ISSN 2319-8133 (Online)
(An International Research Journal), www.compmath-journal.org
Radiation Effects on Unsteady Mixed Convective Flow through a Rotating Vertical Porous Channel
P. Sulochana Professor of Mathematics,
Intell Engg. College, Ananthapuramu, INDIA.
email: arigela.sulochana@gmail.com.
(Received on: November 3, 2016) ABSTRACT
We consider, the effects of magnetic field and rotation on unsteady mixed convection oscillatory flow of viscous incompressible radiative fluid in a vertical channel. The fluid is considered to be gray, absorbing emitting radiation but non- scattering medium. The temperature of the plate at the lower plate is raised or lowered to and the constant temperature is maintained at the upper plate. A uniform transverse magnetic field B 0 acts along the z-axis and the plates are electrically non-conducting.
The channel and the fluid rotate in unision with the uniform angular velocity Ω about
the normal to the plates. The governing equations for the unsteady flow are solved analytically and numerically. The effects of the various flow parameters on the velocity, shear stresses, temperature field and the rate of heat transfer in terms of their amplitudes and tangent of phases are discussed. It is observed that both the primary velocity and the magnitude of the secondary velocity decrease with an increase in radiation parameter. Further, it is seen that the amplitudes of rate of heat transfer at the plates increases with an increase in either Prandtl number or radiation parameter or frequency parameter.
Keywords: Mixed convection, MHD flows, unsteady flows, rotating channels, radiation effects, vertical channels and oscillatory plate temperature.
1. INTRODUCTION
Many practical applications of convection flow exist, for example in the heater and
coolers of mechanical devices, in chemical industries, in nuclear power plants, in the formation
of microstructures during the cooling of molten metal’s, in fluid flows around heat-dissipation
fins, and solar ponds etc. Moreover, MHD mixed convection flow is used frequently in the
field of stellar and planetary magnetospheres, aeronautics, chemical engineering and
electronics. Furthermore, most of the engineering processes are related with a high temperature; accordingly, radiation heat transfer is significant to design the relevant equipment of heat transfer process. In addition, radiation effects on MHD free convection flow and heat transfer are important in the context of space technology. Considering it’s important applications in engineering and industrial fields, a number of theoretical and experimental work have been conducted extensively by many re- searchers. In many of the studies carried on hydro magnetic flow of a radiating fluid inside a vertical channel or over the vertical plate the effect of Hall currents in an unsteady state was often neglected. From practical point of view the effect of Hall currents cannot be ignored because of its important in many flow problems. The combined effects of Hall currents and radiation on the magneto hydro dynamic flows continue to attract the attention of engineering science and applied mathematics researchers owing to extensive applications of such flows in the context of ionized aerodynamics, nuclear energy systems control, improved designs in aerospace MHD energy systems, manufacture of advanced aerospace materials etc. Both analytical and computational solutions have been presented to a wide spectrum of problems. Free and forced convective flow of an electrically conducting fluid through the vertical channel under the influence of magnetic field occurs in many industrial and technical applications which include plasma studies, the boundary layer control in aerodynamics, petroleum industries, MHD power generators, cooling of nuclear reactors, and crystal growth. Helliwell and Mosa 1 reported on thermal radiation effects in buoyancy-driven hydromagnetic flow in a horizontal channel flow with an axial temperature gradient in the presence of Joule and viscous heating. The Hall currents and surface temperature oscillation effects on natural convection magnetohydrodynamic heat-generating flow were considered by Takhar and Ram 2 . Alagoa et al. 3 studied magnetohydrodynamic optically-transparent free-convection flow, with radiative heat transfer in porous media with time-dependent suction using an asymptotic approximation, showing that thermal radiation exerts a significant effect on the flow dynamics.
