical parameters such as M,
and both Pr = 0.7 (air) and Pr = 7.0 (water). The range of
can be taken as5
0 for both air (Pr = 0.7) and water (Pr = 7.0).
Knowing the velocity and temperature field, it is customary to study the skin friction coefficient and Nusselt number in their transient and steady state conditions. Local as well as average values of skin friction, Nusselt number are as follows
) ( )
(
) ( )
(
) (
) (
17 1
0 0
4 1 1
16 0
4 1 1
15 1
0 0
14 0
dX Y Y T Gr T Nu
Y Y T Gr T x
Nu
dX Y Y U f
C
Y Y U f C
The derivatives in (14 - 17) are evaluated using the following five point approximation formula and integrals are evaluated using the Newton-cotes formula.
Figure 7 shows that the steady state local skin friction decreases with increasing magnetic parameter M and thermal conductivity parameter
for Pr = 0.7..It is noted that in the figure 8, the local skin friction decreases with increasing M and for the fixed value of
for higher Prandtl number value Pr=7.0. It is shown in the figures 7 and 8, local skin friction increases suddenly at initial and increases monotonically with time increasing. But from the figure 9, it is observed that the local skin friction increases with increasing the Prandtl numbers and the fixed value of M = 1.0 and 1.0.Figure 10 shows that the average skin friction increases with increasing of M and
for air (Pr=0.7). But for higher Pr = 7.0 the average skin friction decreases with increasing of M and the fixed value of 2.0, it is shown in the figure 11.5. Conclusions
A numerical study has been carried out to study the effects of thermal conductivity and MHD parameters on unsteady isothermal semi infinite vertical plate. The governing partial differential equations are solved by an implicit finite difference scheme of Crank-Nicholson type, which is stable and convergent.
1. The time required for velocity to reach the steady state increases as thermal conductivity parameter increases.
2. The temperature increases with increasing of thermal conductivity and magnetic parameters.
3. The local skin friction and local Nusselt number decrease with increasing of thermal conductivity and magnetic parameters for both Pr = 0.7 and Pr = 7.0.
4. The average skin friction increases with increasing of M and
for Pr = 0.7, but decreases with Pr =7.0.Nomenclature
u
,
v
velocity of the fluid in x, y directions respectivelyU
,V
dimensionless velocity of the fluid in X, Y directions respectivelyG acceleration due to gravity Gr thermal Grashof number
p
c
Specific heat at constant temperaturek
thermal conductivity of the fluid M magnetic parameterB0 magnetic induction
t
timeT
temperature of the fluid in the boundary layer
T
ambient fluid temperaturew
T
plate temperatureT
dimensionless temperaturet
dimensionless timePr Prandtl number
Nux dimensionless local Nusselt number
Nu dimensionless average Nusselt number Cf skin- friction coefficient
Cf average skin-friction coefficient
volumetric coefficient of thermal expansion Kinematic viscosity the fluid density
electrical conductivity of the fluid
thermal conductivity variation parameterSubscripts
w condition of the wall free stream condition
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