Research Article Magnetohydrodynamic Three-Dimensional Couette Flow of a Maxwell Fluid with Periodic Injection/Suction

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1 Hindawi Mathematical Problems in Engineering Volume 2017 Article ID pages Research Article Magnetohdrodnamic Three-Dimensional Couette Flow of a Maxwell Fluid with Periodic Injection/Suction Y. Ali 1 M. A. Rana 1 and M. Shoaib 2 1 Department of Mathematics and Statistics Riphah International Universit Sector I-14 Islamabad Pakistan 2 COMSATS Institute of Information Technolog Kamra Road Attock Pakistan Correspondence should be addressed to Y. Ali; rasir5@gmail.com Received 26 December 2016; Accepted 14 March 2017; Published 13 April 2017 Academic Editor: Eusebio Valero Copright 2017 Y. Ali et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in an medium provided the original work is properl cited. A mathematical model for magnetohdrodnamic (MHD) three-dimensional Couette flow of an incompressible Maxwell fluid is developed and analzed theoreticall. The application of transverse sinusoidal injection at the lower stationar plate and its equivalent removal b suction through the uniforml moving upper plate lead to three-dimensional flow. Approximate solutions for velocit field pressure and skin friction are obtained. The effects of flow parameters such as Hartmann number Renolds number suction/injection parameter and the Deborah number on velocit components skin friction factors along main flow direction and transverse direction and pressure through parallel porous plates are discussed graphicall. It is noted that Hartmann number provides a mechanism to control the skin friction component along the main flow direction. 1. Introduction In recent ears the problem of LFC (laminar flow control) has gained considerable importance due to its importance in the reduction of drag and hence in improving the vehicle power b a considerable amount. To control the boundar laer artificiall several methods have been proposed. One of the effective techniques for the reduction of the drag coefficient which causes large energ losses is the boundar laer suction method. It has been established theoreticall as well as experimentall that the laminarization of boundar laer over a profile reduces the drag and hence the vehicle power requirements b a ver significant amount. Accordingtoboundarlaersuctionmethodslowedthat fluid particles in the boundar laer are removed through the holes and slits in the wall into the interior of the bod and therefore the transition from laminar to turbulent flow causing increase of drag coefficient ma be deferred or prevented [1]. Man workers have considered the numerous aspects of fluid flow problems with suction but most of these studies cope with two-dimensional flows onl. Gersten and Gross [2] considered the viscous fluid and studied the effect of transverse sinusoidal suction velocit on flow with heat transfer over a porous wall. Singh [3] studied the effect of transpiration cooling in the presence of the transverse sinusoidal suction/injection velocit. Chaudhar et al. [4] analzed three-dimensional Couette flow in the presence of transpiration cooling between the plates and reported the effects of suction/injection velocit on the flow field skin friction and heat transfer. Guria and Jana [5] investigated unstead three-dimensional fluctuating Couette flow through porous plates with heat transfer and found that the main flow velocit decreases with increase in frequenc parameter; however the magnitude of the cross flow velocit increases with increase in frequenc parameter. Sharma et al. [6] considered radiation effect in three-dimensional Couette flow with suction/injection on temperature distribution. Chauhan and Kumar [7] investigated heat transient effects in a three-dimensional Couette flow between partl filled channels b a porous material. Various workers [8 11] also investigated three-dimensional flow viscous fluid past a porous plate under different phsical conditions. Man technological problems and natural phenomena are vulnerable to magnetohdrodnamic (MHD) analsis. In the design of heat exchangers and pumps and flow meters thermal protection control and reentr in space propulsion and so

