153 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Three-Dimensional Flow of a Second Grade Fluid along an Infinite Horizontal Plane Wall with Periodic Suction M. Shoaiba, A. M. Siddiquib, M. A. Ranac, A. Imrand a,cDepartment of Mathematics & Statistics, Riphah International University, Sector I-14, Islamabad, Pakistan, Barani Institute of Management Sciences, Rehman Abad, Rawalpindi, Pakistan. bDepartment of Mathematics, York Campus, Pennsylvania State University, York, PA 17403, USA. dCOMSATS Institute of Information Technology, Kamra Road, Post code 43600, Attock Pakistan. aEmail: shoaibmaths@yahoo.com bEmail: ams5@gmail.com cEmail: mafzalrana@gmail.com dEmail: airman_32029@yahoo.com Abstract In this paper, three-dimensional flow of a second grade fluid along a horizontal infinite plate which is subjected to a transverse sinusoidal suction velocity distribution is studied. Due to variable suction velocity distribution the flow becomes three-dimensional and for constant suction the problem becomes two-dimensional. The free stream velocity is uniform and for small perturbation approximation, analytic technique is applied to obtain the expressions for velocity field and components of skin friction. The effect of second-grade parameter, Reynolds number and suction parameter on the velocity in the direction of main flow and on the stress components is investigated with the help of graphs. The existence of backflow is observed and it is noted that the Reynolds number and suction parameter are controlling parameters for the backflow. Keywords: Differential type fluids; Three-dimensional flows; Periodic suction; Regular perturbation method; Series solutions. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 154 1. Introduction The research area of laminar flow control has received attention of many investigators in recent years and this research area is continuously growing. One of the important applications of laminar flow is the calculation of friction drag of bodies in a flow i.e. the drag of a plate at zero incidences, an airfoil and the friction drag. The main purpose is to reduce drag and hence to improve the vehicle power by a considerable amount. The transition from laminar to turbulent flow which results the drag coefficient to increase, may be prevented or deferred by the suction of fluid and heat transfer from boundary layer to the wall [1]. Gersten et al. Reference [2] have investigated the effect of transverse sinusoidal suction velocity on flow and heat transfer along an infinite porous wall. Singh et al. Reference [3] investigated the flow of viscous incompressible fluid along an infinite porous plate when the transverse sinusoidal suction velocity distribution fluctuating with time is applied. Also Singh et al. Reference [4] have examined the effect of buoyancy forces on three-dimensional flow and heat transfer along with porous vertical plate. Singh [5] extended this idea by applying transverse sinusoidal suction velocity in the presence of viscous dissipative heat. Singh et al. Reference [6] studied the effects of magnetic field on the three-dimensional flow past a porous plate. Transient three-dimensional viscous fluid flow along a porous plate has been studied by Singh et al. Reference [7] while Guria et al. Reference [8] have presented hydrodynamics effect on the three-dimensional flow past a vertical porous plate. Gupta et al. Reference [9] observed MHD effect on the three-dimensional flow past a porous plate. All the above problems have been investigated in viscous fluid. Although the Navier--Stokes equations can manage the flows of viscous fluids but such equations are not adequate to describe the