J. Comp. & Math. Sci. Vol. 1(2), 135-144 (2010).
Effect of inclined magnetic field on unsteady MHD flow of an incompressible viscous fluid through a porous medium in parallel plate channel Dr. M. VEERA KRISHNA*, S.V. SUNEETHA* and S. CHAND BASHA** *Department of Mathematics, Rayalaseema University, Kurnool (A.P) - 518002 (India) (veerakrishna_maths@yahoo.com; svsmaths@gmail.com) **Assistant professor, Department of Mathematics, Kottam College of Engineering, Chinnatekuru, Kurnool (A.P)-518218 (India) (chand.mathematics@gmail.com)
ABSTRACT In this paper, we make an initial value investigation of the unsteady flow of incompressible viscous fluid between two rigid non-conducting rotating parallel plates bounded by a porous medium taking hall current into account. We discuss a three dimensional flow in a parallel plate channel in a porous medium under the influence of inclined magnetic field. The perturbations are created by a constant pressure gradient along the plates in addition to the non-torsional oscillations of the upper plate while the lower plate is at rest. The flow in the porous medium is governed by the Brinkman's equations. The exact solution of the velocity in the porous medium consists of steady state and transient state. The time required for the transient state to decay is evaluated in detail and the ultimate quasi-steady state solution has been derived analytically, its behaviour is computationally discussed with reference to the various governing parameters. The shear stresses on the boundaries are also obtained analytically and their behaviour is computationally discussed. Key words : Hall effects, unsteady flows, parallel plate channels, incompressible viscous fluids, Brinkman's model. Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
136
M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010). 1. INTRODUCTION
The flow between parallel plates is a classical problem that has important applications in magneto hydro dynamic (MHD) power generators and pumps, accelerators, aerodynamic heating, electrostatic precipitation polymer technology, petroleum industry, purification of crude oil and fluid droplets, sprays, designing cooling systems with liquid metal, centrifugal separation of matter from fluid and flow meters. Hartman and Lazarus5 studied the influence of a transverse uniform magnetic field on the flow of a viscous incompressible electrically conducting fluid between two infinite parallel stationary and insulating plates. Then the problem was extended in numerous ways. Closed form solutions for the velocity fields were obtained in Ref1,3,12&13 under the different physical effects. Some exact and numerical solutions for the heat transfer problem are found in Ref2&6. In the above mentioned cases the Hall term was ignored in applying Ohm's Law as it has no marked effect for small and moderate values of the magnetic field. However, the current trend for the application of magneto-hydrodynamics is to words a strong magnetic field, so that the influence of electromagnetic force is noticeable by Cramer et al3. Under these conditions, the Hall current is important and it has a marked effect on the magnitude and direction of the current density and consequently on the magnetic force. The unsteady hydro magnetic viscous flow through a non-
porous or porous medium has drawn attention in the recent years for possible applications in Geophysical and Cosmical fluid dynamics. The Hall effects in the unsteady case were discussed by Vatazhin14, Pop7 and Sakhonovski11. Debnath et.al. 4 have studied the effects of Hall current on unsteady hydro magnetic flow past a porous plate in a rotating fluid system and the structure of the steady and unsteady flow fields is investigated. Rao and Krishna8 studied Hall effects on the non-torsionally generated unsteady hydro magnetic flow in semi-infinite expansion of an electrically conducting viscous rotating fluid. Krishna and Rao9 & 10 discussed the Stokes and Eckmann problems in magneto hydro dynamics taking Hall effects into account. In this paper, we make an initial value investigation of the unsteady flow of incompressible viscous fluid between two rigid nonconducting rotating parallel plates bounded by a porous medium taking hall current into account. 2. FORMULATION AND SOLUTION OF THE PROBLEM: We consider the unsteady flow of an incompressible electrically conducting viscous fluid bounded by porous medium with two non-conducting parallel plates. A uniform transverse magnetic field is applied to z-axis. In the presence of strong magnetic field a current is inclined in a direction normal to the both electric and magnetic field viz. the hall current of strength H0 inclined
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010). 137 at angle to the normal to the boundaries in the transverse xz-plane. The inclined magnetic field gives rise to a secondary flow transverse to the channel. The hydro magnetic flow is generated in a fluid system by nontorsional oscillations of the upper plate. The lower plate is at rest. The origin is taken on the lower plate and the x-axis parallel to the direction of the upper plate. Since the plates are infinite in extent, all the physical quantities except the pressure depend on z and t only. In the equation of motion along x-direction, the x-component current density- µe J z H o sin and the z-component current density
µ e J x H o sin .