2 2 Mathematical Problems in Engineering forth MHD principle is used b engineers. It has been proven theoreticall and experimentall that the transition from laminar to turbulent flow which causes the drag coefficient to increase ma be prevented/delaed b suction of the fluid b the application of transverse magnetic field and b heat and mass transfer from the boundar laer to the wall. Das [12] studied three-dimensional MHD Couette flow of a viscous incompressible fluid with heat transfer through a porous plate and reported effects of constant suction and sinusoidal injection on the flow. Sharma and Chaudhar [13] presented MHD effect on viscous incompressible flow between two horizontal parallel porous plates and heat transfer with periodic injection/suction. It was observed that forward flow isdevelopedintheregionnearthestationarplatewhile backward flow is developed in the region near the moving plate. Goal and Narania [14] analzed theoreticall threedimensional free convection Couette flow of a viscous incompressible fluid with transpiration cooling in the presence of transverse magnetic field. The static plate and the plate in uniform motion are subjected to transverse sinusoidal injection and uniform suction of the fluid. Recentl man workers [15 17] studied three-dimensional Couette flow of an incompressible fluid. All the above studies have been performed in viscous fluid. Although the Navier-Stokes equations can cope with the flows of viscous fluids these equations are inadequate to describe the characteristics of non-newtonian fluids. Shoaib et al. [18 22] analzed theoreticall three-dimensional non- Newtonian fluids flow along an infinite plane with periodic suction. However to the best of the authors knowledge the application of transverse sinusoidal injection/suction velocit fortheflowofasecond-gradefluidbetweenparallelplateshas not appeared in the literature. Therefore in the present work magnetohdrodnamic three-dimensional Couette flow of a Maxwell fluid with periodic injection/suction is analzed. A constant suction velocit at the wall leads to two-dimensional flow [2]; however due to variation of suction velocit in transverse direction on wall the problem becomes threedimensional. The solution of the problem is presented using regular perturbation technique. The results obtained are evaluated for various dimensionless parameters such as suction/injection parameter α the Deborah number β Hartmann number M and Renolds number Re. The article is organized as follows: Section 2 presents description of the problem Section 3 gives formulation of the problem Section 4 approximates solutions and Section 5 incorporates results and discussion while Section 6 includes conclusion. 2. Description of the Problem Consider stead three-dimensional full developed laminar Couette flow of an incompressible electricall conducting Maxwell fluid between two parallel porous plates having separation h between them. The x z -plane is taken along the lower plate and the -axis perpendicular to the plates as showninfigure1.themagneticfieldofuniformstrengthb U x 0 B V (z ) V (z ) Figure 1: Diagram of the problem. normal to the plates is applied. The injection/suction velocit distribution [2] of the form V (z ) = V 0 (1+εcos π z h ) (1) is assumed where V 0 is suction/injection velocit and ε is its amplitude. The lower plate is kept stationar while the upper plate is moving with uniform velocit U along the positive x -axis. The transverse sinusoidal injection of the fluid at the lower plate with its corresponding removal b periodic suction through the upper plate is considered. The velocit components along the x - - and z -directions are u V andw respectivel.sincetheflowisassumed to be full developed and laminar all the phsical quantities are independent of x ;ofcoursetheflowremainsthreedimensional due to injection/suction velocit (1). 3. Formulation of the Problem The constitutive equation for a Maxwell fluid model is where S+λ D S Dt =μ A 1 h z T = p I+ S (2) D S Dt = d S dt L S S L T (3) in which p λ μ and I denote the pressure relaxation timethednamicviscositandidentittensorrespectivel; also D/Dt is the convective derivative. The Rivlin-Ericksen tensors A 1 are defined as A 1 = L+ L T L =grad V (4) where T denotes the transpose. So model (2) is compatible with the thermodnamics in the sense that all the