properties of non-Newtonian fluids. Other than viscous fluids there is not a single model which can describe the properties of all non-Newtonian fluids. Therefore, several constitutive relationships of non-Newtonian fluids have been proposed. Generally, non-Newtonian fluids have been classified into three main categories namely the differential, rate and integral types. Second-grade fluid is the simplest subclass of differential type fluids. The aim of present study is to discuss three-dimensional flow of a second-grade fluid along a plane wall which is subjected to the sinusoidally varying velocity distribution. A constant suction velocity at the wall leads to two-dimensional asymptotic suction solution [10], however, due to variation of suction velocity in transverse direction on wall the problem becomes three-dimensional. The regular perturbation method is employed for the solution of the present problem. The results obtained are evaluated for different values of dimensionless parameters such as non-Newtonian elastic parameter ,K Reynolds number eR and suction parameter The article is organized as follows: Section 2 presents the problem description, Section 3 describes the formulation of the problem, Section 4 gives perturbation solutions, Section 5 incorporates results and discussion, while Section 6 includes conclusion. Description of the problem Consider the three-dimensional laminar flow of an incompressible second-grade fluid past an infinite plane wall. A Cartesian coordinate system with the wall lying on xz -plane and the y -axis normal to it is introduced. A suction velocity distribution [2] consisting of a basic steady distribution ( )00 >v with a superimposed weak transversely varying distribution ( ),cos0 l zv πε where l denotes the wave length of the periodic suction American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 155 velocity distribution and ε the amplitude of the suction velocity variation, is taken. Thus, .cos1)( 0       +−= l zvzv πε (1) The constant suction velocity at the wall leads to the well-known two-dimensional asymptotic suction solution [10] while varying suction velocity distributions lead to a cross flow and hence to a three-dimensional flow over the surface. All the physical quantities will be independent of x because of the infinite length of the wall in the x -direction, of course, the flow remains three-dimensional due to variation of suction velocity. Figure 1: Geometry of the problem Formulation of the problem Consider the three-dimensional laminar flow of an incompressible second-grade fluid past an infinite wall, with the x -axis on the wall parallel to the direction of flow. We applied suction velocity distribution [2] of the form )cos1()( 0 l zvzv πε+−= , where ( )00 >v , l and ε are the suction velocity, wave length of the periodic suction velocity distribution and amplitude of the suction velocity distribution. As we have considered asymptotic flow, therefore velocity field is independent of x . In case of constant suction we have well-known two-dimensional asymptotic suction solution and variable suction velocity distribution leads to cross-flow which results in three-dimensional flow. The constitutive expression for second-grade fluid model is ,2 12211 AAAIT αα +++−= µp (2) in which ( )2,1=iiα denote the pressure the identity tensor the dynamic viscosity and material American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 156 constants respectively. The Rivlin-Ericksen tensors 1A and 2A are defined as , , , 11 1 2 1 V AAAA A ∇= += = L LL dt d LL T T + + where ∇ is the operator, V is the velocity field. For the model ( )2 required to be compatible with the thermodynamics in