Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering             Research and Applications...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering                 Research and Applicat...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering                    Research and Appli...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering                Research and Applicati...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering              Research and Application...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering                Research and Applicati...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering                 Research and Applicat...
Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering                Research and Applicati...
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  1. 1. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367 Unsteady MHD Flow Induced by a Porous Flat Plate in a Rotating System Maitree Jana1, Sanatan Das2 and Rabindra Nath Jana3 1,3 (Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India) 2 (Department of Applied Mathematics, University of Gour Banga, Malda 732 103, India)ABSTRACT The unsteady magnetohydrodynamic uniform transverse magnetic field. Recently, Das etflow of a viscous incompressible electrically al.[8] have studied the unsteady viscousconducting fluid bounded by an infinite porous incompressible flow induced by a porous plate in aflat plate in a rotating system in the presence of a rotating system.uniform transverse magnetic field has been This paper is devoted to study the effect of theanalyzed. Initially ( t  = 0 ) the fluid at infinity magnetic field on the unsteady MHD flow of amoves with uniform velocity U 0 . At time t  > 0 , viscous incompressible electrically conducting fluid induced by an infinite porous flat plate in a rotatingthe plate suddenly starts to move with uniform system. Initially (t  = 0) , the fluid at infinity movesvelocity U 0 in the direction of the flow. The with uniform velocity U 0 . At time t  > 0 , the platevelocity field and the shear stresses at the platehave been derived using the Laplace transform suddenly starts to move with a uniform velocity U 0technique. Solutions are also obtained for small in the direction of the flow. The velocitytime. It is observed that the primary velocity distributions and the shear stresses at the plate due toincreases whereas the secondary velocity the primary and secondary flows are also obtaineddecreases with an increase in either rotation for small time t . To demonstrate the effects ofparameter or magnetic parameter. It is also rotation and applied magnetic field on the flow field,found that the shear stress at the plate due to the the the velocity distributions and shear stresses dueprimary flow decreases with an increase in either to the primary and the secondary flows are depictedrotation parameter or magnetic parameter. On graphically. It is observed that the primary velocitythe other hand, the shear stress at the plate due increases whereas the secondary velocity decreasesto the secondary flow decreases with an increase with an increase in either rotation parameter K 2 orin rotation parameter while it increases with anincrease in magnetic parameter. magnetic parameter M 2 . Further, it is found that theKeywords: Magnetohydrodynamic, rotation, series solution obtained for small time convergesinertial oscillation and porous plate. more rapidly than the general solution.I. INTRODUCTION II. FORMULATION OF THE PROBLEM When a vast expanse of viscous fluid AND ITS SOLUTIONbounded by an infinite flat plate is rotating about an Consider the flow of a viscousaxis normal to the plate, a