An Unsteady MHD Flow Past a Porous Flat Plate on Taking Hall Currents into account

An Unsteady MHD Flow Past a Porous Flat Plate on Taking Hall Currents into account

Bhaskar Chandra Sarkar

Department of Mathematics, Ramananda College, Bishnupur 722122, India

Abstract- study of an unsteady hydro-magnetic flow of a viscous incompressible electrically conducting fluid bounded by an infinitely long porous flat plate in the presence of Hall currents taken into account has been investigated. Initially ( = ), the fluid at infinity moves with a uniform velocity in the direction of the flow. At time > , the plate suddenly moves with the same uniform velocity in the direction of the flow. The governing equations are solved analytically using the Laplace transform technique. The solutions are also obtained for small and large times. The effects of pertinent parameters on the velocity field and the shear stress at the porous plate are exhibited with the help of graphs and tables. It is interesting to note that the series solution of fluid motion converge more quickly than the exact solution for small times.

Keywords: Magnetic Parameter, Hall Currents, Suction parameter, time, general solution and solutions for small time.

1. INTRODUCTION

   The effects of transversely applied magnetic field on the flow of an electrically conducting viscous fluids have been discussed widely owing to their astrophysics, geophysical and engineering applications. When the strength of the magnetic field is strong, one cannot neglect the effects of Hall current. In an ionized gas where the density is low and/or the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before suffering collisions and a current is induced in a direction normal to both the electric and the magnetic fields. This phenomenon, well known in the literature, is called the Hall effect. The study of hydromagnetic viscous flows with Hall currents has important engineering applications in problems of magnetohydrodynamic generators and of Hall accelerators as well as in flight magnetohydrodynamics. The unsteady hydromagnetic flow of an incompressible electrically conducting viscous fluid induced by a porous plate is of considerable interest in the technical field due to its frequent occurrence in industrial and technological applications. Pop and Soundalgekar [1] have investigated the effects of Hall currents on hydromagnetic flow near a porous plate. The hydromagnetic flow past a porous flat plate with Hall effects has been studied by Gupta[2]. Debnath et al.[3] have discussed the effects of Hall current on an unsteady hydromagnetic flow past a porous plate in a rotating fluid system. Hossain[4] has studied the effect of Hall current on an unsteady hydromagnetic free convection flow near an infinite vertical porous plate. The effect of Hall current on hydromagnetic free convection flow near an accelerated porous

   plate has been studied by Hossain and Mohammad [5]. Mazumder[6] has studied the combined effect of Hall current and rotation on hydromagnetic flow over an oscillating porous plate. Maji et al. [7] have studied the Hall effects on hydromagnetic flow on an oscillatory porous plate. Hall effects on the magnetohydrodynamic shear flow past an infinite porous flat plate subjected to uniform suction or blowing have been investigated by Gupta et al. [8]. Hall effects on unsteady hydromagnetic flow past an accelerated porous plate in a rotating system have been studied by Das et al. [9]. Das et al.

   [10] have investigated Hall effects on unsteady MHD rotating flow past a periodically accelerated porous plate with slippage. Hall effect on MHD transient flow past an impulsively started infinite horizontal porous plate in a rotating system has been studied by Reddy [11]. Characteristics of MHD Casson fluid past an inclined vertical porous plate have been investigated by Reddy et al. [12].

   The objective of the present paper is to analyze the effects of Hall currents on an unsteady hydromagnetic flow of a viscous incompressible electrically conducting fluid induced by an infinitely long porous flat plate in the presence of a uniform transverse magnetic field. Initially t = 0 the fluid at infinity moves with uniform velocity U0. At time t > 0, the plate suddenly moves with uniform velocity U0 in the direction of the flow. An exact solution of the governing equation has been obtained by using Laplace transform technique. In order to verify the results obtained exactly another solution which is valid for small times is also obtained. The solution for the flow describing the large time is also derived.

2. FORMULATION

   Consider an unsteady MHD flow of a viscous incompressible electrically conducting fluid filling the semi infinite space z 0 confined to the an infinitely long flat porous plate of finite dimension. Consider the cartesian coordinates system with -axis along the plate , y-axis is perpendicular to the plate and z axis is normal to the xy plane in the vertically upward direction [See Fig.1]. A uniform magnetic field of strength B0 is imposed perpendicular to the plane of the plate. Initially, the fluid flows past an infinitely long porous flat plate with free-stream velocity U0 along -axis. At time t > 0, the plate suddenly starts to move with same uniform velocity as that of the free stream velocity U0. Since the plates are infinitely long, all physical variables, except pressure, depend on z only.