3 Mathematical Problems in Engineering 3 motions satisf the Clausius-Duhen inequalit and the specific Helmholtz free energ is minimum in equilibrium. The laws of conservation of mass and momentum are given b div V =0 ρ V (5) t = div T+J B. Thus the problem is governed b the following sstem of differential equations: V + w z =0 ρ(v u u 2 u 2 u +w z )+λ(v 2 2 +w 2 z 2 +2V w 2 u u z )=μ( u z 2 ) σb2 o (u U) λ( u p + u z p z ) ρ(v V V 2 V 2 V +w z )+λ(v 2 2 +w 2 z 2 The following dimensionless parameters have been introduced: = h z= z h u= u U V = V U w= w U α= V o U Re = hu υ β= λu h (8) +2V w 2 V V z )= p +μ( V z 2 ) λ(v 2 p 2 p +w 2 z V p + V z p z ) ρ(v w w 2 w 2 w +w z )+λ(v 2 2 +w 2 z 2 +2V w 2 w w z )= p z +μ( w z 2 ) σb 2 o w λ(v 2 p 2 p z +w w p z 2 + w z p z ) subject to boundar conditions u =0 V (z )=V 0 (1 + ε cos π z h ) w =0 at =0; u =U V (z )=V 0 (1 + ε cos π z h ) w =0; at =h. (6) (7) p= p ρu 2 M= σb2 o h2 ρυ where α Reβ and M denote suction/injection parameter Renolds number the Deborah number and Hartmann number respectivel. Then (6) (7) become V + w =0 (9) z V u +w u z +β(v2 2 u 2 +w2 2 u z 2 +2Vw 2 u z ) = 1 Re ( 2 u u z 2 M2 (u 1)) β( u p + u z p z ) V V +w V z +β(v2 2 V 2 +w2 2 V z 2 +2Vw 2 V z ) = p + 1 Re ( 2 V V z 2 ) β(v 2 p 2 +w 2 p z V p + V p z z ) (10) (11)

4 4 Mathematical Problems in Engineering V w +w w z +β(v2 2 w 2 +w2 2 w z 2 +2Vw 2 w z ) = p z + 1 Re ( 2 w w z 2 M2 w) β(v 2 p p z +w 2 z 2 w p + w p z z ) subject to nondimensional boundar conditions u=0 V (z) =α(1+εcos πz) w=0 u=1 V (z) =α(1+εcos πz) at =0; (12) (13) And the boundar conditions are V 1 (0 z) =αcos πz w 1 (0 z) =0 V 1 (1 z) =αcos πz w 1 (1 z) =0. (18) The injection/suction velocit consists of basic uniform distribution V o with a superimposed weak sinusoidal distribution εv o cos πz; therefore the velocit components V 1 ( z) w 1 ( z) and pressure component p 1 ( z) are also separated into main and small sinusoidal components. Hence we assume that V 1 ( z) = V 11 () cos πz (19) w 1 ( z) = 1 π V 11 sin πz (20) p 1 ( z) =p 11 () cos πz. (21) w=0; at =1. u V and w denote the velocities in the x- - and zdirections respectivel. 4. Solution of the Problem 4.1. Cross-Flow Solution. Since ε 1 and is positive we assumethesolutionoftheform H(z)=H 0 () + εh 1 ( z) + ε 2 H 2 (z)+ (14) where H stands for an of u V wandp. The set of cross-flow solutions V 1 ( z) w 1 ( z)andp 1 ( z) is independent of the main flow velocit component u.the equations governing the fluid flow are V 1 + w 1 =0 (15) z α V 1 2 V 1 +βα2 2 = p 1 βα 2 p Re ( 2 V V 1 z 2 ) (16) Here denotes the differentiation with respect to. It is worth mentioning that the velocit components (19)-(20) identicall satisf the continuit equation (15). Set (18) (20) into (16) and (17) to have αv 11 +βα2 V 11 = p 11 βαp Re (V 11 π2 V 11 ) (22) αv 11 +βα2 V 11 = π2 p 11 βαπ 2 p Re (V 11 (π2 +M 2 ) V 11 ). (23) Eliminate the pressure p 11 from (22) and (23) to get V V 11 +π4 V 11 2π 2 V 11 M2 V 11 R(V 11 π2 V 11 ) (24) =βα 2 Re (V V 11 π2 V 11 ) where α Re =R. Assuming that β 1and taking (24) becomes V 11 () = V 110 () + βv 111 () + O (β 2 ) (25) V V 110 +π4 V 110 2π 2 V 110 M2 V 110 R(V 110 π2 V 110 ) =0. The corresponding boundar conditions are (26) α w 1 2 w 1 +βα2 2 = p 1 z βα 2 p 1 z + 1 Re ( 2 w w 1 z 2 M 2 w 1 ). (17) V 110 (0) =α=v 110 (1) (27) V 110 (0) =0=V 110 (1). In view of boundar conditions (27) the general solution of (26) ields V 110 () = L 7 e L 3 +L 8 e L 4 +L 9 e L 5 +0 e L 6. (28)