layer is formed near the incompressible electrically conducting fluid fillingplate where the coriolis and viscous forces are of the the semi infinite space z  0 in a cartesiansame order of magnitude. This layer is known as coordinate system. Initially, the fluid flows past anEkman layer. The study of flow of a viscous infinitely long porous flat plate when both the plateincompressible electrically conducting fluid induced and the fluid rotate in unison with an uniformby a porous flat plate in rotating system under the angular velocity  about an axis normal to theinfluence of a magnetic field has attracted the plate. Initially (t  = 0) , the fluid at infinity movesinterest of many researchers in view of its wide with uniform velocity U 0 along x -axis. A uniformapplications in many engineering problems such as magnetic field H 0 is imposed along z -axis [Seeoil exploration, geothermal energy extractions andthe boundary layer control in the field of Fig.1] and the plate is taken electrically non-aerodynamics. The unsteady flow of a viscous conducting. At time t  > 0 , the plate suddenly startsincompressible fluid in a rotating system has been to move with the same uniform velocity as that ofstudied by Thornley[2], Pop and Soundalgeker[3], the free stream velocity U 0 .Gupta and Gupta[4], Deka et al.[5] and many otherresearchers. Flow in the Ekman layer on anoscillating plate has been studied by Gupta et al.[6].Guria and Jana[7] have studied the hydromagneticflow in the Ekman layer on an oscillating porousplate in the presence of a 2360 | P a g e
  2. 2. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367 Ex E y which in turn gives = 0 and = 0 . This z z implies that Ex = constant and E y = constant everywhere in the flow. In view of these conditions, equation (5) yields jx =  ( Ex  e H0v),ˆ (6) j y =  ( E y  e H 0u ˆ ). (7) As the magnetic field is uniform in the free stream,   we have from j =  H that jx  0 , j y  0 as z   . Further, u  0 , v  0 as z   gives ˆ ˆ Ex = 0, E y = e H 0U 0 . Substituting the values of E x and E y , equations (6) Fig.1: Geometry of the problem and (7) become jx = e H0v, j y = e H0 (u  U0 ). ˆ ˆ (8) At time (t  = 0) , the equations of motion are On the use of (8) and usual boundary layer ˆ du 1 p d 2u  H 0 ˆ 2 approximations, equations (1) and (2) becomew0  2v =  ˆ  2  j , (1) dz  x dz  y duˆ d 2u  2 H 2 ˆ w0  2v =  2  e 0  u  U 0  , ˆ ˆ (9) dvˆ 1 p d 2v  H 0 ˆ 2 dz dz   w0  2 u =  ˆ  2  j , (2) dz  y dz  x ˆ dv d 2 v  2 H 2 ˆ w0  2  u  U 0  =  2  e 0 v. ˆ ˆ (10) 1 p dz dz  0= , (3)  z where (u, v, w0 ) are the velocity components along ˆ ˆ Introducing the non-dimensional variables U z ˆ (u  iv) ˆ ˆ x , y and z -directions, p the pressure including = 0 , F= , i = 1, (11) centrifugal force,  the fluid density,  the  U0 kinematic coefficient of viscosity,  the electrical equations (9) and (10) become conductivity and w0 being the suction velocity at ˆ ˆ the plate. ˆ ˆ d 2F d 2 S dF d   ˆ    M 2  2iK 2 F   M 2  2iK 2 , 12) The boundary conditions for u and v are u = 0, v = 0, w = w0 at z  0 ˆ ˆ ˆ w0  where S = is the suction parameter, K 2 = 2 u  U0 , v  0 as z   . ˆ ˆ (4) U0 U0 The Ohms law is     e H 0 2 2 the rotation parameter and M 2 = the j =  ( E  e q  H ), (5) U 02   where H is the magnetic field vector, E the magnetic parameter. electric field vector, e the magnetic permeability. ˆ The corresponding boundary conditions for F ( ) We shall assume that the magnetic Reynolds number are for the flow is small so that the induced magnetic ˆ ˆ F  0 at   0 and F  1 as   . (13) field can be neglected. The solenoidal relation  .H = 0 for the magnetic field gives H z = H 0 = The solution of (12) subject to the boundary constant everywhere in the fluid where conditions (13) can be obtained, on using (11), as  H  ( H x , H y , H z ) . The equation of conservation of uˆ a   = 1  e 1 cos b1 , (14) the charge   j = 0 gives jz = constant where U0  j  ( jx , j y , jz ) . This constant is zero since jz = 0 ˆ v a  = e 1 sin b1 , (15) at the plate which is electrically non-conducting. U0 Thus jz = 0 everywhere in the flow. Since the where induced magnetic field is neglected, the Maxwells   H  equation   E =  e becomes  E = 0 t 2361 | P a g e