the sense that all motions satisfy the Clasius-Duhen inequality and assumption that the specific Helmholtz free energy is a minimum in equilibrium, then the material parameters must meet the following conditions [11] .0 and 0,0 211 =+≥≥ αααµ (4) The laws of conservation of mass and momentum for the present flow problem are given by ,0= ∂ ∂ + ∂ ∂ ∗ ∗ ∗ ∗ z w y v (5) ,3223 22 3333 1 22       ∂ ∂ + ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ +       ∂ ∂ + ∂ ∂ =      ∂ ∂ + ∂ ∂ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ z uw zy uv zy uw y uv z u y u z uw y uv α µρ (6) , 2 25 2 2 2 2 2 2 2 2 2 2 2 22 2 3 3 2 3 2 3 3 3 22 1 22             +++ ++++ ++++ +       ∂ ∂ + ∂ ∂ + ∂ ∂ −=      ∂ ∂ + ∂ ∂ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ ∗ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∗ ∂∂ ∂∗ ∂∂ ∂∗ ∂ ∂∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ z v y v z u y u y w y w y u y u y w z v y v y v zy u z u zy v z v z v zy v zy v y v wvwv z v y v y p z vw y vv α µρ (7) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 157 , 2 25 2 2 2 2 2 2 2 2 2 2 2 22 2 3 3 2 3 3 3 3 3 22 1 22             +++ ++++ ++++ +       ∂ ∂ + ∂ ∂ + ∂ ∂ −=      ∂ ∂ + ∂ ∂ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∗ ∂∂ ∂∗ ∂ ∂∗ ∂ ∂∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ y w z w y u z u z v z v z u z u z v y w z w z w zy u y u zy w y w z w zy w y w y v wvvw z w y w z p z ww y wv α µρ (8) with the boundary conditions [2] , as ,0 , , ,0at 0 ),cos1( ,0 0 0 ∞→==−== ==+−== ∗∗ ∞ ∗∗∗∗ ∗∗ ∗ ∗∗ yppwvvUu yw l zvvu πε (9) in which ∗∗ vu , and ∗w denote the velocities in the ∗x -, ∗y - and ∗z -directions, respectively. We now introduce the following non-dimensional variables [9]: . , , , , , 2U pp U ww U vv U uu l zz l yy ρ ∗∗∗∗∗∗ ====== (10) Then the Eqs. ( ) ( )95 − become ,0= ∂ ∂ + ∂ ∂ z w y v (11) , 1 3 3 2 3 2 3 3 3 2 2 2 2       ∂ ∂ + ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ +       ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ z uw zy uv zy uw y uvK z u y u Rz uw y uv e (12) , 2 25 1 2 2 2 2 2 2 2 2 2 2 2 22 2 3 3 2 3 2 3 3 3 2 2 2 2             +++ ++++ ++++ +       ∂ ∂ + ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ z v y v z u y u y w y w y u y u y w z v y v y v zy u z u zy v z v z v zy v zy v y v e wvwv K z v y v Ry p z vw y vv (13) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 158 , 2 25 1 2 2 2 2 2 2 2 2 2 2 2 22 2 3 3 2 3 3 3 3 3 2 2 2 2             +++ ++++ ++++ +       ∂ ∂ + ∂ ∂ + ∂ ∂ −= ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ y w z w y u z u z v z v z u z u z v y w z w z w zy u y u zy w y w z w zy w y w y v e wvvw K z w y w Rz p z ww y wv (14) and the boundary conditions take forms , as ,0 , ,1 ,0at 0 ),cos1()( ,0 ∞→=−== ==+−=== ywvu yw l zzvvu α πεα (15) where . , , 2 10 l K U vUlRe ρ αα ν === (16) Solution of the problem Since ε is very small, therefore we assume solution in such a way +++= 2 2 10 FFFF εε (17) where F stands for any of wvu ,, and p . For ,0=ε the problem becomes two-dimensional, so we have ,01 0 2 0 2 3 0 3 =+      − dy du dy ud Rdy udK e αα (18) subject to boundary conditions . as 1 ,0at ,0 0 0 ∞→= == yu yu (19) The order of differential equation is increased from 2 to 3 due to presence of elasticity parameter. We are required three boundary conditions for unique solution of Eq. ( )18 . To remove this difficulty we assume the solution of the form American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 159 ),( 2 01000 KOKuuu ++= (20) Where K is very small parameter. Using Eq. ( )20 in Eqs. ( )18 - ( )19 and comparing coefficients of )( 0KO and )(KO , we get the following boundary value problems: ( ) ( ) .1 ,00 ,0 0000 00 2 00 2 =∞= =+ uu dy duR dy ud eα (21) ( ) ( ) .0 ,00 ,0 0101 01 2 01 2 3 00 3 =∞= =−− uu dy duR dy ud dy udR ee αα (22) Solving the boundary value problems ( )21 ( )22− to obtain ( ),1)(00 yReeyu α−−= ………. (23) ( ) .)