The magnetohydrodynamic free convection heat and mass transfer of a heat generating fluid past an impulsively started infinite vertical porous plate with Hall current and radiation absorption was studied by Kinyanjui 4 . The thermal radiation interaction with unsteady MHD flow past a vertical porous plate immersed in a porous medium was investigated by Samad and Rahman 5 . Chaudhary and Jain 6 studied the behaviours of unsteady hydromagnetic flow of a viscoelastic fluid from a radiative vertical porous plate. The effects of thermal radiation and Hall currents on magnetohydrodynamic free-convective flow and mass transfer over a stretching sheet with variable viscosity in the presence of heat generation/absorption were investigated Shit and Haldar 7 . Israel-Cookey 8 studied the MHD oscillatory Couette flow of a radiating viscous fluid in a porous medium with periodic wall temperature. An analytical model of MHD mixed convective radiating fluid with viscous dissipative heat has been presented by Ahmed and Batin 9 . The effects of thermal radiation, Hall currents, Soret and Dufour on MHD flow by mixed convection over a vertical surface in porous media where described by Shateyi 10 . Singh and Pathak 11 has conducted an analysis of an oscillatory rotating
MHD Poiseuille flow with injection/suction and Hall Currents. Aurangzaib and
Sharidan Shafie 12 have studied the effects of Soret and Dufour on unsteady MHD flow by mixed convection over a vertical surface in porous media with internal heat generation, chemical reaction and Hall currents. The exact solution of MHD mixed convection periodic flow in a rotating vertical channel with heat radiation has been presented by Singh 13 . Singh and Pathak 14 have discussed the effect of rotation and Hall currents on mixed convection MHD flow through a porous medium in a vertical channel in the presence of thermal radiation.
Inview of this, the effects of magnetic field and rotation on unsteady mixed convection oscillatory flow of viscous incompressible radiative fluid in a vertical channel have been studied. The effects of the various flow parameters on the velocity, shear stresses, temperature field and the rate of heat transfer in terms of their amplitudes and tangent of phases are discussed.
2. FORMULATION AND SOLUTION OF THE PROBLEM
We consider an unsteady MHD flow of a viscous incompressible electrically conducting radiative fluid in a vertical plate channel under the influence of uniform transverse magnetic field B 0 acts along the z-axis and the plates are electrically non-conducting.. The fluid is considered to be gray, absorbing emitting radiation but non-scattering medium. The temperature of the plate at the lower plate is raised or lowered to and the constant temperature is maintained at the upper plate. Both the channel and the fluid rotate in unision with the uniform angular velocity Ω about the normal to the plates. We choose a Cartesian co-ordinate system with x-axis in the direction of the flow, the z-axis is normal to the plate and the y-axis is perpendicular to xz-plane (Fig.1).
Fig. 1 Physical Configuration of the Problem
The equation of continuity . q 0 gives w 0 where q ( u , v , 0 ) u, v and 0 are
the velocity components along the co-ordinate axes. Since the channel walls are infinite in
extent and the flow is unsteady, the physical variables are the function of z and t only. The
Boussinesq approximation is assumed to hold and for the evaluation of the gravitational body
force, the density is assumed to be dependent on the temperature according to the equation of
state 0 1 ( T T 2 ) , where T is the fluid temperature, is the fluid density, the
coefficient of thermal expansion and 0 the reference fluid density. Using Boussinesq approximation, the unsteady governing equations of motion of the flow along x and y- directions in a rotating frame of reference are
) (
2 B u g T T 2
z u x
p ρ v 1 t
u 0 2
2 2
(2.1)
B v z
v y
p ρ u 1 t
v 0 2
2 2
2 (2.2)
The energy equation is
) ( 16 k * * T 2 3 T T 2 z
k T t
C T 2
2
p
(2.3)
Where, (u, v) is the velocity components along O(x, y) directions respectively. is the density of the fluid, μ e ( B 0 μ e H 0 ) is the magnetic permeability, is the coefficient of kinematic viscosity, H 0 is the applied magnetic field and p the fluid pressure. * is the Stefan-Boltzman constant, k * the spectral mean absorption coefficient of the medium, k the thermal conductivity C p is the specific heat at constant temperature
The boundary conditions are
h z t T
T T T v
u 0 , 0 , 2 ( 1 2 ) cos , (2.4) h
z T T v
u 0 , 0 , 2 (2.5)
Introducing the non-dimensional variables
2 1
2 2
2
* 1 2
1
*
* , , , , , ,
T T
T T p ph
h t v vh
u uh h y y h x x
Making use of non-dimensional variables, the governing equations reduces to (dropping asterisks)
M u Gr
u x v P
u K
2 1 2
1 2 1
1 2
2 (2.6)
1 2 1
1 2
2 v M v
y u P
v K
2 1 2
(2.7)
2 R
2
Pr (2.8)
The boundary conditions are
1 cos
, 0 , 0 1
1 v n at
u (2.9)
1 0
, 0 ,
0
v at
u (2.10)
Where
ρ
h σ B M
2 2
2 0 is the Hartmann number,
2
2 h
K the rotation parameter,
2 2 2
1 )
(
T T h
Gr g the Grashof number,
k Pr C p
the Prandtl number,
k h T R k
2 3 2
*
16 *
the radiation parameter
h 2
n is the frequency parameter and
2 2
p
P h is the non-dimensional fluid pressure.