5 Mathematical Problems in Engineering 5 Similarl the first-order equation is V V 111 +π4 V 111 2π 2 V 111 M2 V 111 R(V 111 π2 V 111 ) (29) =αr(v V 110 π2 V 110 ). And corresponding boundar conditions are V 111 (0) =0 V 111 (1) =0 V 111 (0) =0 V 111 (1) =0. The solution of the boundar value problem (29)-(30) is V 111 () =1 e L 3 +2 e L 4 +3 e L 5 +4 e L 6 +(5 e L 3 +6 e L 4 +7 e L 5 +8 e L 6 ). (30) (31) Thus (25) (19) and (20) after substituting (28) and (31) respectivel become V 11 () = L 7 e L 3 +L 8 e L 4 +L 9 e L 5 +0 e L 6 +β(1 e L 3 +2 e L 4 +3 e L 5 +4 e L 6 +(5 e L 3 +6 e L 4 +7 e L 5 +8 e L 6 )) V 1 ( z) = (L 7 e L 3 +L 8 e L 4 +L 9 e L 5 +0 e L 6 +β(1 e L 3 +2 e L 4 +3 e L 5 +4 e L 6 +(5 e L 3 +6 e L 4 +7 e L 5 +8 e L 6 ))) cos πz w 1 ( z) = 1 π (L 7L 3 e L 3 +L 8 L 4 e L 4 +L 9 L 5 e L 5 +0 L 6 e L 6 +β(1 L 3 e L 3 +2 L 4 e L 4 +3 L 5 e L 5 +4 L 6 e L 6 +5 e L 3 +6 e L 4 +7 e L 5 +8 e L 6 +(5 L 3 e L 3 +6 L 4 e L 4 +7 L 5 e L 5 +8 L 6 e L 6 ))) sin πz (32) (33) (34) where constants L i (i = ) are defined in the Appendix Main Flow Solution. When ε = 0theproblemis reduced to two-dimensional flow and we have d 2 u 0 d 2 M2 (u 0 1)=R du 0 u 0 d +βαrd2 d 2 (36) and corresponding boundar conditions are u 0 (0) =0 (37) u 0 (1) =1 since β 1soweassumethat Then u 0 =u 00 +βu 01 +O(β 2 ). (38) d 2 u 00 d 2 R du 00 d M2 (u 00 1)=0 the solution of problem (39) is u 00 (0) =0 u 00 (1) =1; (39) u 00 () = 1 + e+l 2 e L 2+ e L. (40) 2 e Similarl solution of the first-order boundar value problem is Thus d 2 u 01 d 2 R du 01 d M2 u 01 =αr d2 u 00 d 2 u 01 (0) =0 u 01 (1) =0 u 01 () = L 31 e +L 32 e L 2 +(L 33 e +L 2 +L 34 e L 2+ ). u 0 () = 1 + e+l 2 e L 2+ e L +β(l 2 e 31 e +L 32 e L2 +(L 33 e +L 2 +L 34 e L 2+L 1 )). (41) (42) (43) When ε =0theequationsofmotiongoverningthefloware perturbedbsubstituting(14)in(10)and(13);wehavethe first-order equation and corresponding boundar conditions. Then the first-order equation and corresponding boundar conditions are 4.2. Pressure. Substituting (32) in (23) we have p 11 () = 9 e L 3 +L 20 e L 4 +L 21 e L 5 +L 22 e L 6 +β(l 23 e L 3 +L 24 e L 4 +L 25 e L 5 +L 26 e L 6 (35) α u 1 + V du 0 2 u 1 1 d +β(α2 2 +2αV d 2 u 0 1 d 2 ) = 1 u 1 Re ( u 1 z 2 ) βdu 0 p 1 d (44) +(L 27 e L 3 +L 28 e L 4 +L 29 e L 5 +L 30 e L 6 )). u 1 (0 z) =0=u 1 (1 z). (45)

6 6 Mathematical Problems in Engineering The solution of (44) can be expressed as u 1 ( z) = u 11 () cos πz.then d 2 u 11 d 2 π 2 u 11 R du 11 d = Re V 11 du 0 d +β(αr d2 u 11 d 2 u d 2 +2RV 0 11 d 2 + Redu 0 dp 11 d d ). The corresponding boundar conditions (45) become (46) u 11 (0) =0=u 11 (1). (47) Equation (46) is of third order while we have onl two conditions. To overcome this difficult we express the solution of (46) as follows: u 11 () = u 110 () + βu 111 () + O (β 2 ). (48) Then zeroth-order problem ields d 2 u 110 d 2 R du 110 d (π2 +M 2 )u 110 = Re V 110 du 00 d The solution of problem (49) is u 110 (0) =0=u 110 (1). u 110 () = L 37 e L 35 +L 38 e L 36 +L 39 e (L 2+L 3 ) L 40 e (+L 3 ) +L 41 e (L 2+L 4 ) L 42 e (+L 4 ) +L 43 e (L 2+L 5 ) L 44 e (+L 5 ) +L 45 e (L 2+L 6 ) L 46 e (+L 6 ). (49) (50) Similarl the first-order problem and corresponding boundar conditions are d 2 u 111 d 2 R du 111 d (π2 +M 2 )u 111 and hence u 111 () = L 55 e L 35 +L 56 e L L 57 L 35 L 36 e L 35 L 58 L 36 L 35 e L 36 + L 59 L 47 e (L 2+L 3 ) + L 60 L 48 e (+L 3 ) + L 61 e (L 2+L 4 ) + L 62 e (+L 4 ) + L 63 e (L 2+L 5 ) + L 64 L 52 e (+L 5 ) + L 65 L 53 e (L 2+L 6 ) + L 66 e (+L 6 ) + L 67 ( 2(L 2 +L 3 ) R L 47 L 47 + L 68 ( 2( +L 3 ) R L 48 L 48 + L 69 ( 2(L 2 +L 4 ) R + L 70 ( 2( +L 4 ) R + L 71 ( 2(L 2 +L 5 ) R + L 72 ( 2( +L 5 ) R L 52 L 52 + L 73 ( 2(L 2 +L 6 ) R L 53 L 53 )e (L 2+L 3 ) )e (+L 3 ) )e (L 2+L 4 ) )e (+L 4 ) )e (L 2+L 5 ) )e (+L 5 ) )e (L 2+L 6 ) + L 74 ( 2( +L 6 ) R )e (+L 6 ). In view of (43) (50) (52) and (48) to (14) one has u(z)=1+ e+l 2 e L 2+ +β(l 31 e +L 32 e L 2 +(L 33 e +L 2 +L 34 e L 2+ )) (52) = Re (V 110 ( du 01 d +2αd2 u 00 d 2 ) + du 00 d (V dp 110 d )+α2 d 2 u 00 d 2 ) u 111 (0) =0 u 111 (1) =0 (51) +ε(l 37 e L35 +L 38 e L36 +L 39 e (L 2+L 3 ) L 40 e (+L ) 3 +L 41 e (L 2+L ) 4 L 42 e (+L 4 ) +L 43 e (L 2+L ) 5 L 44 e (+L ) 5 +L 45 e (L 2+L 6 ) L 46 e (+L ) 6 +β(l 55 e L35 +L 56 e L 36