( 3 01 Ry e yeRyu αα −−= ………. (24) Therefore, in view of Eqs. ( )23 and ( ),24 Eq. ( )20 yields ( ) .1)( 3 0 yR e yR ee yeRKeyu αα α −− −−= (25) When ,0≠ε the solution of the problem is obtained by the perturbation method ),(),()( 2 10 εε Ozyuyuu ++= (26) ),(),( 2 10 εε Ozyvvv ++= (27) ).(),( 2 10 εε Ozywww ++= (28) Using Eqs. ( )26 - ( )28 into Eqs. ( )11 - ( )15 to obtain differential equations corresponding to first order terms ,011 = ∂ ∂ + ∂ ∂ z w y v (29) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 160 ,1 3 0 3 12 1 3 3 1 3 2 1 2 2 1 2 0 1 1       ∂ ∂ + ∂∂ ∂ − ∂ ∂ −+      ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ − y uv zy u y uK z u y u Ry uv y u e ααα (30) ,1 2 1 3 3 1 3 2 1 2 2 1 2 11       ∂∂ ∂ + ∂ ∂ −      ∂ ∂ + ∂ ∂ + ∂ ∂ −= ∂ ∂ − zy v y vK z v y v Ry p y v e αα (31) ,1 2 1 3 3 1 3 2 1 2 2 1 2 11       ∂∂ ∂ + ∂ ∂ −      ∂ ∂ + ∂ ∂ + ∂ ∂ −= ∂ ∂ − zy w y wK z w y w Rz p y w e αα (32) and the boundary conditions . as ,0 ,0 ,0 ,0at 0 ,cos ,0 111 111 ∞→=== ==−== ywvu yw l zvu πα (33) The set of linear differential equations ( ) ( )3329 − describe the three-dimensional flow. Cross flow Solution In this section the set of cross-flow solutions ),,(1 zyv ),(1 zyw and ),(1 zyp are considered. This set of solution is independent of the main flow component u. The suction velocity consists of basic uniform distribution 0v with a superimposed weak sinusoidal distribution ( ),cos0 zv πε therefore the velocity components ),,(1 zyv ),(1 zyw and pressure ),(1 zyp are also separated into main and small sinusoidal components. Therefore, assume the following forms for ),,(1 zyv ),(1 zyw and ),(1 zyp : ,cos)(),( 111 zyvzyv π= (34) ,sin)(1),( 111 zyvzyw π π ′ −= (35) .cos)(),( 111 zypzyp π= (36) In Eq. ( )35 the dash ՛ denotes differentiation with respect to y. We note that the velocity components ( ) ( )3534 − identically satisfy the continuity equation ( ).29 Substituting Eqs. ( )34 - ( )36 in Eqs. ( )31 and ( )32 , we have ,)( 111111 2 1111 2 11 ′′′′′′′′ −=−+−− pRvRvvvvRK eee αππα (37) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 161 ,)( 11 2 1111 2 1111 2 11 pRvRvvvvRK eee παππα =−+−− ′′′′′′′′′′′′ (38) and the boundary conditions are .0)0(,)0( 1111 =−= ′ vv α (39) On eliminating the pressure from Eqs. ( )37 and ( )38 we get the following differential equation: ( ) .022 11 4 11 2 11 2 111111 4 11 2 11 =−++−−+− ′′′′′′′′′′′′′′′′′′′ vvRvvRvvvvRK eee παππαππα …. (40) We assume ),( 2 11111011 KOKvvv ++= ….. (41) using Eq. ( )41 in Eq. ( )40 and solving resulting equation, we get the following solutions: ),( )(110 yy eev πλ λπ λπ α −− − − − = (42) ),)(( ))(2( )(2 111 yyy e e yeee R Rv λπλ λπ πλλα πλπλα −−− −−− −− +− = (43) where . 22 2 2 πααλ +     += ee RR Substitution of Eqs. ( )42 and ( )43 in Eq. ( )41 , yields ).)(( ))(2( )()( )( 2 11 yyy e eyy yeee R RKeev λπλπλ λπ πλλα πλπλαλπ λπ α −−−−− −−− −− + −− − − = (44) Similarly from Eqs. ( )44 and ( )38 , we get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y e ee e y ee eeee R y eee e ye R RKRK R K e RR RKRKRKRK K KK eRRKRK R Kp e λ λ αλ πλα π πλλππλλαπλλ πλπλαπλλα αλ πλα πλπαλαπλλ λπαλαπλαλα αλπαλ πλπ αλ παππαπα αλ πλααπ πλπ αλ − − − + −       −+−+−− −+−− − + +                 +−−−+ +−− + − − −       −−− − + + − − = 223 2242 3223 234 )2( )( 23 2324 2 11 )2( )( 232 243 2 )2( )( 2 (45) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 162 Substituting Eqs. ( )44 and ( )45 in Eqs. ( )34 - ( )36 , we get ,cos ))( ()( )( ),( )2( )( 1 z ye eeKeezyv y yy