  3. 3. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367 1 U0 t 2  1 2 On the use of (11) together with t = , equation 1   S 2  2 2  2   S  S a1 =     M 2   4K 4     M 2  , (24) yields 2  4 2     4           F S F  2 F =  ( M 2  2iK 2 ) F . (26) t   2 1 The corresponding initial and the boundary  1 2 conditions for F ( , ) are 1   S 2  2 2  2  b1 =    M 2   4K 4    S  M 2   . (16) ˆ F ( ,0) = F ( ),   0, (27)  4 2     4      F (0, t ) = 0 for t > 0, F (, t ) = 0 for t > 0, (28)      ˆ where F ( ) is given by (11). The solutions given by (14) and (15) are valid for both suction ( S > 0) and blowing ( S < 0) at the By defining F ( , t ) = H ( , t )e t , (29) plate. If M 2 = 0 , the equations (14) and (15) are identical with the equation(12) of Gupta[9]. equation (26) becomes At time t > 0 , the plate suddenly starts to move H H  2 H S = , (30) with a uniform velocity U 0 along x -axis in the t   2 direction of the flow. Assuming the velocity where components (u, v, w0 ) along the coordinate axes,  = M 2  iK 2 , (31) we have the equations of motion as with initial and boundary conditions u u 1 p  2u ˆ H ( ,0)  F ( ) for   0, (32)  w0  2 v    2 t  z  x z H (0, t ) = 0 for t > 0, H (, t ) = 0 for t > 0. (33) e H 0 2 2  (u  U 0 ), (17) On the use of Laplace transform, equation (30)  becomes v v 1 p  2 v  2 H 2 d 2H  w0  2 u =   2  e 0 v, (18) s dH  sH = e( i ) , (34) t  z  y z  d 2 d 1 p where 0 . (19)  z  The initials and boundary conditions are H ( , s) =  0 H ( , t )e st dt. (35) u  u, v  v at t   0 for z  0, ˆ ˆ The boundary condition for H ( , s) are u  U0 , v  0 at z  0, t  > 0 H (0, s) = 0 for t > 0 and H (, s) = 0 for t  0. u  U0 , v  0 as z  , t  > 0 (20) (36) It is observed from the equation (19) that the The solution of the equation (34) subject to the pressure p is independent of z . Using equations boundary conditions (36) is   (17) and (18) together with conditions u  U 0 and   S S2  s  2  v  0 as z   , we have  e  4   e  ( a1  ib1 ) 1 p 1 p H ( , s) = .  (37)   0 and   0. (21) s s  x  y The inverse Laplace transform of (37) and on the On the use of (20), equations (17) and (18) become use of (29) and (25), we get u u  2u  2 H 2 u  iv ( a ib ) 1   S  w0  2v   2  e 0 (u  U 0 ), (22) = 1 e 1 1  e 2 t  z z  U0 2 v v  2 v  2 H 2  ( a ib )     w0  2   u  U 0    2  e 0 v. (23)  e 2 2 erfc   (a2  ib2 ) t  t  z z   2 t  Equations (22) and (23) can be written in combined ( a  ib )    form as e 2 2 erfc   (a2  ib2 ) t  , (38) F F  2 F  2 H 2 2 t   w0  2i  F =  2  e 0 F , (24) t  z z  where where (u  iv) F=  1. (25) U0 2362 | P a g e
  4. 4. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367 1  1 2 1    S 2  2 2  2  S  a2 , b2 =    M 2   4K 4     M 2  . 