Combining the equations (2.6) and (2.7), Let q u 1 iv 1 , we have
iK M q Gr
q P q
2 2
( 2 2 2 ) (2.11)
0
Pr
R
2 2
(2.12) The corresponding boundary conditions are
1 cos
,
0
n at
q (2.13)
1 0
,
0
at
q (2.14)
We assume
) 2 (
1
in
in e
P e
,
, ) ( )
( ) ,
( f e in g e in
q ( , ) 1 ( ) e in 2 ( ) e in (2.15) Where f ( ), g ( ), 1 ( ) and 2 ( ) are unknown functions.
Using the equation (2.15) the equations (2.11) and (2.12) reduces to f
M iK in q Gr
d
2 2
) 2
2 (
1 2 2
1
(2.16)
g M iK in q Gr
d
2 2
) 2
2 (
1 2 2
2
(2.17)
0 Pr)
( 1
1
R in d
2 2
(2.18)
0 Pr)
( 2
2
R in d
2 2
(2.19) Corresponding boundary conditions are
1 2
/ 1 ,
0 1 2
g at
f (2.20)
1 0
,
0 1 2
g at
f (2.21)
Solving the equations (2.16) - (2.19) making use of boundary conditions (2.20) and (2.21) we obtain,
1 1
3 3 2
3 2 1 3
3
3 sinh 2
) 1 ( sinh 2
sinh ) 1 ( sinh ) (
cosh 1 cosh 1 2 ) 1
( a
a a
a a
a Gr a
a
f a (2.22)
2 2
4 4 2
4 2 2 4
4
4 sinh 2
) 1 ( sinh 2
sinh ) 1 ( sinh ) (
cosh 1 cosh 1 2 ) 1
( a
a a
a a
a Gr a
a
g a (2.23)
1 1
1 2 sinh 2
) 1 ( ) sinh
( a
a
(2.24)
2 2
2 2 sinh 2
) 1 ( ) sinh
( a
a
(2.25)
Substituting equations (2.22) - (2.25) in equation (2.15), we obtain velocity and the temperature distributions. On separating into real and imaginary parts, one can easily obtained from the solutions.