R Ryy e e π λπ λπ λπ α λ πλ αλ πλπλαπλ       −− −+− − − = − −− − +−− (46) ,sin ))( )(()( )( ),( )2( )( 1 z ye eeKeezyw y yy R Ryy e e π λπλ π λπ αλ λ πλ αλ πλαπλ       −− −+− − − = − −− − +−− (47) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .cos )2( )( cos 23 22 43 cos2 )2( )(),( 22 322 4 2 322 32 34 )2( )( 23 2324 2 1 2 zye R RK RK R K ze RR RKRK RKRK K KK zeRRKRK R Kzyp y e e e e y ee ee ee R y eee e e π πλλππλλα πλλπλπλα πλλα αλ πλα π πλπαλαπλ λλπαλα πλαλα αλπαλ πλπ αλ ππαππαπα λ πλααπ πλπ αλ λ λ αλ πλα π − − − + −           −+−+ −−−+ −− − + +                       +−−− ++− − + − − −       −−− − + + − − = (48) The Eqs. ( )46 and ( )47 present the cross-flow velocity distribution and pressure in Eq. ( )48 provide the input for the solution to the axial velocity. The viscous results [2] are recovered when Main flow solution The solution for the Eq. ( )30 can be expressed as .cos)(),( 111 zyuzyu π= (49) The corresponding boundary conditions ( )33 are reduced to . 0 ,0 0 11 11 ∞→= == yasu yatu (50) Further we assume that ).( 2 11111011 KOKuuu ++= (51) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 163 Then the boundary conditions ( )50 yield . as0 ,0at 0 110111 110111 ∞→== === yuu yuu (52) Using Eqs. ( ),25 ( )46 and Eqs. ( )49 - ( )52 in Eq. ( )30 , we get [ ] ( ) ( ) ( ) ( ) ,cos )( cos 22)( cos )( ),( )( 7 )( 5 )( 6 )( 42 2 1 64 32 2 )( 3 )( 211 32 z yeEyeE eEeERK zye R EeEE R E R ERK zeAeAeARzyu yRyR yR R EyR R E e y e y ee e yRyRye ee e e e e ee π λπ α π λαπαλαλπ α π λπ α απαλ απ πα αλ λα λλ απαλλ       ++ +++ − −             − −      +++ − + +− − = +−+− +−+− −− +−+−− (53) where ( ) , 2 , , 2 , 2 23 1321       −−===      −= π λ λ πλπλ π λ λ π π λ λ π EAAA , 2 )( 2 )( 2 )(2 33 22 2 λ αλπ λ αλπ αλ πλππα ee e e RR R RE + + + − − + +−= ( ) ),()( 2 )(2 3 2 3 e e e e RR R RE απλπ π απλ αλ πλπλα +− + + − + −= ( ) ( ) ( ) ( ) ( ) , 22 1 , 24 2 3 2 5 3 2 224       + − =      + − + = e e e e e e e e e R R R R ER R R R RE απ αλ λαπ λα απ αλ λαπ αλ αλ ( ) ( )( ) ( )( ).1 ,2 3 7 3 226 e e e e e R R ER R RE αλ πα αλ απ απ −=− + = Substituting Eqs. ( )25 and ( )53 in Eq. ( ),26 we get ( ) [ ] ( ) ( ) ( ) ( ) ,cos )( cos 22)( cos )( 1),( )( 7 )( 5 )( 6 )( 42 2 1 64 32 2 )( 3 )( 21 3 32 z yeEyeE eEeERK zye R EeEE R E R ERK zeAeAeARyeRKezyu yRyR yR R EyR R E e y e y ee e yRyRyeyR e yR ee e e e e eeee π λπ αε π λαπαλαλπ αε π λπ αεα απαλ απ πα αλ λα λλ απαλλαα       ++ +++ − −             − −      +++ − + +− − +−−= +−+− +−+− −− +−+−−−− (54) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 164 It should be noted that the limiting velocity 1u as 0→K , differs from that computed by Gersten and Gross [2]. This is due to some calculation mistake in their work. Shear stress components The expressions for the shear stress components in the x -direction and z -direction can be expressed as follows: ,cos)(cos )( 10 0 zRFzF R C e e yy u fx πεπε α −+= = =∂ ∂ (55) and .sin)( )( 2 0 zRF C e yy w fz πε α µ −= = =∂ ∂ (56) The functions )(1 eRF and )(2 eRF are given by . )()( 22)(2 )()( 7654 )( 2 )( 1 64 32 21 32    −++−++ ++    − −      +++− − − + = ++ EEREER R EEE R E R EKRRF ee R ER R ER eee e e e e e απαλ λαπαλα λ λπλ ππλ πα απ λα αλ (57) It is worth mentioning that the skin friction factor ( )eRF1 when 0→K reduces to steady state value of [7]. It is also indicated that limiting result as 0→K differs from that found by Gersten and Gross [2]. This happens due to some calculation mistake in their work. ( ) . 2 1)( 2 2       − += λα αλλ e e e R RKRF (58) The limiting result of )(2 eRF as 0→K is identical to that obtained by Gersten and Gross [2] and steady state value presented by Singh et al. [7]. 