2   4   4          (39) On separating into a real and imaginary parts one u can easily obtain the velocity components and U0 v from equation (38). The solution given by (38) U0 is valid for both suction ( S > 0) and blowing ( S < 0) at the plate. If M 2 = 0 , then the equation (38) is identical with equation (28) of Das et al. [8]. III. DISCUSSIONS Fig.2: Primary velocity for different K 2 and To study the flow situations due to the M 2 when t = 0.2 and S = 1 impulsive starts of a porous plate for different values of rotation parameter K 2 , magnetic parameter M 2 , suction parameter S and time t , the velocities are u shown in Figs.2-5. The primary velocity and U0 v the secondary velocity are shown in Figs.2 and U0 3 against the distance  from the plate for several values of K 2 and M 2 with t = 0.2 and S = 1.0 . It u is observed that the primary velocity increases U0 v whereas the secondary velocity decreases with U0 an increase in either rotation parameter K 2 or magnetic parameter M 2 . It is observed that for v Fig.3:Secondary velocity for different K 2 and M 2 = 2 and K 2  5 , the secondary velocity U0 M 2 when t = 0.2 and S = 1 shows incipient flow reversal near the plate although the primary flow does not. Figs.4 and5 show the effect of time t on the primary and the secondary velocities with M 2 = 5 , K 2 = 2 and S = 1 . It is found that the primary velocity increases whereas the secondary velocity decreases with an increase in time t . Fig.4: Primary velocity for different time t when M 2 = 5 , K 2 = 2 and S = 1 2363 | P a g e
  5. 5. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367 S  S2         M 2 t  v a   2  4  = e 1 sin b1 e    U0   B( , t ) cos 2K 2t  A( , t )sin 2K 2t  , (44)   where  S2  A( , t ) = T0    M 2   4t  T2  4     2  2   S     M 2   4 K 4  (4t )2 T4  4        2  3  S2   S   M 2    4t  T6  , 3    M 2   12 K 4   4   4      Fig.5: Secondary velocity for different time t when (45)M 2 = 5 , K 2 = 2 and S = 1  S2  2 B( , t ) = 2 K 2 4t T2  4 K 2   M 2 4t  T4  4 We now consider the case when time t is small  which correspond to large s( 1) . In this case,method used by Carslaw and Jaeger [9] is used since  2   2S   2  M   8K 6   4t  T6  . (46)it converges rapidly for small times. Hence, for 3  6 K  2small times, the inverse Laplace transform of the  4       equation (37) gives     n with T2n = i n erfc   , n  0, 2, 4,. S 1    S 2t  S2  2 t H ( , t ) = e 2 4    4 n =0     (4t ) n i 2n   Equations (43) and (44) show that the effects of rotation on the unsteady part of the secondary flow    ( i )  t become important only when terms of order t is erfc  e , (40) 2 t  taken into account.where  is given by (31). For small values of time, we have drawn the velocity u v components and on using the exact On the use of (29), equation (40) yields U0 U0 solution given by equation (38) and the series S 2t  n solutions given by equations (43) and (44) in Figs.6 S    t  S2 F ( , t ) = e 2 4    4 n =0     (4t ) n i 2n   and 7. It is seen that the series solutions given by (43) and (44) converge more rapidly than the exact    ( i ) solution given by (38) for small times. Hence, weerfc  e , (41) conclude that for small times, the numerical values 2 t  of the velocity components can be evaluated from where i n erfc(.) denotes the repeated integrals of the the equations (43) and (44) instead of equation (38).complementary error function given by  i n 1i n erfc( x) = erfc( )d , n  0, 1, 2,, x 2 2i0 erfc( x)  erfc( x), i 1erfc( x) = e x . (42) On the use of (25) and separating into a real andimaginary parts, equation (41) gives  S  S2         M 2 t u a   2  4    = 1  e 1 cos b1 e U0  A( , t ) cos 2K 2t  B( , t )sin 2 K 2t  , (43)   2364 | P a g e