   Ex = 0, Ey = 0, (9)

   everywhere in the flow. Substituting the above values of Ex and Ey in the equations (6) and (7) and solving for jx and jy, we get

   x 2

   j = B0 (v + m u), (10)

   1+m

   y 2

   j = B0 (u m v). (11)

   1+m

   Under usual boundary layer approximations and on the use of

   (10) and (11), equations (1) and (2) become

   Fig.1: Geometry of the problem.

   At time t = 0, for the velocity components u, v, w in

   du

   w0 dz

   dv

   w0 dz

   d2u

   = dz2

   d2v

   = dz2

   B2

   0 [( 0) ] (12)

   u U m v ,

   (1+m2)

   0 [ ( 0)] (13)

   B2

   v + m u U .

   (1+m2)

   the directions x, y and z axes, the momentum equations for the steady flow are

   Introducing the non-dimensional variables

   = U0z , (u , v ) = (u,v) , i = 1, (14)

   du 1 p

   d2u B

   1 1 U0

   w0 dz

   =

   x

   + dz2

   + 0 j , (1)

   y

   equations (12) and (13) become

   dv 1 p

   d2v B

   du1

   d2u1 M2

   w0 dz

   =

   y

   + dz2

   0 j , (2)

   x

   S =

   d

   d2 (1+m2) [(u1 1) m v1], (15)

   1 p

   S dv1 = d2v1 M2

   [v + m (u

   1)], (16)

   0 = z, (3)

   d d2 w0

   (1+m2) 1 1

   B2

   where p is the modified pressure including centrifugal force,

   the density of the fluid, the kinematic coefficient of viscosity

   where S =

   U0

   is the suction parameter and M2 = 0

   U2

   0

   the

   and j (jx, jy, jz) the current density vector.

   The boundary conditions are

   u = 0, v = 0 at z = 0 and u U0, v 0 as z .

   magnetic parameter.

   Corresponding boundary conditions become

   u1 = 0 = v1 at = 0 and u1 1, v1 0 as .

   (4)

   Combining equations (15) and (16), we get

   The generalized Ohms law, on taking Hall currents into account and neglecting ion-slip and thermo-electric effect, is (see Cowling [13])

   j + ee (j × B) = (E + q × B), (5)

   B0

   S dF = d2F

   d d2

   where

   M2(1+im) (F 1), (18)

   1+m2

   F = u1 + i v1. (19)

   where B is the magnetic field vector, E the electric field vector, e the cyclotron frequency, e the collision time of electron and the electrical conductivity.

   We shall assume that the magnetic Reynolds number for the flow is small so that the induced magnetic field can be neglected. This assumption is justified since the magnetic

   Reynolds number is generally very small for partially ionized

   Corresponding boundary conditions are

   F = 0 at = 0 and F 1 as . (20)

   The solutions of the equation (18) subject to the boundary conditions (20) are given by

   u1 = 1 ecos, (21)

   gases. The solenoidal relation . B = 0 for the magnetic field

   gives Bz = B0 = constant everywhere in the fluid where B (Bx, By, Bz). The equation of conservation of the charge j =

   where

   v1 = e

   S 1 S2

   = + [{( +

   sin, (22)

   1

   M2 2 mM2 2 2

   ) + ( ) }

   0 gives jz = constant. This constant is zero since jz = 0 at the plate which is electrically non-conducting. Thus jz = 0 everywhere in the flow. Since the induced magnetic field is

   2 2

   4 1 + m2

   1 + m2

   1

   2

   neglected, the Maxwells equation × E = B becomes

   S2 M2

   t

   × E = 0 which in turn gives Ex = 0 and Ey = 0. This

   + ( 4 + 1 + m2)] ,

   z z

   1

   implies that Ex = constant and Ey = constant everywhere in

   the flow.