7 Mathematical Problems in Engineering 7 + L 57 e L35 L + 58 e L 36 L 35 L 36 L 36 L 35 + L 59 L 47 e (L 2+L 3 ) + L 60 L 48 e (+L 3 ) + L 61 e (L 2+L 4 ) + L 62 e (+L 4 ) + L 63 e (L 2+L 5 ) + L 64 L 52 e (+L 5 ) + L 65 L 53 e (L 2+L 6 ) + L 66 e (+L 6 ) + L 67 ( 2(L 2 +L 3 ) R L 47 L 47 + L 68 ( 2 ( +L 3 ) R L 48 L 48 + L 69 ( 2(L 2 +L 4 ) R + L 70 ( 2( +L 4 ) R + L 71 ( 2(L 2 +L 5 ) R + L 72 ( 2( +L 5 ) R L 52 L 52 + L 73 ( 2(L 2 +L 6 ) R L 53 L 53 )e (L 2+L 3 ) )e (+L 3 ) )e (L 2+L 4 ) )e (+L 4 ) )e (L 2+L 5 ) )e (+L 5 ) )e (L 2+L 6 ) + L 74 ( 2( +L 6 ) R )e (+L ) 6 )) cos πz. (53) 4.3. Shear Stress Components. The components of shear stress F 1 and F 2 in the x-direction and z-direction respectivel on the lower plate are as follows: τ z = ε π (dv 11 d ) sin πz =0 τ x =( du 0 d ) +ε( du 11 =0 d ) cos πz =0 F 2 = 1 π (dv 11 d ) =0 F 1 =( du 11 d ). =0 So from (32) and (53) respectivel we have F 2 = 1 π (L 7L 2 3 +L 8L 2 4 +L 9L L 2 6 +β(1l L L L (5 L 3 +6 L 4 +7 L 5 +8 L 6 ))) (54) F 1 =L 25 L 23 +L 26 L 24 +L 27 (L 2 +L 3 ) L 28 ( +L 3 )+L 29 (L 2 +L 4 ) L 30 ( +L 4 )+L 31 (L 2 +L 5 ) L 32 ( +L 5 )+L 33 (L 2 +L 6 ) L 34 ( L 45 +L 6 )+β(l 43 L 23 +L 44 L 24 + L 23 L 24 + L 46 L 24 L 23 + L 47 L 35 (L 2 +L 3 )+ L 48 L 36 ( +L 3 ) + L 37 (L 2 +L 4 )+ L 38 ( +L 4 )+ L 39 (L 2 +L 5 )+ L 52 L 40 ( +L 5 )+ L 53 L 41 (L 2 +L 6 ) + L 42 ( +L 6 )+ L 55 L 35 (1 (L 2 +L 3 )(2(L 2 +L 3 ) R) L 35 )+ L 56 L 36 (1 ( +L 3 )(2( +L 3 ) R) L 36 )+ L 57 L 37 (1 (L 2 +L 4 )(2(L 2 +L 4 ) R) L 37 )+ L 58 L 38 (1 ( +L 4 )(2( +L 4 ) R) L 38 )+ L 59 L 39 (1 (L 2 +L 5 )(2(L 2 +L 5 ) R) L 39 )+ L 60 L 40 (1 ( +L 5 )(2( +L 5 ) R) L 40 )+ L 61 L 41 (1 (L 2 +L 6 )(2(L 2 +L 6 ) R) L 41 )+ L 62 L 42 (1 ( +L 6 )(2( +L 6 ) R) L 42 )) (55) where L i (i= )are defined in the Appendix. 5. Results and Discussion In this work stead and full developed laminar Couette flow of an incompressible Maxwell fluid through porous plates with periodic suction/injection is modelled and investigated analticall. The application of transverse sinusoidal injection at the lower plate remained stationar and its equivalent confiscation b suction through the uniforml moving upper plate leads to three-dimensional flow. The coupled highl nonlinear equations of motion are solved engaging perturbation method. The effects of various nondimensional

8 8 Mathematical Problems in Engineering 1.0 Re =2 z=0 = 2 M=1 1.0 z=0 = 2 M= u u Figure 2: Variation of u along for different values of α. Re =1 Re =2 Re =3 Figure 4: Variation of u along for different values of Re. 1.0 Re =2 z=0 = 2 M=1 1.0 z=0 = 2 Re= u u Figure 3: Variation of u along for different values of β. M=1 M=2 M=3 Figure 5: Variation of u along for different values of M. parameters on velocit field skin friction components and pressure are presented graphicall in Figures The main flow velocit profiles are depicted in Figures 2 5. The effects of injection/suction parameter α thedeborah number β and Renolds number Re are shown in Figures 2 4 respectivel. It is evident from Figure 2 that injection/suction parameter causes decrease in the main flow velocit component. In fact fluid experiences greater viscosit with the porous boundaries and hence offers resistance to flow resulting in reduction in the velocit. It is noted that the velocit decreases exponentiall with increasing the DeborahnumberorRenoldsnumber.Forhighervalueof the Deborah number or Renolds number the deca is more. The minimum and maximum velocities occur on the plates which are the velocities of the plates. It is noted that main flow velocit increases with the increase in Hartmann number M (Figure 5). v Re =2 z=0 = 2 M=1 Figure 6: Variation of V along for different values of α.

