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# Analysis of three dimensional couette flow and heat

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IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology

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### Analysis of three dimensional couette flow and heat

1. 1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 85 ANALYSIS OF THREE DIMENSIONAL COUETTE FLOW AND HEAT TRANSFER IN POROUS MEDIUM BETWEEN TWO PERMEABLE PLATES WITH SINUSOIDAL TEMPERATURE AND MAGNETIC FIELD V.P.Rathod1 , Ravi.M2 1 CNCS, Department of Mathematics, Haramaya University 2 Asst.Professor, Department of Mathematics, Govt First Grade College, Raichur, Karnataka State, India Abstract This paper constitutes the analysis of three dimensional couette flow and heat transfer through porous medium under magnetic field. There are two permeable plates in the porous medium, viz., stationery plate and moving plate. The stationery plate is maintained at sinusoidal surface temperature while the moving plate is kept at uniform injection velocity under isothermal condition. The expressions for velocity and temperature fields are obtained. With this, skin-friction components and Nusselt number at both the plates will be derived and are plotted against Reynolds number. Keywords: Three dimensional, Heat transfer, porous medium, Permeable plates, Sinusoidal, Magnetic field --------------------------------------------------------------------***---------------------------------------------------------------------- 1. INTRODUCTION The unsteady laminar flow of an incompressible viscous electrically conducting second order fluid between infinite parallel plates subject to a transverse magnetic field is investigated by Lalitha Jayaraman and Ramanaiah . Raptis et al, [5,4,2,13] have discussed the free convection flow through a porous medium bounded by an infinite plate, when there is no stream velocity. Raptis and Perdikis  have studied the effect of free convection and mass transfer flow through a porous medium bounded by infinite vertical porous plates, when there is a free stream velocity. Analysis of three dimensional couette flow and heat transfer in porous medium between two permeable plates with sinusoidal temperature is studied by Tak and Vyas . Raptis and Perdikis  have studied the combined free and forced convective flow through a porous medium bounded by a semi-infinite vertical porous plate. Gersten and Gross  have studied the effect of transverse sinusoidal suction velocity on flow and heat transfer along an infinite vertical porous wall. Raptis and Takhar  have studied the forced flow of a viscous incompressible fluid through the forced flow of a viscous incompressible fluid through a highly porous medium bounded by a semi-infinite vertical plate in the presence of mass transfer. Oscillatory flow through a porous medium by the presence of free convective flow is studied by Raptis, Perdikis, and C.P. Magneto hydro dynamic free convective effect for an incompressible viscous fluid past an infinite limiting surface is studied by Raptis and Tzivandis. The MHD flow and heat transform in a channel with porous walls of different permeability has been investigated by the method of quasi linearization by Rama Bhargava and Meena Rani . Soundalgekar and Bhat have studied an approximate analysis of an oscillatory MHD channel flow and heat transfer under transverse magnetic field. In the present problem, by taking viscous dissipation in to account, the three dimensional couette flow and heat transfer in a porous medium under magnetic field are analysed .The stationery plate is subjected to both sinusoidal temperature and suction velocity while the moving plate is isothermal with uniform injection velocity. 