fluid, µe is the magnetic permeability,, is the coefficient of kinematic viscosity, k is the permeability of the medium,
H o is the applied magnetic field. When the strength of the magnetic field is very large, the generalized Ohm's law is modified to include the Hall current, so that
J
ωe τ e JxH σ (E µ e qxH) H0
(2.3)
Where, q is the velocity vector, H is the magnetic field intensity vector, E is the electric field, J is the current density vector, e is the cyclotron frequency, e is the electron collision time,,
We choose a Cartesian system 0(x, y, z) such that the boundary walls are at z=0 and z=l . Z-axis being the axis of rotation of the plates. The flow through porous medium governed by the Brinkman equations. The unsteady hydro magnetic equations governing flow through porous medium under the influence of a transverse magnetic field with reference to a frame are
is the fluid conductivity and, µe is the magnetic permeability.
u 1 P d 2 u µ J H sin 2 e z o u u t ρ x dz k (2.1)
J z m J x sin σμ e H 0 sin u
In equation (2.3) the electron pressure gradient, the ion-slip and thermo-electric effects are neglected. We also assume that the electric field E=0 under assumptions reduces to
J x m J z sin σμ e H 0 sin w (2.4) (2.5)
where m e e is the Hall parameter..
2
w d w µ J H sin 2 e x o w w (2.2) t k dz Where, (u, w) is the velocity components along O(x, z) directions respectively. is the density of the
On solving equations (2.4) and (2.5) we obtain
Jx
σμ e H 0 sin (mu sin w ) (2.6) 1 m 2 sin 2
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
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M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010).
σμ e H 0 sin Jz (u mw sin ) 1 m 2 sin 2
(2.7)
Using the equations (2.6) and (2.7), the equations of the motion with reference to rotating frame are given by 2
2
2
2
u 1 P d u σμ e H 0 sin 2 t ρ x dz ρ(1 m 2 sin 2 )
(u mw sin )
u k
(2.8)
Pl 2 * , P * 2 , l Using non-dimensional variables, the governing equations are (dropping asterisks)
q P 2 q M 2 sin 2 2 q D 1 q t x z (1 im sin ) (2.13) where, 2
2
H l 2 M e 0 is the Hartmann number v 2
2
2
w d 2 w µ e H 0 sin 2 2 t dz ρ(1 m 2 sin 2 ) )
(mu sin w )
D 1
w k
(2.9)
Now combining the equations (2.8) and (2.9), we obtain Let q u iw , 2
t 0,
q ae i t be i t ,
z0 t 0,
(2.11)
z l (2.12)
We introduce the following non dimensional variables are
q*
is the effect of inclined magnetic field
2
(2.10) The boundary and initial conditions are
z z* , l
m ωe e is the Hall parameter
Also the equation (2.13) reduces to
q 1 P 2 q H o sin 2 2 e q q t x z (1 im sin ) k
q 0,
l2 is inverse Darcy parameter k
q l ,
t *
t l 2 * , , l2
q 2 q M 2 sin 2 P 2 D 1 q t z (1 im sin ) (2.14) Corresponding initial and boundary conditions are
q 0, q ae i t be i t ,
t 0,
z 0 (2.15)
, t 0, z 1 (2.16)
Taking Laplace transform of equation (2.14) using initial condition (2.15) the governing equations in terms of the transformed variable reduces to
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010). 139
d 2 q M 2 sin 2 P D 1 s q 2 s dz (1 im sin )
b
Sinh a3 z Sinh a 3
e i t
P Cosh a1 z a1
e a1 t
(2.17) The relevant transformed boundary conditions are
q 0, q
z 1,
(2.18)
a b , z=0 s i s i
(2.19)
P 2
1 s (2.20)
1 a P Where A= , B= Sinh s i 2 1 1 s
P Cosh1 b P 2 2 s i 1 s 1 s
M 2 sin 2 1 s D 1 (1 im sin ) Taking inverse Laplace transform to the equation (2.20), we obtain q
P P Sinh a1 z P Cosh a1 Sinh a1 z a1 a1 Sinh a1 a1 Sinh a1 a
Sinh a 2 z Sinh a 2
ei t
P z (Cos n 1) ( a1 n 2 2 ) t e (2.21) n 2 2 (n 2 2 a1 )
The shear stresses on the upper plate and the lower plate are given by
Solving the equation (2.17) and making use of the boundary conditions (2.18) and (2.19), we obtain
q A Cosh 1 z B Sinh 1 z
az bz 2 2 2 2 n a3 n 1 n a 2
dq dq τ U and τ L dz z 1 dz z 0 3. RESULTS AND DISCUSSION The flow is governed by the non-dimensional parameters M the Hartman number, D-1 the inverse Darcy parameter and m is the Hall parameter. The velocity field in the porous region is evaluated analytically its behaviour with reference to variations in the governing parameters has been computationally analyzed. The profiles for u and v have been plotted in the entire flow field in the porous medium. The solution for the velocity consists of three kinds of terms 1. Steady state 2. The quasi-steady state terms associated with non-torsional oscillations in the boundary, 3. the transient term involving exponentially varying time dependence. From the expression (2.21), it follows that the transient component in the velocity in the fluid region decays in dimensionless time.