9 Mathematical Problems in Engineering Re =2 z=0 = 2 M=1 03 Re =2 z = 0.5 = 2 M= v 125 w Figure 7: Variation of V along for different values of β. Figure 10: Variation of w along for different values of α. 140 z=0 = 2 M=1 03 Re =2 z = 0.5 = 2 M= v w Re =1 Re =2 Re =3 Figure 8: Variation of V along for different values of Re. Figure 11: Variation of w along for different values of β. 140 z=0 = 2 Re=2 03 z = 0.5 = 2 M= v w M=1 M=2 M=3 Re =1 Re =2 Re =3 Figure 9: Variation of V along for different values of M. Figure 12: Variation of w along for different values of Re.

10 10 Mathematical Problems in Engineering z = 0.5 = 2 Re=2 8 6 w F Re M=1 M=2 M=3 M=1 M=2 M=3 Figure 13: Variation of w along for different values of M. Figure 16: Variation of F 1 along Re for different values of M. 8 M=1 1.5 M= F F Re Figure 14: Variation of F 1 along Re for different values of α Re Figure 17: Variation of F 2 along Re for different values of α. 60 M=1 M= F F Re Re Figure 15: Variation of F 1 along Re for different values of β. Figure 18: Variation of F 2 along Re for different values of β.

11 Mathematical Problems in Engineering 11 8 z=0 = 2 M= F p Re M=1 M=2 M=3 Figure 19: Variation of F 2 along Re for different values of M. Re =1 Re =2 Re =3 Figure 22: Variation of p along for different values of Re. Re =2 z=0 = 2 M=1 6 z=0 = 2 Re= p 1 0 p Figure 20: Variation of p along for different values of α. M=1 M=2 M=3 Figure 23: Variation of p along for different values of M. p Re =2 z=0 = 2 M=1 Figure 21: Variation of p along for different values of β. The influences of injection/suction parameter α the Deborah number β Renolds number Re and Hartmann number M on velocit component V are shown in Figures 6 9 respectivel. It is observed from Figures 6 and 9 that the velocit increases with increasing the injection/suction parameter α and Hartmann number M. Thismeansthat injection/suction and Hartmann number provide a mechanism to enhance the velocit. Moreover for α 0.1 the velocit profile behaves as a linear function; of course for α 0.4 the velocit profile is parabolic. Figures 7 and 8 illustrate the effect of the Deborah number β and Renolds number Re on the velocit component. The velocit as expected decreases with an increase in the Deborah number or Renolds number. Smmetric velocit profiles about the middle of the plates are obtained. The transverse velocit component w is studied for different values of injection/suction parameter α thedeborah

12 12 Mathematical Problems in Engineering number β Renolds number Re and Hartmann number M in Figures It is noted from Figure 10 that there is forward flow from = 0 to about = 0.5 andthen onwards there is backward flow. In fact the dragging effect of the faster laer exerted on the fluid particles near the lower plate (stationar plate) is sufficient to overcome the adverse pressure gradient and hence there is forward flow. On the contrar due to the periodic suction at the upper plate (moving plate) the dragging effect of the faster laer exerted on the fluid particles will be reduced and hence this dragging effect is insufficient to overcome the adverse pressure gradient and there is backflow. It is also observed that the backflow is just the optical image of the forward flow. It is obvious from Figures that the velocit component w increases with an increase of α β andreinforwardflow;howevera reverse effect is seen in the backward flow. On the contrar the velocit component w decreases with an increase in M in forward flow while it increases in backflow as shown in Figure 13. The variations of skin friction components at the lower plateversusrenoldsnumberreinthemainflowdirection and transverse directions are presented in Figures Figures depict the effect of injection/suction the Deborah number and Hartmann number on the skin friction component F 1. Depending upon the value of α and β the value of F 1 decreasesforsmallvaluesofrenoldsnumberand then increases for large values of Renolds numbers (Figures 14 and 15). Phsicall it seems that for small values of Re viscous forces are dominant over the inertial forces causing decrease in skin friction along the main flow direction and skin friction is exerted b the plate on the fluid. On the contrar for