2. Results and discussion The effects of dimensionless parameters such as elastic parameter K , Reynolds number eR and suction parameter α on velocity component u are shown in Figures 2-4 Skin friction factors ),(1 eRF )(2 eRF are American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 165 presented graphically in figures 5-8. Figure 2 shows that the velocity component u decreases with the increase of dimensionless parameter K which was expected naturally. For a particular value of K, the velocity component u increases gradually to attain maximum value equal to unity. The figure 3 shows the effect of Reynolds number eR on the main flow velocity component u. It is observed from this figure that velocity is increasing function of .eR However, velocity decreases in the vicinity of the plate. Moreover, backflow is observed for .30>eR The influence of suction parameter α on the velocity component u is demonstrated in figure 4 The main flow velocity component u increases as the suction parameter α increases which was expected naturally. However, it decreases near the plate and then increases exponentially. Backflow near the plate is observed for 3.0>α . Furthermore, 1→u as .∞→y The effect of dimensionless parameters K and α on the shear stress component )(1 eRF is depicted in Figures 5 and 6 respectively. The figure 5 shows that the shear stress component )(1 eRF increases with an increase in .K It decreases as eR increases from zero to some value (depending upon K) of ,eR then increases exponentially and tends to infinity. Similar effect of α on 1F is noted in figure 6 Of course, 1F tends to be linearized as .1.0→α Moreover, 1)(1 →eRF as .0→eR Figure 2: Variation of u at α=0.1, Re=10, ε=0.1 and z=0 for different values of K American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 166 The )(2 eRF and its asymptotic limits are shown in Figures 7 and 8 . In figure 7 the dimensionless parameter α is fixed and K is varied. In figure 8 the role of these dimensionless parameters is interchanged. Figure 7 shows great influence of elastic parameter on )(2 eRF which is decreasing function of elastic parameter K . Moreover, 2F increases as eR increases from zero to some value ( )Kupon depending of ,eR then decreases for higher values. It is shown in figure 8 that )(2 eRF initially increases and then decreases for any fix value of .α Also, it can be perceived that 2F tends to linearized as 1.0→α . The figure 9 demonstrates that the transverse wall shear stress, which results from the secondary flow normal to the main flow direction, disappears due to symmetry at the points of maximum and minimum suction velocity. The effect of elastic parameter K and suction parameter α on the velocity component 1w are tabulated in Table 1. It is observed that 1w increases as α increases. However an opposite effect of K on 1w is noted. It also decreases in the y-direction. Figure 3: Variation of u at α=0.1, K=0.1, ε=0.1 and z=0 for different values of Re Figure 4: Variation of u at K=0.1, Re =10, ε=0.1 and z=0 for different values of α American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 167 Figure 5: Variation of F₁(Re) at α=0.1 for different values of K Figure 6: Variation of F₁(Re) at K=0.1 for different values of α Figure 7: Variation of F₂(Re) at α=0.1 for different values of K American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 168 Figure 8: Variation of F₂(Re) at K=0.1 for different values of α Figure 9: Flow streamlines on the surface of the flat plate for K=0.1 and α=0.1 Table 1: Effects of K and α on transverse velocity component w for ,1.0=ε 5.0−=z and . y K=0.1, α=0.1 K=0.1, 3.0=α K=0.1, 5.0=α K=0.5, 1.0=α K=0.5, 3.0=α K=0.5, 5.0=α 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.007656 