  6. 6. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367 plate ( = 0) due to the primary and the secondary flows decrease with an increase in K 2 . Fig.9 reveals that for fixed value of K 2 , both the shear stresses  x and  y decrease with an increase in time t . 0 0 Further, it is observed that  x steadily decreases 0 while  y steadily increases with an increase in 0 M2. Fig.6: Primary velocity for general solutionand solution for small time with M 2 = 5 , K 2 = 2and S = 1 Fig.8: Shear stress  x and  y when t = 0.2 and 0 0 S =1 Fig.7: Secondary velocity for general solutionand solution for small time with M 2 = 5 , K 2 = 2and S = 1 The non-dimensional shear stresses at theplate ( = 0) due to the primary and the secondary Fig.9:Shear stress  x and  y when K 2 = 2 andflows are given by [from equation (38)] 0 0 u   iv S =1 x  i y = = a1  ib1 The steady state shear stress components at the plate 0 0 U0 ( = 0) are obtained by letting t   as S     a2  ib2  erf  a2  ib2  t S  x  i y = (a1  ib1 )   (a2  ib2 ). (48) 2 2   t Estimation of the time which elapses from the 2 1  a2  ib2 e . (47) starting of the impulsive motion of the plate till the t   steady state is reached can be obtained as follows. ItThe numerical results of the shear stresses  x and is observed from (47) that the steady state is reached 0 after time t0 when erf (a2  ib2 ) t = 1 . Since y at the plate ( = 0) are shown in Figs.8 and 9 0 erf (a2  ib2 ) = 1 when a2  ib2 t0 = 2 , it followsagainst the magnetic parameter M 2 for several thatvalues of the rotation parameter K 2 , time t withS = 1 . It is seen that both the shear stresses at the 2365 | P a g e
  7. 7. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367  1  2  2   2   S   M   4 K 4   4t  ST4  Y3 / t   2 S  2   2 2 t0 = 4    M 2   4K 4  . (49)  4      4        It is seen that t0 decreases with increase in either  2  3  S2   S  2 2 S or K or M . This means that the system with    M   12 K 4  2  M 2   4   4 suction or rotation or magnetic field takes less time       to reach the steady state than the case without   4t  ST6  Y5 / t  (52) 3suction or rotation or magnetic field.For small time, the shear stresses at the plate  Y  ( = 0) due to primary and the secondary flows can Q( , t ) = 2 K 2  4t   ST2  1 be obtained as  t  S2  S 2  2 Y    M 2 t 4 K 2   M 2   4t   ST4  3  u (0, t ) 1  4   4 x = =  e      t 0 U0 2   S2  2    3 Y   P(0, t ) cos 2K 2t  Q(0, t )sin 2K 2t  , (50)  6 K 2   M 2   8K 6   4t   ST6  5   ,    4         t  S2    M 2 t (53) v(0, t ) 1  4 y = = e   with 0 U0 2 Q(0, t ) cos 2 K 2t  P(0, t )sin 2 K 2t  , (51) dT2n Y        2n 1 , where Y2n 1  i 2n 1erfc  .where d 2 t 2 t   S2  (54) P( , t ) = ST0  Y1/ t    4   M 2   4t  ST2  Y1/ t     Table 1. Shear stress at the plate ( = 0) due to primary flow when M 2 = 5 and S = 1  x (For General solution)  x (Solution for small times) 0 0 2 K t 0.002 0.004 0.006 0.002 0.004 0.006 0 10.456620 6.816017 5.220609 10.45661 6.816015 5.220608 4 10.026410 6.386783 4.792615 10.02646 6.387048 4.793337 8 9.426691 5.789964 4.199518 9.426891 5.791084 4.202537 12 8.891527 5.259642 3.675391 8.891985 5.262150 3.682057 Table 2. Shear stress at the plate ( = 0) due to secondary flow when M 2 = 5 and S = 1  y (For General Solution)  y (Solution for small times) 0 0 2 K t 0.002 0.004 0.006 0.002 0.004 0.006 0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 4 1.268350 1.186032 1.123545 1.268379 1.186199 1.124013 8 2.004932 1.840343 1.715468 2.004992 1.840708 1.716536 12 2.504465 2.257698 2.070635 2.504561 2.258326 2.072551For small time, the numerical values of the shear equations (50) and (51) instead of equation (47).stresses calculated from equations (47), (50) and We also consider the case when t is large. For(51) are entered in Tables 1 and 2 for several values large times, the asymptotic formula for theof rotation parameter K 2 and time t . It is complementary error function with complexobserved that for small time the shear stresses argument z 2calculated from the equations (50) and (51) give e zbetter result than those calculated from the equation erfc( z )  as | z | , (55)(47). Hence, for small time, one should derived the znumerical results of the shear stresses from the with erfc( z) = 2  erfc( z) enables us to derive the 2366 | P a g e