   S2 M2 2

   mM2

   1 2

   2 2

   =

   1 {( 4 + 1+m2)

   + (1+m2) } . (23)

   In view of the above assumption, equation (5) gives

   jx + mjy = (Ex + vB0), (6)

   2

   [

   (S2 + M2 )

   2 ]

   4 1+m

   jy mjx = (Ey uB0), (7)

   where m = ee is the Hall parameter. Since the magnetic field is uniform in the free stream so that there is no current and hence, we have

   jx 0, jy 0 as z . (8) On the use of (8), equations (6) and (7) give

   The solutions given by (21) and (22) is valid for both suction

   (S > 0) and blowing (S < 0) at the plate.

   At time t > 0, the plate suddenly moves with uniform velocity U0 along x-axis in the direction of flow. Then the unsteady fluid flow be governed by the following system of equations:

   u u

   t w0 z

   = 1 p

   x

   + 2u

   z2

   2

   B

   0 (u mv), (24)

   (1+m2)

   u1 + i v1 = 1 e

   (+i)

   1 S

   v v

   t w0 z

   = 1 p

   y

   2v

   + z2

   2

   B

   0 (v + mu), (25)

   (1+m2)

   + e 2 [e(a+ib)erfc ( 2

   2

   + (a + ib))

   . (26)

   0 = 1 p

   z

   Corresponding initials and boundary conditions are

   where

   +e(a+ib) erfc (

   2

   (a + ib))],

   (44)

   1

   u = U0, v = 0 at z = 0, t > 0 and 2 2 1 2

   u U , v 0 as z , t 0. (27)

   a, b = 1 [{(S2 + M2 )

   mM2 2 S2 M2

   +

   ( ) } ± ( + )] .

   0

   Using infinity conditions equations (24) and (25) become

   2 4

   1+m2

   1+m2

   4 1+m2

   (45)

   u u

   t w0 z

   2u

   = z2

   B2

   0 [( 0) ] (28)

   u U mv ,

   (1+m2)

   On separating into real and imaginary parts one can easily obtain the velocity components u1 and v1 from the equation

   v w

   v 2v B2

   0

   =

   [v + m(u

   (44). The solution given by (44) is valid for both suction (S >

   t 0 z

   z2

   (1+m2)

   0) and blowing (S < 0) at the plate.

   U0)]. (29)

   Introducing the non-dimensional variables

   0 1 1 0

   = U z/, (u , v ) = (u,v) , = U2t/, i = 1, (30)

   U0

   Equations (28) and (29) become

   SOLUTION AT SMALL TIMES:

   In this case, we use the method which is previously used by Carslaw and Jaegar [14] since it converges rapidly for small

   u1 S u1 = 2u1 M2

   [(u

   1) mv ], (31)

   times. For small time (<< 1) and large S(>> 1) on taking

   2

   (1+m2) 1 1

   inverse Laplace transformation of (42) we get,

   v1

   v1 2v1 M2

   S 1 2 S2

   S =

   [v + m(u

   1)], (32)

   2 S

   n n 2n

   2

   (1+m2) 1 1

   H(, ) = e

   4 n=0 ( 4 + ) (4) i

   erfc(/2)

   Combining equations (31) and (32) we get

   e(+i)+. (46)

   (

   On the use of (37), equation (46) becomes

   F F 2F M2(1+im)

   S S2 2

   S = 2

   where

   1+m2 F, (33)

   F(, ) = e2 4

   S + )n(4)ni2nerfc(/

   n=0

   4

   F = u1 + iv1 1. (34)

   2) e(+i), where inerfc(. ) denotes the repeated integrals of the complementary error function given by

   Corresponding initial and the boundary conditions for F(, )

   in erfc(x) =

   in1erfc()d, n = 0,1,2, ,

   are

   x

   0 ( ) ( )

   (47)

   F(, 0) = F() 0, (35)

   F(0, ) = 0 for > 0,

   F(, ) = 0 for 0, (36)

   i erfc x

   = erfc x ,

   i1erfc(x) = 2

   ex2

   where F() is given by (19).