2. MATHEMATICAL ANALYSIS Consider the three dimensional couette flow through porous medium between two infinite porous plates under magnetic field. So that, the stationery plate is kept along x-z plane and the other plate is at a distance c from the stationery plate which is moving with velocity U along x-direction. The stationery plate is kept at sinusoidal temperature and suction velocity, varying in z-direction while the moving plate is subjected to the uniform injection velocity under isothermal condition. Taking viscous dissipation in to account, the flow and heat transfer are governed by the following equations: 0 v w y z           (1)
2. 2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 86 2 2 2 2 2 0 v +w = ( ) u u u u y z y z Bu u K                            (2) 2 2 2 2 2 0 1 w = - ( ) v v p v v v y z y y z Bv v K                                   (3) 2 2 2 2 2 0 1 v +w ( ) w w p w w y z z y z Bw w K                                    (4) 2 2 2 2 [v +w ]= [ ]p T T T T C K y z y z                         (5) Where 2 2 2 2 2 2 [( ) ( ) ] [( ) ( ) ( ) ] v w u y z y w w u y z z                                   (6) Where u* ,v* ,w* are respectively the velocity components in the directions of x, y and z axes. , K ,T , p ,  , pC , K,  , and 0B are respectively the kinematic viscosity, permeability of porous medium, temperature, pressure, density, Specific heat of the fluid at constant pressure, thermal conductivity, viscosity of the fluid, electrical conductivity, uniform magnetic field of the fluid concerned. The boundary conditions: 0u  , v =-V(1+ cos ) z c     , 0w  , 1T =T (1+ cos ) z c    , 0P  at 0y  u U  , v =-V , 0w  , 2T =T , p = V c K   , T2>T1 at y c  (7) Where 1 ,U, V are constants with dimension of velocity and c,T1 are constants with dimensions of length and temperature respectively. Introducing the following non-dimensional quantities: y= y c  , z= z c  , u= u U  , v= v U  , w= w U  , 2 p= p U  , 1 2 1 T T T T      , pC P K    , 2 K K c   , V U   , 2 2 1 E= ( )p U C T T , d RK   , 1 2 1 a= T T T and 2 0 M= B c U   (8) Where R, P, K, and E are respectively, Reynolds number, Prandtl number, permeability parameter, suction parameter and Eckert number With this, the equations (1) to (5) reduce to 0 v w y z       (9) 2 2 2 2 1 ( ) -Mu u u u u u v w y z R y z RK             (10) Where 2 2 2 2 1 ( ) Mv v v P v v v w y z y R y z v RK                  (11) Uc R  
3. 3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 87 2 2 2 2 1 ( ) -Mw w w P w w v w y z z R y z w RK                 (12) (13) Where 2 2 2 2 2 2{( ) +( ) }+( ) +( ) +( ) v w u w w u y z y y z z                with boundary conditions: y=0: u=0, v=- (1+ cos )z   , w=0, p=0 and za  cos (14) y=1: u=1, v=- , w=0, p=d, 1 In order to solve above equations (10) to (13), it is assumed that 0 1u(y,z)=u (y)+ ( , )u y z (15) 0 1v(y,z)=v (y)+ ( , )v y z w(y, z)= w1(y, z) 0 1p(y,z)=p (y)+ (y,z)p 0 1( , ) ( ) ( , )y z y y z    Since, amplitude )1( of sinusoidal suction velocity is small compared to its wavelength. Employing equations (15) in equations (9) to (14) and taking 0  , the unperturbed quantities satisfies the following equations: 0v 0  (16) 0)( 0 0 00  Mu K u uRu  (17) MdP 0 (18) 0)( 2 000  uEPPR  (19) with boundary conditions: y=0: u 0 =0, 0v =- , 0 =0 , p0=0, y=1: u 0 =1, 0v =- , 0 1  , p0=d (20) Where prime denotes the derivatives with respect to