The non-dimensional shear stresses at the plates 1 are )
cos( 3
3 1
n q R
(2.26)
)
cos( 4
4 1
n q R
(2.27) The rate of heat transfer at the plates 1 are respectively given by
)
cos( 1
1 1
n
R (2.28)
)
cos( 2
2 1
n
R (2.29)
3. RESULTS AND DISCUSSION
We have represented graphically the non-dimensional velocity components u 1 and v 1
and temperature distribution θ against η for several values of magnetic parameter M, radiation
parameter R, rotation parameter K, Prandtl number Pr, Grashof number Gr, frequency
parameter n and phase angle nτ in Figs.2-18. It is seen from Figures (2-4) that both the primary
velocity and the magnitude of the secondary velocity decrease with an increase in either
magnetic parameter M or radiation parameter R or rotation parameter K. The application of the
transverse magnetic field plays the important role of a resistive type force (Lorentz force)
similar to drag force (that acts in the opposite direction of the fluid motion) which tends to resist the flow thereby reducing its velocity. There is a fall in velocity in the presence of radiation. It is observed from Figure (5) that both the primary velocity and the magnitude of the secondary velocity increase with an increase in Prandtl number Pr. An increase in Grashof number leads to rise both the primary velocity and the magnitude of the secondary velocity shown in Figure (6). This is because, increase in Grashof number means more heating and less density. Figure (7) illustrates that both the primary velocity and the magnitude of the secondary velocity decrease with an increase in phase angle nτ. It is seen from Figure (8) the primary velocity increases while the magnitude of the secondary velocity decreases with an increase in frequency parameter n.
It is illustrated from Figure (9) that the fluid temperature decreases with an increase in radiation parameter R. This result qualitatively agrees with expectations, since the effect of radiation is to decrease the rate of energy transport to the fluid, thereby decreasing the temperature of the fluid. Figure (10) displays that the fluid temperature increases with an increase in Prandtl number Pr. This implies that an increase in Prandtl number Pr leads to fall the thermal boundary layer flow. This is because fluids with large have low thermal diffusivity which causes low heat penetration resulting in reduced thermal boundary layer. Figure (11) reveals that the fluid temperature decreases with an increase in phase angle nτ. It is illustrated from Figure (12) that the fluid temperature increases near the plate at 1 and it decreases away from the plate at 1 with an increase in frequency parameter n.
Numerical results of the amplitude and the tangent of phase of the shear stress at plate due to the primary flow and the amplitude and the tangent of phase of the shear stress at plate due to the secondary flow are presented in Figures (13-18). Figure (13) illustrates that both the amplitudes R 3 and R 4 decrease with an increase in magnetic parameter M. We noticed that, from Figure (14) that both the magnitude of the tangent phases, tan 3 and tan 4 decrease with an increase in magnetic parameter M. It is seen from Figure (15) that both the amplitudes R 3 and R 4 decrease with an increase in rotation parameter K. Figure (16) shown that the magnitude of the tangent phase tan 3 enhances whereas the tangent of phase tan 4 decreases with an increase in rotation parameter K. It is seen from Figure (17) that both the amplitudes R 3 and R 4 increase with an increase in frequency parameter n. Figure (18) depicts that the magnitude of the tangent phase, tan 3 enhances whereas the tangent of phase tan 4 reduces with an increase in frequency parameter n .
Numerical results of the R 1 and R 2 and the tangent phases tan 1 and tan 2 of rate of
heat transfer at the plates 1 and 1 respectively are presented in the tables (1-4) and
discussed with reference to different variations in the governing parameters. We noticed that,
from Table (1) the amplitude R 1 increases with an increase in either radiation parameter R or
Prandtl number Pr or frequency parameter n. It is revealed from Table (2) that for fixed values
of radiation parameter R, the tangent of phase tan 1 increases with an increase in either
Prandtl number Pr or frequency parameter n. It is seen that for fixed values of and , the tangent of phase tan 1 decreases with an increase in radiation parameter R. Table (3) displays that the amplitude R 2 increases with an increase in either radiation parameter R or Prandtl number Pr or frequency parameter n. It is illustrated from Table (4) that for fixed values of radiation parameter R, the tangent of phase tan 2 increases with an increase in either Prandtl number Pr or frequency parameter n. Also, it is seen that for fixed values of Pr and n, the tangent of phase tan 2 decreases with an increase in radiation parameter R.
Fig 2: Velocity profiles for different M 2 4
/ ,
1 , 71 . 0 Pr , 4 , 5 ,
2 3 Gr R n n
K
Fig 3: Velocity profiles for different R
4 / ,
1 , 71 . 0 Pr , 5 ,
5 ,
3
22