large values of Renolds number the inertial forces become dominant over the viscous forces resulting in the change in direction of the skin friction; that is the skin friction is exerted b the fluid on the plate which enhances b increasing the Renolds number. Depending upon the value of Hartmann number M the skin friction component F 1 decreases for small values of M and then increases rapidl for large values of M (Figure 16). It reflects that viscous forces are dominant over the electromagnetic forces for small Hartmann number resulting in reduction in skin friction exerted b the plate on the fluid along the x-direction. In contrast the electromagnetic forces become dominant over the viscous forces for large M leadingtochangeofroleofskin friction; that is in this case skin friction is exerted b the fluid on the plate which increases exponentiall b increasing the Hartmann number. Figures are drawn for skin friction component along z-direction versus the Renolds number for different values of injection/suction parameter α the Deborah number βandhartmannnumbermrespectivel.itisobservedthat the skin friction component F 2 attains maximum value for particular value of α β orm and then reduces rapidl and approaches zero. The reduction in the skin friction exerted bthefluidontheplateforlargevaluesofrenoldsnumber happened due to the dominance of viscous forces over the inertial forces. Moreover the skin friction exerted b the fluid on the plate increases b increasing suction parameter (for small value of Re). Then of course it approaches zero duetodominantroleofviscousforces.similareffectof the Deborah number β on the skin friction component is observed. However weak dependence of skin friction component along the z-direction is recorded. It is observed from Figure 19 that with the increase of Hartmann number M skinfrictioncomponentf 2 decreases for small values of Renolds number and increases for large values of Renolds number and approaches zero for Re 100. The effects of injection/suction parameter αthedeborah number β Renolds number Re and Hartmann number M on pressure are shown in Figures respectivel. It is noted from Figure 20 that for an increase in injection/suction parameter α adverse pressure increases near the stationar plate; of course favourable pressure increases near the moving plate. Phsicall this means that injection at the lower plate promotes thickening of boundar laer which in turn enhances the adverse pressure near the lower plate. On the contrar suction through the upper moving plate causes thinning of boundar laer resulting in enhancement in favourable pressure near the upper plate. Figure 21 indicates that adverse pressure increases near the stationar plate with increasing the Deborah number; however it decreases near the moving plate and favourable pressure is noted near the moving plate for β 0.1. Such behavior is expected because injection enhances adverse pressure (Figure 20) and the Deborah number also promotes the pressure and therefore ultimatel large development in adverse pressure near the stationar plate. On the other hand suction enhances favourable pressure (Figure 20) while the Deborah number enhances adverse pressure resulting in combined effect in the form of favourable pressure for β 0.1 near the moving plate. Figure 22 shows that adverse pressure decreases with an increase in Renolds number in the vicinit of the lower plate and favourable pressure also decreases b increasing Renolds number near the upper plate. It indicates that the dominance of inertial forces reduces the injection (at lower plate) and suction (through upper plate) effects causing reduction in pressure in the neighborhood of the plates. Figure 23 indicates that pressure increases with an increase in Hartmann number and reverse effect can be observed near the moving plate. 6. Concluding Remarks On the basis of above discussion the following conclusions are made: (1) The main flow velocit decreases with increasing either injection/suction parameter or Renolds number. It decreases with an increase in the Deborah number. (2) The velocit component V and transverse velocit component w increase with increasing injection/suction parameter; however a reverse effect is observed with an increase in the Deborah number and Renolds number. (3) Renolds number provides a mechanism to stabilize the skin friction components F 1 and F 2.