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M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010).
1 1 , t> max 2 2 When the / a1 / / a1 n / transient terms decay the steady oscillatory solution in the fluid region is given by
(q)steady =
P P Sinh a1 z a1 a1 Sinh a1 P Cosh a1 Sinh a1 z
(q) oscillator y a
a1 Sinh a1 Sinh a 2 z Sinh a 2 b
ei t
Sinh a3 z Sinh a3
e i t
We now discuss the quasi steady solution for the velocity for different sets of governing parameters namely viz. M the Hartman number and D-1 the inverse Darcy parameter m is the Hall parameter, P the non dimensional pressure gradient, the frequency oscillations , a and b the constants related to non torsional oscillations of the boundary, for computational analysis purpose we are fixing the axial pressure gradient as well as a and b,
. Figures (1-6) corresponding 3 to the velocity components u and w along the imposed pressure gradient for different sets of governing parameters and
when the upper boundary plate executes non-torsional oscillations. The magnitude of the velocity u and w increases for the sets of values 0.1 z 0.3 as well as which reduces for all values of z with increase in the intensity of the magnetic field (Fig 1 & 4). The resultant velocity q decreases with increasing the Hartmann number M. The magnitude of the velocity u decreases in the upper part of the fluid region 0.1 z 0.2 while it experiences enhancement lower part 0.3 z 0.9 with increasing the inverse Darcy parameter D-1(Fig. 2). The magnitude of the velocity w increases in the upper part of the fluid region 0.1 z 0.3, while it reduces in lower part 0.4 z 0.9 with increasing the inverse Darcy parameter D-1 (Fig 5). The resultant velocity q reduces with increasing the inverse Darcy parameter D-1. The magnitude of velocity u decreases in the upper part of the fluid region while it experiences enhancement lower part 0.3 z 0.9 and also the magnitude of velocity v increases through out the fluid region (Fig. 3 & 6). However the resultant velocity q enhances with increasing the Hall parameter m. The shear stresses x and y on the upper plate have been calculated for the different variations in the governing parameters and are tabulated in the tables (1-2). On the upper plate we notice that the magnitudes of x enhances the inverse Darcy parameter D-1 and the hall parameter m decreases
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010). 141 with increase in the Hartmann number M (table 1). The magnitude of y decreases with increase in the Hartmann number M, the inverse Darcy parameter D-1 and the Hall parameter m fixing the
other parameters (table 2). The similar behaviour is observed on the lower plate. We also notice that the magnitude of the shear stresses on the lower plate is very small compare to its values of the upper plate.
1.5
1.35
1.3
1.15
1.1 0.9 u
0.95
M=2
0.7
M=5
0.5
M=8
D‾¹=3000
0.35
D‾¹=4000
0.15
0.1 -0.1
D‾¹=2000
u 0.55
M=10
0.3
D‾¹=1000
0.75
-0.05
0
0.2
0.4
0.6
0.8
1
-0.25
-0.3
0
0.2
0.6
0.8
1
z
z
Fig. 1. The velocity profile for u with M.
Fig. 2. The velocity profile for u with D-1.