0.02290 0.03796 0.007200 0.01869 0.02597 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 169 0.8 0.004250 0.01205 0.01887 0.003993 0.00973 0.01238 1.2 0.001769 0.00475 0.00701 0.001660 0.00379 0.00437 1.6 0.000654 0.00166 0.00230 0.000613 0.00131 0.00135 2.0 0.000227 0.00054 0.00070 0.000212 0.00042 0.00038 3. Conclusion The three-dimensional incompressible laminar flow of a second grade fluid past a wall is analyzed. A suction with a slightly sinusoidal transverse suction velocity distribution at the wall is employed. Approximate solutions for main flow, cross flow and pressure are presented. For the asymptotic flow condition far downstream the components of the wall shear stress are computed. The major findings of the present study are as follow: • When K increases the main flow velocity u decreases. In the limiting case, when ,∞→y it ( ) velocityflowmain approaches to unity • When eR increases the main flow velocity u also increases • Shear stress components tend to be linearized as 1.0→α • The shear stress components in the direction of main flow )(1 eRF and the function )(2 eRF which characterizes the wall shear stress in the z -direction, strongly depend upon both elastic parameter K and suction parameter α • Reynolds number eR and suction parameter α provide a mechanism to control the backflow • When ,0→K the viscous results for cross flow [2] are recovered • The limiting main flow velocity u when 0→K differs from that obtained by Gersten and Gross [2] due to some calculation mistake in their work • The steady state value of skin friction factor in main flow direction [7] is recovered when .0→K It, however, differs from that obtained by Gersten and Gross due to some computational mistake in their work • The limiting result of )(2 eRF as 0→K is identical to that obtained by Gersten and Gross [2] and steady state value presented by Singh et al [7]. References [1]. G. V. Lachmann, Boundary layer and flow control. Its principles and application, Vol. I and II, Pergamon Press 1961. [2]. K. Gersten and J. F. Gross, Flow and heat transfer along a plane wall with periodic suction, ZAMP 25 (1974), 399-408. [3]. P. Singh, V. P. Sharma and U. N. Misra, Three dimensional fluctuating flow and heat transfer along a American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 153-170 170 plate with suction, Int. J. Heat Mass Transfer. 21 (1978), 1117-1123. [4]. P. Singh, V. P. Sharma and U. N. Misra, Three dimensional free convection flow and heat transfer along a porous vertical plate, Appl. Sci. Res. 34 (1978), no. 1, 105-115. [5]. K. D. Singh, Three dimensional viscous flow and heat transfer along a porous plate, Z. Angew. Math. Mech. 73 (1993), no. 1, 58-61. [6]. K. D. Singh, Hydromagnetic effects on the three-dimensional flow past a porous plate, Z. Angew. Math. Phys. 41 (1990), no. 13, 441-446. [7]. P. Singh, V. P. Sharma and U. N. Misra, Transient three-dimensional flow along a porous plate, Acta Mechanica. 38,183-190 (1981). [8]. M. Guria and R. N. Jana, Hydrodynamic effect on the three-dimensional flow past a vertical porous plate, Int. J of Mathematical Sciences. 2005:20 (2005) 3359-3372. [9]. G. D. Gupta and Rajesh Johari, MHD three-dimensional flow past a porous plate, Indian J. pure appl. 32(3):377-386, March 2001. [10]. H. Schlichting, Boundary layer theory, McGraw-Hill 1968. [11]. R. S. Rivlin, J. L. Ericksen, Stress deformation Relations for isotropic material, J. Rat. Mech. Anal. 4:323-425, 1955. 1. Introduction Description of the problem Formulation of the problem Solution of the problem Cross flow Solution Main flow solution Shear stress components 2. Results and discussion 3. Conclusion