y. The solutions of equations (16) to (19) satisfying the boundary conditions (20) can be expressed as follows: 0v = - (21) 1 2 1 2 0u = m y m y m m e e e e   (22) 0P =(d+M )y (23) And 1 ( ) 1 2 2 1 2 2 1 0 1 2 2 1 2 ( ) 2 1 2 2 2 1 2 1 2 e [ ( ) 2(2 ) 2 ] 2(2 ) ( ) ( ) PR y m y m m m y m m y meEP c c e e m PR m e m m e m PR m m PR m m                    (24) Where 2 2 1 1 1 m = [-R + 4( ) 2 R M K     2 2 2 1 1 = [-R - 4( ) 2 m R M K      1 1 2 2 2 2 1 1 1 2 1 2 2 1 2 2 2 1 2 1 2 1 [ { ( ) 2(2 ) ( ) 2 ( ) }-1] 2(2 ) ( ) ( ) mPR m mPR m m mPR PR m e eEP c e e e m PR m e e m m e e m PR m m PR m m                          2 2 2 2 1 ( ) E v w y z PR y z R                 
4. 4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 88 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 1 2 1 2 ( 1)1 [ { 1 ( ) 2(2 ) ( 1) 2 ( 1) }+1]} 2(2 ) ( ) ( ) m m mPR m m m m eEP c e e e m PR m e m m e m PR m m PR m m                     Also, the perturbed quantities satisfies the following. 1 1 0 v w y z       (25) 2 2 0 1 1 1 1 1 12 2 1 [ ] u u u u u v Mu y y R y z RK               (26) 2 2 1 1 1 1 1 12 2 1 [ ] v p v v v Mv y y R y z RK                 (27) 2 2 1 1 1 1 1 12 2 1 [ ] w p w w w Mw y z R y z RK                 (28) And 2 2 0 01 1 1 1 1 2 2 1 2 [ ] . . u uE v y y PR y z R y y                     (29) with boundary conditions: y=0; 1 0u  , 1 cosv z   , 1 0w  , 1 cosa z  , 1 0p  y=1 ; 1 0u  1 0v  1 0w  , 1 0  , 1 0p  (30) Equations (25) to (29) are linear partial differential equations, which describes perturbed three dimentional flow due to variation of suction velocity stationery surface temperature along z-direction. The form of suction velocity and surface temperature suggests the following forms of 1u , 1v , 1w , 1P and 1 : 1 2( )cosu u y z (31) (32) (33) 1 2( )cosp p y z (34) 1 2( )cosy z   (35) The expression for 1v and 1w have been chosen so that the equation of continuity (25) is satisfied. The equations (27) and (28) , being independent of the main flow and the temperature field, can be solved first. Therefore, substituting (32), (33) and (34) in equations (27) and (28), the following ordinary simultaneous differential equations are obtained: 2 2 2 2 2 2( ) ( 1 ) ( )kv RK v k RKM v RK P         (36) 2 2 2 2 2( ) ( 1 )kv RK v k RKM v RKP        (37) With corresponding boundary conditions: 2 2 2 2 0: , 0, 1: 0, 0 y v v y v v         (38) Where a prime denotes derivative with respect to y. The solution of system of equations (36) and (37) is substituted in equations (32) to (34) to obtain the values of 1v , 1w and 1p : 1 2 1 1 2 3 4[ ]cosm y m y y y v c e c e c e c e z          (39) 1 2 1 1 1 2 2 3 4 1 [ ]sinm y m y y y w c m e c m e c e c e z               (40) 1 2 1 1 22 3 4 1 [ ]cos m y m y y y P A e A e RK A e A e z             (41) Where 2 2 2 1 1 4( ) 2 R R RM Km          And 1 2( ) sv v y co z 1 2 1 ( )sinw v y z   
5. 5. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 89 2 2 2 2 1 4( ) 2 R R RM Km          31 2 4 1 2 3 4, , , dd d d c c c c d d d d     1 2 1 2 1 2 1 2 1 1 1 1 d= m m m m e e e e m m m e m e e e                     ; 2 2 1 2 2 1 1 1 0 d = 0 0 m m e e e m m e e e                   ; 1 1 2 1 1 1 1 1 0 d = 0 0 m m e e e m m e e e                   ; 1 2 1 2 3 1 2 1 2 1 1 1 0 d = 0 0 m m m m e e e m m m e m e e                   ; 1 2 1 2 4 1 2 1 2 1 1 1 0 d = 0 0 m m m m e e e m m m e m e e               3 2 2 1 1 1 1 1 1 1= ( 1 )A Kc m RK c m k RKM c m        , 3 2 2 2 2 2 2 2 2 2( 1 )A Kc m RK c m k RKM c m         , 3 2 2 3 3 3 3( 1 )A Kc RK c k RKM c          , 3 2 2 4 4 4 4( 1 )A Kc RK c k RKM c           Now, for the main flow and the temperature field, Substitute the expressions (31) and (35) in equations (26) & (29), we get, 2 2 2 2 2 0( ) ( 1 )Ku RK u k RKM u RKv u        (42)        20022 2 22 2)( uPEuPRvPR  (43) With boundary conditions. 