1.55
0.02
1.35
-0.03 0
1.15
u
0.4
0.95
m=1
0.75
m=2
0.55
m=3
0.35
m=4
v
0.2
0.4
0.6
0.8
1
-0.08
M=2
-0.13
M=5
-0.18
M=8 M=10
-0.23
0.15 -0.28
-0.05
-0.33
-0.25 z
Fig. 3. The velocity profile for u with m.
z
Fig. 4. The velocity profile for v with M.
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
142
M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010).
0 -0.05
0
0.2
0.4
0.6
0.8
1 D‾¹=1000
-0.1 v
D‾¹=2000 -0.15
D‾¹=3000
-0.2
D‾¹=4000
-0.25 -0.3 z Fig. 5. The velocity profile for v with D-1.
0.05 0 -0.05 0
0.2
0.4
0.6
0.8
1 m=1
-0.1
m=2
v -0.15
m=3
-0.2
m=4
-0.25 -0.3 -0.35 z
Fig. 6. The velocity profile for u with m. Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010). 143 M
I
II
III
IV
V
2
0.045256
0.052353
0.668275
0.052876
0.060992
5
0.032828
0.043526
0.050346
0.043387
0.051144
I
II
III
IV
V
D
1000
2000
3000
1000
1000
m
1
1
1
2
3
-1
Table 1. The shear stress ( x ) on the upper plate
M
I
II
III
IV
V
2
-0.05393
-0.04055
-0.03495
-0.04804
-0.32568
5
-0.04525
-0.03425
-0.02434
-0.03345
-0.02245
I
II
III
IV
V
D
1000
2000
3000
1000
1000
m
1
1
1
2
3
-1
Table 2: The shear stress ( y ) on the upper plate 4. CONCLUSIONS 1. The resultant velocity q enhances with increasing hall parameter m and decreases with increasing inverse Darcy parameter D-1 as well as the Hartmann number M. 2. On the upper plate the magnitudes of x enhances with increasing the hall parameter m and the inverse Darcy parameter D-1 decreases with increase in the Hartmann number M. The magnitude of y decreases with increase
in the Hartmann number M, the hall parameter m and the inverse Darcy parameter D-1 fixing the other parameters. 3. The similar behaviour is observed on the lower plate. 4. The magnitude of the shear stresses on the lower plate is very small compare to its values of the upper plate. REFERENCES 1. Alpher R.A. "Heat transfer in magneto dynamic flow between parallel plates". Int. J. Heat and mass transfer,
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
144 M. Veera Krishna et al., J.Comp.&Math.Sci. Vol.1(2), 135-144 (2010).
2.
3.
4. 5. 6.
7. 8.
9.
10.
3, 108 (1961). Attia H.A. and N.A. Kotb., "MHD flow between two parallel plates with heat transfer", Acta. Mech., 117, 215-220 (1996). Cramer K. and S. Pai., "Magneto hydro dynamics for engineers and applied physicists". Mc.Graw Hill, New York (1973). Debnath. L., Ray. S.C. and Chatterjee. A.K., ZAMM, 59, 469-471 (1979). Hartmann J. and F. Lazarus., Mat. Fys. Medd., 15, 6-7 (1937). Nigam. S.D. and S.N. Singh., "Heat transfer by laminar flow between parallel plates under the action of transverse magnetic field". Q. J. Mech. Appl. Math.,13, 85 (1960). Pop. I. J. Math. Phys. Sci., 5, 375 (1971). Prasad Rao. D.R.V. and Krishna. D.V. Ind.J.Pure and Appl.Math.12, 12, 270 (1981). Rao, D. R. V., Krishna. D.V. and Debnath. L., Acta. Mech., 43, 49 (1982). Rao. D.R.V., SivaPrasad. R and U. Rajeshwara Rao., Bulletin of pure
and applied sciences, 22E, 2, 501510 (2003). 11. Sakhnovski. E. G., Zh.Telchon Fiz., 33, 631 (1963). 12. Sutton. G. W. and A. Sherman, "Engineering Magneto hydro dynamics". Mc. Graw Hill, New York (1965). 13. Tao. L. N., "Steady motion of conducting fluids in channels under transverse magnetic field with consideration of Hall effects", J. Aerospace Sci. Vol. 29, p. 287 (1960). 14. Vatazhin. Z., Angew. Math. Mech. 43, 127 (1963). Appendix:
a1
M 2 sin 2 D 1 , (1 im sin )
a2
M 2 sin 2 D 1 i, (1 im sin )
a3
M 2 sin 2 P D 1 i, P (1 im sin )
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)