   To solve the equation (33), we assume

   On separating real and imaginary parts, we have the velocity components as

   S

   F(, ) = H(, )e , (37)

   u1 = 1 ecos + e (2+1)[A(, ) cos1 +

   where

   B(, ) sin1], (48)

   = M2(1+im). (38)

   (S+1)

   1+m2

   v1 = e

   sin + e 2

   [B(, ) cos1

   Using (37), the equation (33) becomes

   H S H = 2H, (39)

   2

   with the initial and boundary conditions

   H(, 0) = F() for 0, (40)

   H(0, ) = 0 for > 0,

   H(, ) = 0 for 0. (41)

   Taking Laplace transformation of (39) and solving the

   A(, ) sin1], (49)

   where

   A(, ) = T0 + 1(4)T2 + (12 12)(4)2T4

   + (13 3112)(4)3T6 + , B(, ) = 1(4)T2 + 211(4)2T4

   2 2 2

   + (3121 13)(4)3T6 + ,

   = S + M , = mM . (50)

   1

   resulting equation subject to the initial and boundary conditions

   4 1+m2 1

   1+m2

   (40) and (41), we get

   (S/2+S2/4+s)

   H(, s) = e

   s

   (+i)

   e

   s

   . (42)

   The above equations show that the Hall effects become important only when terms of order is taken into account.

   SOLUTION AT LARGE TIMES

   where

   H(, s) = H(, )es d (43)

   S

   For large time , the expression (44) can be written as

   0

   u1 + i v1 = 1 +

   1 e2[e(a+ib) erfc((a + ib) +

   )

   The inverse Laplace transformation of (42) on using

   (37) and (34), we get

   2

   e(a+ib) erfc ((a + ib)

   2

   )]. (51)

   2

   Further, if 2, 1 then the solution becomes

   S 2 2

   frequency of oscillations at first increases, reaches a maximum at m = 1 and then decreases.

   e

   (a b )

   2

   u1 = 1 + (a2 + b2) [(acos 2ab

   bsin 2ab ) sinh acos b

   +(bcos 2ab +

   asin 2ab )cosha sinb], (52)

3. RESULTS AND DISCUSSION

   4

   In order to gain a clear insight of the physical problem, we have discussed the effects of various physical parameters such

   S 2 2 2

   e2(a b )

   v1 = (a2 + b2) [(acos2ab bsin2ab )coshasinb

   (bcos2ab + asin2ab )sinhacosb]. (53)

   The above equations (52) and (53) show the existence of inertial oscillations. The frequency of these oscillations is given by

   = 2ab = mM2 , (54)

   1+m2

   which does not occur in the absence of Hall currents.

   It is observe from equations (52) and (53) that the Hall parameter not only induced a cross flow but also occurs inertial oscillations of the fluid velocity.

   Fig.2: Frequency of oscillations for M2 and m

   It is seen from Fig.2 that the frequency of these oscillations increases with increase in magnetic field M2. On the other hand, with an increase in Hall Currents m, the

   as magnetic parameter M , Hall parameter m, suction parameter S and time t on the fluid velocity profiles and shear stress at the porous plate. The fluids velocity profiles are shown in Figs.3-12. It is seen from the Figs. 3-12 that the optimum fluid velocity occur in the vicinity of the plate and asymptotically approaches to zero in the free stream region for both the primary and secondary velocity components. It is seen from Figs.3 and 4 that the primary fluid velocity component u1 increases whereas the secondary fluid velocity component v1 decreases with an increase in magnetic parameter M2. That means Magnetic field has accelerating influence on primary flow and retarding influence on secondary flow. Magnetic field regulates the motion. It is observed from Figs.5 and 6 that the primary velocity u1 decreases whereas the secondary velocity v1 increases with increase in Hall parameter m. This indicates that the activity of Hall currents on the velocity components opposite to the activity of magnetic field on the velocity components. Figs.7 and 8 indicate the variations of suction parameter S on the primary and secondary flows. It is found that the primary velocity u1 increases whereas the secondary velocity v1 decreases with increase in suction parameter S. It is seen from Figs.9 and 10 that the primary velocity u1 increases whereas the secondary velocity v1 decreases as time progresses. For small values of time, we have drawn the velocity components u1 and v1 on using the exact solution given by equation (44) and the series solution given by equations (48) and (49) in Figs.11 and 12. It is seen that the series solution given by (48) and (49) converge more quickly than the exact solution given by (44) for small time.