2 2 2 2 0: 0, 1: 0, 0 y u a y u         (44) Solving (42) & (43) with boundary conditions (44) & substituting the solutions in equations (31) & (35), the expressions for 1 1u and can be given as: 1 2 1 1 1 2 1 2 1 1 2 1 2 2 2 2 ( ) 1 5 6 1 ( ) ( ) ( ) 2 3 4 ( ) ( ) ( ) 5 6 7 ( ) 8 [ { }]cos m y m y m m y m m m m y m y m y m m y m m y m y m y R u D e D e B e e e B e B e B e B e B e B e B e z                            (45) 2 1 2 1 1 2 1 2 1 ( ) [( ) ( ) ( ) +f(y)]cos a a y a a a a y a a G e F a e e e G e F a e z e e           (46) Where )1()()( 2 11 2 11 11 1 RKMKmmRKmmK mKc B        )1()()( 2 21 2 21 12 2 RKMKmmRKmmK mKc B       
6. 6. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 90 )1()()( 2 1 2 1 13 3 RKMKmRKmK mKc B    )1()()( 2 1 2 1 14 4 RKMKmRKmK mKc B    1 2 5 2 2 2 1 2 1( ) ( ) ( 1 ) Kc m B K m m RK m m K RKM          )1()()( 2 22 2 22 22 6 RKMKmmRKmmK mKc B        )1()()( 2 2 2 2 23 7 RKMKmRKmK mKc B    )1()()( 2 2 2 2 24 8 RKMKmRKmK mKc B    ))(( )( 2112 2 5 mmmm m eeee AeBR D      ; ))(( )( 2121 1 6 mmmm m eeee AeBR D      87654321 BBBBBBBBA  1 1 1 2 1 1 2 1 2 2 2 2 ( ) ( ) ( ) ( ) 1 2 3 4 ( ) ( ) ( ) ( ) 5 6 7 8 m m m m m m m m m m m m B B e B e B e B e B e B e B e B e                       2 2 2 1 1 ( ) ( ) 2 4 R R m RM K          , ) 1 ( 4 ) 2 ( 2 22 2 RM K RR m     1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 F A A A A A A A A A A A A A A A A A A A A                      1 2 1 1 1 2 2 1 2 2 1 2 1 2 1 2 1 1 2 2 ( ) ( ) ( ) 1 2 3 ( 2 ) ( 2 )( ) 4 5 6 ( 2 ) ( 2 ) ( 2 ) 7 8 9 ( 2 ) ( 2 ) ( 2 ) 10 11 12 ( ) ( ) 13 14 ( ) m PR y m PR y PR y m m y m m yPR y m m y m m y m y m y m y m y m m m y m m m f y Ae A e A e A e A e A e A e A e A e A e A e A e A e A e                                                1 2 1 2 1 1 1 2 2 1 2 2 ( ) 15 ( ) ( ) ( ) 16 17 18 ( ) ( ) 19 20 y m m y m m y m m y m m y m m y m m y A e A e A e A e A e A e                     1 2 1 1 1 2 2 1 2 2 1 2 1 2 1 2 1 1 2 2 1 2 ( ) ( ) ( ) ( ) 1 2 3 4 ( 2 ) ( 2 ) ( 2 ) ( 2 ) 5 6 7 8 ( 2 ) ( 2 ) ( 2 ) ( 2 ) 9 10 11 12 ( ) ( ) ( ) 13 14 15 G= m PR m PR PR PR m m m m m m m m m m m m m m m m m m m m Ae A e A e A e A e A e A e A e A e A e A e A e A e A e A e A                                                      1 2 1 1 1 2 2 1 2 2 ( ) ( ) ( ) 16 17 18 ( ) ( ) 19 20 m m m m m m m m m m e A e A e A e A e               2 4)( 2222 1    RPPR a ; 2 4)( 2222 2    RPPR a 2 1 2 1 21 22 1 )()(         PRmPRPRm ccRP A ; 2 2 2 2 22 22 2 )()(         PRmPRPRm ccRP A ; 22 23 22 3 )()(      PRPRPR ccRP A ; 2 2 4 2 4 2 2 ( ) ( ) P R c c A PR PR PR              ;
7. 7. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 91 1 2 1 1 1 5 2 1 1 1 1 2 2 1 1 1 1 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m EPRm Pc m A e e m PR B m m m m PR m m              ; 1 2 2 1 2 6 2 2 5 2 1 2 2 1 2 1 2 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m EPRm Pc m A e e m PR B m m m m PR m m              ; 1 2 1 2 1 7 2 1 2 1 2 2 2 2 1 2 1 ( ( ) 2 1 2 ( ))( ( 2 ) ( 2 ) m m EPRm Pc m A e e m PR B m m m m PR m m              1 2 2 2 2 8 2 2 6 2 2 2 2 2 2 2 2 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m EPRm Pc m A e e m PR B m m m m PR m m              1 2 3 11 9 2 1 3 1 2 2 1 1 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m Pc mEPRm A e e m PR B m m PR m                1 2 3 22 10 2 2 7 2 2 2 2 2 