  8. 8. Maitree Jana, Sanatan Das, Rabindra Nath Jana / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2360-2367asymptotic solution from the equation (38) in the rotation parameter K 2  5 , the secondary flowfollowing form shows an incipient flow reversal near the plate 1    ( a ib ) S u  iv    although the primary flow does not. It is seen that = 1  e 2 e 2 2 erfc (a2  ib2 ) t   the shear stress components decrease with an U0 2   2 t increase in rotation parameter.  ( a2  ib2 )    REFERENCESe erfc (a2  ib2 ) t   . (56)  2 t  [1] C. Thornley. On Stokes and Rayleigh layers in a rotating system, Qurat. J. Mech. and Appl.Further, if   2 t , t  1 then the equation (56) Math., 21, 1968, 451-461.yields [2] I. Pop and V. M. Soundalgekar, On unsteady S boundary layers in a rotating flow, J. inst, 2 2   ( a 2 b2 )tu e 2 Maths, Applics., 15, 1975, 343-346.  1 2U0 (a2  b2 )  t 2 [3] P. Puri. Fluctuating flow of a viscous fluid on a porous plate in a rotating medium, Acta    a2 cos 2K 2t  b2 sin 2K 2t sinh a2 cos b2  Mechanica, 21, 1975, 153-158.  b cos 2K t  a sin 2K t  cosh a  sin b  ], (57) 2 2 [4] A. S. Gupta, and P. S. Gupta, Ekman layer on a 2 2 2 2 porous oscillating plate, Bulletine de S Lacademie Plomise des sciences, 23, 1975,    ( a 2 b2 )t v e 2 2 2 225-230.  2 [5] R. K. Deka, A. S. Gupta, H. S. Takhar, and V.U 0 (a2  b2 )  t 2 M. Soundelgekar, Flow past an accelerated    a2 cos 2K 2t  b2 sin 2K 2t cosh a2 sin b2 horizontal plate in a rotating system, Acta mechanica, 165, 1999, 13-19.  b cos 2K t  a sin 2K t  sinh a  cos b  ]. 2 2 2 2 2 2 [6] A. S. Gupta, J. C. Misra, M. Reza, and V. M. Soundalgekar, Flow in the Ekman layer on an(58) oscillating porous plate, Acta Mechanica, 165,Equations (57) and (58) show the existence of 2003, 1-16.inertial oscillations. The frequency of these [7] M. Guria and R. N. Jana, Hydromagnetic flowoscillations is 2K 2 . It is observed that the rotation in the Ekman layer on an oscilating porousnot only induced a cross flow but also occurs plate, Magnetohydrodynamics, 43, 2007, 3-11.inertial oscillations of the fluid velocity. The [8] S. Das, M. Guria and R. N. Jana, Unsteadyfrequency of these oscillations increases with an hydrodynamic flow induced by a porous plateincrease in the rotation parameter K 2 . It may be in a rotating system, Int. J. Fluid Mech. Res.,noted that the inertial oscillations do not occur in 36, 2009, 289-299.the absence of the rotation. It is interesting to note [9] A.S. Gupta, Ekman layer on a porous plate,that the frequency of these oscillations is Acta Mechanica, 15, 1972, 930-931.independent of the magnetic field as well as [10] H. S. Carslaw and J. C. Jaeger, Conduction insuction/blowing at the plate. solids (Oxford University Press, Oxford, 1959).IV. CONCLUSION An unsteady MHD flow of anincompressible electrically conducting viscousfluid bounded by an infinitely long porous flat platein a rotating system has been studied. Initially, attime (t   0) , the fluid at infinity moves with auniform velocity U 0 . After time t  > 0 , the platesuddenly starts to move with the same uniformvelocity U 0 as that of the fluid at infinity in thedirection of the flow. It is observed that the primaryvelocity increases with an increase in magneticparameter M 2 . On the other hand, the secondaryvelocity decreases with an increase in magneticparameter which is expected as the magnetic fieldhas a retarding influence on the flow field. It isinteresting to note that the series solution obtainedfor small time converges more rapidly than thegeneral solution. Further, for large values of the 2367 | P a g e

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