   Fig.3: Primary velocity for M2 when

   = 0.5, = 0.2 and S = 1.0

   Fig.4: Secondary velocity for M2 when

   m = 0.5, = 0.2 and S = 1.0

   Fig.5: Primary velocity for m when

   M2 = 5, = 0.2 and S = 1.0

   Fig.6: Secondary velocity for m when

   M2 = 5, = 0.2 and S = 1.0

   Fig.7: Primary velocity for S when

   M2 = 5, = 0.2 and m = 0.5

   Fig.8: Secondary velocity for S when

   M2 = 5, = 0.2 and m = 0.5

   Fig.9: Primary velocity for when

   M2 = 5, m = 0.5 and S = 1.0

   Fig.10: Secondary velocity for

   when M2 = 5, m = 0.5 and S = 1.0

   Fig.11: Primary velocity for for the general solution and solution for small times when M2 = 5, m = 0.5 and S = 1.0

   The shear stresses at the plate = 0 due to the primary and secondary flow are given by

   Fig.12: Secondary velocity for for the general solution and solution for small times when M2 = 5, m = 0.5 and S = 1.0

   are shown in Figs.13 and 14 for different values of Hall parameter and time . It is seen that the magnitude of the

   F

   x + iy = []

   S

   = + i [ +

   2

   (a + ib)

   erf(a +

   shear stress component

   and the component

   decrease as

   ib) + 1

   =0

   e(a+ib)2]. (55)

   time progresses whereas they increase with an increase in Hall parameter .

   The numerical results of the shear stress components and

   Fig.13: shear stress x at the plate = 0 due to the primary flow when M2 = 5 and S = 1.0

   Fig.14: shear stress y at the plate = 0 due to the Secondary flow when M2 = 5 and S = 1.0

   For small times, the shear stress at the plate = 0

   due to primary and the secondary flows can be obtained as

   = [1] = 1 1[(0, )cos +

   (0, ) = 1(4)(2 + 1/)

   + 211(4)2(4 + 3/)

   +(3 2 3)(4)3( + /) +

   =0 2 1

   1 1 1 6 5

   (0, )sin1], (56)

   , (59)

   = [1]

   =0

   1

   =

   2

   1[(0, )cos1

   with

   2

   = 21,

   (0, )sin1], (57)

   2

   where

   (0, ) = (0 + 1/) + 1(4)(2 + 1/)

   + (12 12)(4)2(4 + 3/)

   +(13 3112)(4)3(6 + 5/) +

   , (58)

   where 21 = 21e(/2). (60)

   We compare the the numerical values of the shear stress components for general solution and the solution for small times. It is observed from Tables-1 and 2 that for small time, values of shear stresses give higher result than the values of shear stresses for general solution.

   Table-1

   Shear stress due to primary flow for M2 = 5, S = 1.0

   	x(For General solution)			x(Solution for small times)
   m\	0.005	0.010	0.015	0.005	0.010	0.015
   0.0	5.896093	3.644242	2.673381	5.896093	3.644238	2.673358
   0.5	6.032860	3.765285	2.782767	6.032872	3.765358	2.782959
   1.0	6.290388	3.998794	2.998242	6.290404	3.998886	2.998493
   1.5	6.515985	4.208713	3.196242	6.515995	4.208762	3.196378

   Table-2

   Shear stress due to secondary flow for M2 = 5, S = 1.0

   	y(For General solution)			y(Solution for small times)
   m\	0.005	0.010	0.015	0.005	0.010	0.015
   0.0	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
   0.5	0.393570	0.361540	0.337486	0.393615	0.361790	0.338174
   1.0	0.595888	0.555407	0.524776	0.595908	0.555517	0.525087
   1.5	0.660252	0.622620	0.594005	0.660258	0.622650	0.594089

4. CONCLUSION

   The goal of this paper is to investigate the effect of Hall currents on an unsteady hydromagnetic flow of a viscous incompressible electrically conducting fluid bounded by an infinitely long flat porous plate in the presence of a uniform transverse magnetic field. The non-dimensional form of the governing equations of the fluid flow have been solved by the Laplace transform technique. Some conclusions of the study are as below:

   • The optimum fluid velocity occur in the vicinity of the plate and asymptotically approaches to zero in the free stream region for both the velocity components.

   • Magnetic field regulates the fluid motion.

   • Hall currents reduces the primary flow whereas it

     accelerates the secondary flow.

   • As time progresses or if we enhance the porosity of the porous plate the primary flow increases but the secondary flow decreases.

   • The series solution of fluid motion converge more

     quickly than the exact solution for small time.

   The magnitude of the shear stress component due to primary flow and the component due to secondary flow decrease as time progresses whereas they increase as Hall current increases.

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