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m Pc mEPRm A e e m PR B m m PR m                1 2 1 4 1 11 2 1 4 1 2 2 1 1 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m EPRm Pc m A e e m PR B m m PR m                  1 2 2 4 2 12 2 2 8 2 2 2 2 2 ( ( ) 2 1 2 ( ))( ) ( 2 ) ( 2 ) m m EPRm Pc m A e e m PR B m m PR m                  1 2 1 1 2 13 1 5 2 12 1 2 2 1 1 1 2 2 1 2 1 1 2 1 2 ( ( ) ( ) ( ) ( )) 1 ( ) ( ) ( ) m m Pc m mEPR A m B m m e e m m PR m B m m m m m PR m m m                   1 2 2 1 2 14 1 6 2 22 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 ( ( ) ( ) ( ) ( )) 1 ( ) ( ) ( ) m m Pc m mEPR A m B m m e e m m PR m B m m m m m PR m m m                   1 2 3 1 2 15 1 7 22 1 2 2 3 1 2 2 1 2 1 2 2 ( ( ) ( ) ( ) ( )) 1 ( ) ( ) ( ) m m Pc m mEPR A m B m e e m m PR m B m m m PR m m                      1 2 4 1 2 16 1 8 22 1 2 2 4 1 2 2 1 2 1 2 2 ( ( ) ( ) ( ) ( )) 1 ( ) ( ) ( ) m m Pc m mEPR A m B m e e m m PR m B m m m PR m m                      1 2 1 1 5 17 2 2 1 1 1 1 2 1 ( ) ( ) ( )m m EPm m D A e e m m PR m m           1 2 1 2 6 18 2 2 1 2 1 2 2 1 ( ) ( ) ( )m m EPm m D A e e m m PR m m           1 2 2 1 5 19 2 2 2 1 2 1 2 1 ( ) ( ) ( )m m EPm m D A e e m m PR m m           1 2 2 2 6 20 2 2 2 2 2 2 2 1 ( ) ( ) ( )m m EPm m D A e e m m PR m m          
8. 8. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 92 3. RESULTS AND DISCUSSION Knowing velocity and temperature fields, the important characteristic parameters namely, skin-friction components x and z along main flow and transverse direction, non- dimensional rate of heat transfer (Nusselt number) at both the plates can be calculated. 1 2 1 2 1 2 1 5 2 6 1 1 1 1 2 2 1 3 1 4 2 1 5 2 2 6 2 7 2 8 [ {( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) }]cos x m m m m m m m D m D e e R m m B m m B e e m B m B m m B m m B m B m B z                                    (47) 2 21 1 1 2 2 2 2 3 4 ( ) [ ]sin z z d dw c m c m v dy c c z                   (48) 3.1 Nusselts Number at the Stationary Plate: * * * 0 2 1 ( )y c T Nu T T y       1 2 2 1 2 1 1 2 2 2 1 2 2 2 1 2 1 2 1 2 1 2 Nu=-[ ( ) ( ( ) 2 2 2 ( ) ( ) ( ) {( ) ( ) (0)}cos ] m m a a a a a a m mEP PR c e e m PR m PR m m m m PR G e F a G e F a a a f z e e e e                         (49) 3.2 Nusselt Number at the Moving Plate * * 2 1 ( ) * y c c T Nu T T y       [ 1 1 2 2 1 2 2 1 1 2 2 1 1 2 22 ( ) 1 2 2 1 2 ( )2 2 1 2 2 1 2 1 2 Nu = - [ ( ) ( ( ) 2 2 2 ( ) ( ) ( ) {( ) ( ) (1)}cos ] m PR m m m m m a a a a a a a a m eEP PR c e e e m PR m e mm e m PR m m PR G e F a G e F a ae a e e e e e f z                            (50) The velocity components u, v and w in representative plane z=0 and z=1/2,are plotted against y in figures 1,2 and 3 respectively, for various values of Reynolds number R, Suction parameters  and permeability parameters K at constant magnetic field M. It is observed from these figures, that u and w both increase as R,  or K. Also , when R or K increase v increases. In figure4, the temperature distribution function  is plotted against y, in representative plane z=0 at constant magnetic field. From the fig, temperature increases as Eckert number E or Prandtl number P increases but the same decreases as  increases. The absolute values of the skin friction components x in plane z=0 and z in plane z=1/2are plotted against Reynolds number R at constant magnetic field M in figures 5 and 6 respectively, for various values of  and K. It is noted that both x and z increase as Reynolds number R increases or  increases. Further, permeability parameter K increases x increases whereas z decreases. Fig7.shows the variation of Nusselt number lNul, at the plate y=1, against R at constant magnetic field M. Here, lNul decreases significantly with R. Also when  increases lNul increases. The velocity components u, v and w in representative plane z=0 and z=1/2,are plotted against y in figures 8,9 and 10 respectively, for the fixed values of Reynolds number R, Suction parameters  and permeability parameters K at different magnetic fields. It is observed from these figures, that if Magnetic field parameter M increases, both u and v decreases, but w increases. As R increases, u, v and w increases. In figure11, the temperature distribution function  is plotted against y, in representative plane z=0 for different magnetic 0 2 0 0( ) ( ) cosx x d du du z u dy dy         
9. 9. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 93 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 M=0.2; R  K B 50 0.01 50 C 50 0.02 50 D 50 0.03 50 E 50 0.04 50 F 60 0.05 1000 G 300 0.01 50 H 400 0.01 50 I 1000 0.01 50 Fig1.velocity component u against y in plane z=0 for  I H G F E D C B u y 0.0 0.2 0.4 0.6 0.8 1.0 0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 M=0.2; R  K B 50 0.01 50 C 50 0.02 50 D 50 0.03 50 E 50 0.04 50 F 300 0.01 50 G 300 0.01 50 H 400 0.01 50 I 1000 0.01 50 Fig.3.Velocity component w against y in plane z=1/2 for  I H G F E D C B w*10 y 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 M=0.2; Fig.4.Temperature  against y in plane z=0 for R=50,,K=10,and a=0.78 I H G F E D C B P E B 1.51 0.1 C 2.00 0.1 D 2.50 0.1 E 3.00 0.1 F 3.00 0.05 G 4.00 0.05 H 5.00 0.05 I 6.00 0.05  y 0 20 40 60 80 100 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Fig.5.Skin-friction coeffecient x against Reynolds- number R in plane z=0 for =0.1 M=0.2; G F E D C B  K B 0.05 1 C 0.05 5 D 0.05 100 E 0.02 100 F 0.03 100 G 0.04 100 x R 0.0 0.2 0.4 0.6 0.8 1.0 -0.210 -0.208 -0.206 -0.204 -0.202 -0.200 Fig2.Velocity component v against y in plane z=0 for , D C B M=0.2; R K B 50 10 C 100 50 D 500 100 v y fields. As y and M increases  also increases for fixed Eckert number E, Prandtl number P, Reynolds number R, Suction parameters  and permeability parameters K. The absolute values of the skin friction components x in plane z=0 and z in plane z=1/2 are plotted against Reynolds number R in figures 12 and 13 respectively, for fixed values of  and K at different magnetic fields. It is noted that x increases and z decreases as Reynolds number R increases, while x and z decreases, as M increases for fixed  , K. Figure14.shows the variation of Nusselt number |Nu|, at the plate y=1, against R. Here, |Nu| increases significantly with R. Also, |Nu| increases as M increases.
10. 10. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 94 0 20 40 60 80 100 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 -0.040 -0.045 F E G D C BM=0.2; K B 0.05 1 C 0.05 6 D 0.05 100 E 0.02 100 F 0.03 100 G 0.04 100 Fig.6.Skin-friction coeffecient z against Reynolds- number R in plane z=1/2 for =0.1 z R 0 20 40 60 80 100 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 Sl.No.  K P E I 0.02 10 0.44 0.01 II 0.04 10 0.44 0.01 III 0.01 50 0.71 0.01 IV 0.02 50 0.71 0.01 V 0.05 10 0.78 0.05 VI 0.05 100 0.78 0.05 M=0.2; Fig.7.Nusselt number |Nu| against Reynolds- number R inplane z=0 for =0.1 VI V IV III II I |Nu| R 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 M=0 M=0.4 M=0.2 M=0.0 M=0.2 M=0.4 Fig.8.Velocity component u against y in plane z=0 for =0.1,R=50,=0.01,k=50 u y 0.0 0.2 0.4 0.6 0.8 1.0 -0.210 -0.208 -0.206 -0.204 -0.202 -0.200 M=0.0 M=0.2 M=0.4 Fig.9.velocity component v against y in plane z=0 for =0.2,=0.05,R=500,K=100 M=0.2 M=0.4 M=0 v y 0.0 0.2 0.4 0.6 0.8 1.0 0.000 -0.001 -0.002 -0.003 -0.004 -0.005 M=0.0 M=0.2 M=0.4 M=0 M=0.4 M=0.2 Fig.10.Velocity component w against y in plae z=1/2 for =0.1,R=50,=0.02,K=50 w*10 y 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fig.11.Temperature function against y in plane z=0 R=50,=0.05,K=10,,a=0.78, P=1.51,E=0.1 M=0.0 M=0.2 M=0.4 M=0.4 M=0.2 M=0  y
11. 11. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 95 0 20 40 60 80 100 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 M=0.0 M=0.2 M=0.4 Fig.12.Skin-friction coefficient x against Reynolds number R in plane z=0 for =0.1,,K=1 M=0.4 M=0.2 M=0 x R 0 20 40 60 80 100 -0.054 -0.052 -0.050 -0.048 -0.046 -0.044 -0.042 -0.040 -0.038 -0.036 -0.034 -0.032 -0.030 -0.028 -0.026 -0.024 -0.022 -0.020 -0.018 -0.016 M=0.4 M=0.2 M=0 Fig.13.Skin-friction coefficient z against Reynolds number in plane z=1/2 for ,=0.05,K=100 M=0.0 M=0.2 M=0.4 z R 0 20 40 60 80 100 -0.38 -0.37 -0.36 -0.35 -0.34 -0.33 -0.32 -0.31 M=0.0 M=0.2 M=0.4 M=0.0 M=0.4 M=0.2 Fig.14.Nusselt number |Nu|against Reynolds number R in plane z=0 for ,, K=10,P=0.44,E=0.01,y=1,a=0.78 |Nu| R REFERENCES . A. A. Raptis and H. S. Takhar, 1986, Combined mass transfer and forced flow through a porous medium, Int. Comm. Heat and Mass Transfer, 13, pp. 599-603. . A. A. Raptis, N. G. Kafousias and C. V. Massalas, 1982, Free convection and mass transfer flow through a porous medium bounded by an infinite vertical porous plate with constant heat flux, ZAMM, 62, pp. 489-491. . A. Raptis and G. Tzivanidis, 1983, Magnetohydrodynamic free convective effect for an incompressible viscous fluid past an infinite limiting surface, Astro Physics and Space Science, 94(2), pp. 311-317. . A. Raptis, C. Perdikis and G. Tzivanidis, 1976, Free convection flow through a porous medium bounded by a vertical surface, Journal of Physics D:Applied Physics, 14(7)L99. . A. Raptis, G.Tzivanidis and N. G. Kafousias, 1981, Free convection and mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction, Letters in Heat and Mass Transfer, 8, pp. 417-424. . K. Gersten and J. F. Gross, 1974, Three dimensional flow and heat transfer, ZAMP, 25, pp. 399-408. . S. S. Tak and M. K. Vyas, 2006, Analysis of three dimensional couette flow and heat transfer in porous medium between two permeable plates with sinusoidal temperature, Ultra Science, 18(3)M, pp. 489-500. . V. M. Soundalgekar and J. P. Bhat, 1971, Oscillatory MHD channel flow and heat transfer, Ind. J. Pure. Appl. Math., 15, pp. 819-828. . Lalitha Jayaraman and G. Ramanaiah, 1984, Second-order fluid-transient MHD couette flow with constant stress at upper plate, Indian Journal of Pure and Applied Mathematics, 15(8), pp. 927-934. . Raptis, A.A., Perdikis, C.P., 1985 Oscillatory flow through a porous medium by the presence of free convective flow, Int. J. Engng. Sci., pp. 23, 51-55. . A. A. Raptis and C. P. Pfrdikis, 1988, Combined free and forced convection flow through a porous medium, International Journal of Energy Research, 12(3), pp. 557-560. . A. A. Raptis and C. Perdikis, 1987, Mass transfer and free convection flow through a porous medium, Energy Research, 11, pp. 423-428. . A. A. Raptis, 1983, Unsteady free convective flow through a porous medium, International Journal of Engineering Science, 21(4), pp. 345-348. . Rama Bhargava and Meena Rani, 1984, MHD flow and heat transfer in a channel with porous walls of different permeability(eng), Indian Journal of Pure and Applied Mathematics, 15 (4), pp. 397-408.