13 Mathematical Problems in Engineering 13 (4) The present analsis gives a better result as variable injection/suction velocit is considered at both plates because in natural practice injection/suction cannot be uniform in all cases. Appendix Constants involved in this paper are = R R 2 +4M 2 2 L 2 = R+ R 2 +4M 2 2 L 3 = ( ) 2 +4π 2 2 L 4 = + ( ) 2 +4π 2 2 L 5 = L 2 (L 2 ) 2 +4π 2 2 L 6 = L 2 + (L 2 ) 2 +4π 2 2 L 7 =( e L 4 L 4 L 5 α+e L 5 L 4 L 5 α+l 6 L 4 L 5 α e L 5+L 6 L 4 L 5 α L 4 L 6 α e L 4+L 5 L 4 L 6 α e L 6 L 4 L 6 α+e L 5+L 6 L 4 L 6 α e L 5 L 5 L 6 α+l 5 L 5 L 6 α+e L 6 L 5 L 6 α e L 4+L 6 L 5 L 6 α) ( e L 3+L 4 L 3 L 4 +L 5 L 3 +L 6 L 3 L 4 e L 4+L 6 L 3 L 4 +e L 3+L 4 L 3 +L 5 L 3 L 5 e L 3+L 6 L 3 L 5 +e L 5+L 6 L 3 L 5 e L 3+L 4 L 4 L 5 +e L 3+L 5 L 4 L 5 +L 6 L 4 L 5 e L 5+L 6 L 4 L 5 e L 3+L 4 L 3 L 6 +e L 3+L 5 L 3 L 6 +L 6 L 3 L 6 e L 5+L 6 L 3 L 6 +e L 3+L 4 L 4 L 6 e L 4+L 5 L 4 L 6 e L 3+L 6 L 4 L 6 +e L 5+L 6 L 4 L 6 e L 3+L 5 L 5 L 6 +L 5 L 5 L 6 +e L 3+L 6 L 5 L 6 e L 4+L 6 L 5 L 6 ) 1 L 8 =( e L 3 L 3 L 5 α+e L 5 L 3 L 5 α+e L 3+L 6 L 3 L 5 α e L 5+L 6 L 3 L 5 α+e L 3 L 3 L 6 α e L 3+L 5 L 3 L 6 α e L 6 L 3 L 6 α+e L 5+L 6 L 3 L 6 α e L 5 L 5 L 6 α+e L 3+L 5 L 5 L 6 α+e L 6 L 5 L 6 α e L 3+L 6 L 5 L 6 α) (e L 3+L 5 L 3 L 4 e L 4+L 5 L 3 L 4 e L 3+L 6 L 3 L 4 +L 6 L 3 L 4 e L 3+L 4 L 3 L 5 +L 5 L 3 L 5 +e L 3+L 6 L 3 L 5 e L 5+L 6 L 3 L 5 +e L 3+L 4 L 4 L 5 e L 3+L 5 L 4 +L 6 L 4 L 5 +e L 5+L 6 L 4 L 5 +e L 3+L 4 L 3 L 6 e L 3+L 5 L 3 L 6 e L 4+L 6 L 3 L 6 +e L 5+L 6 L 3 L 6 e L 3+L 4 L 4 L 6 +L 5 L 4 L 6 +e L 3+L 6 L 4 L 6 e L 5+L 6 L 4 L 6 +e L 3+L 5 L 5 L 6 e L 4+L 5 L 5 L 6 e L 3+L 6 L 5 L 6 +L 6 L 5 L 6 ) 1 L 9 =( e L 3 L 3 L 4 α L 3 L 4 α+e L 3+L 6 L 3 L 4 α e L 4+L 6 L 3 L 4 α+e L 3 L 3 L 6 α e L 3+L 4 L 3 L 6 α e L 6 L 3 L 6 α+l 6 L 3 L 6 α e L 4 L 4 L 6 α+e L 3+L 4 L 4 L 6 α+e L 6 L 4 L 6 α e L 3+L 6 L 4 L 6 α) ( e L 3+L 5 L 3 L 4 +L 5 L 3 +L 6 L 3 L 4 e L 4+L 6 L 3 L 4 +e L 3+L 4 L 3 +L 5 L 3 L 5 e L 3+L 6 L 3 L 5 +e L 5+L 6 L 3 L 5 e L 3+L 4 L 4 L 5 +e L 3+L 5 L 4 L 5 +L 6 L 4 L 5 e L 5+L 6 L 4 L 5 e L 3+L 4 L 3 L 6 +e L 3+L 5 L 3 L 6 +L 6 L 3 L 6 e L 5+L 6 L 3 L 6 +e L 3+L 4 L 4 L 6 e L 4+L 5 L 4 L 6 e L 3+L 6 L 4 L 6 +e L 5+L 6 L 4 L 6 e L 3+L 5 L 5 L 6 +L 5 L 5 L 6 +e L 3+L 6 L 5 L 6 e L 4+L 6 L 5 L 6 ) 1 0 =( (( e L 3 +e L 5 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 5 )) ( e L 3 L 3 ( L 3 +L 4 )α+l 3 ( e L 3 L 3 L 4 )α) +( e L 3 L 3 L 4 L 3 L 3 L 5 e L 5 L 3 L 4 L 5 +e L 5 L 4 L 5 )(( e L 3 )L 3 α+( L 3 +L 4 )(α L 3 α))) (( e L 3 L 3 L 4 L 3 L 3 L 5 e L 5 L 3 L 4 L 5 +e L 5 L 4 L 5 )(( e L 3 +e L 6 )( L 3 +L 4 ) ( e L 3 )( L 3

14 14 Mathematical Problems in Engineering +L 6 )) (( e L 3 +e L 5 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 5 )) ( ( e L 3 L 3 L 4 )( L 3 +L 6 )+( L 3 +L 4 ) ( e L 3 L 3 +e L 6 L 6 ))) 1 1 = L 3 L 4 (( ( e L 3 )( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 ) ( L 3 +L 4 )(1 L 3 +L 5 L 3 +L 4 )) (e L 4 L 5 ) 1 +(( e L 5 L 4 +e L 6 L 4 L 5 e L 6 L 6 +e L 5 L 6 ) ( ( ( e L 3 )( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 )( L 3 +L 4 )) ( e L 3 L 3 L 4 )( L 3 +L 5 )+( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) + (e L 4 L 5 e L 4 L 5 )( ( )( e L 3 L 3 L 4 )+( L 3 +L 4 ) ( e L 3 5 (1 + L 3 ) e L 4 6 (1 + L 4 ) e L 5 7 (1 + L 5 ) e L 6 8 (1 + L 6 ))))) ((e L 4 L 4 +e L 3 L 5 )( ( ( e L 3 L 3 L 4 )( L 3 +L 5 )+( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) (( e L 3 +e L 6 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 6 )) + (e L 4 L 5 )( ( e L 3 L 3 L 4 )( L 3 +L 6 ) +( L 3 +L 4 )( e L 3 L 3 +e L 6 L 6 )))) 1 2 = L 3 L 4 (( ( e L 3 )( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 ) ( L 3 +L 4 )( L 3 +L 5 )) (( L 3 +L 4 )(e L 4 L 5 ) + ((e L 5 L 3 e L 6 L 5 +e L 6 L 5 +e L 3 L 6 e L 5 L 6 )) ( ( ( e L 3 )( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 )( L 3 +L 4 )) ( e L 3 L 3 L 4 )( L 3 +L 5 )+( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) + (e L 4 L 5 e L 4 L 5 )( ( )( e L 3 L 3 L 4 )+( L 3 +L 4 ) ( e L 3 5 (1 + L 3 ) e L 4 6 (1 + L 4 ) e L 5 7 (1 + L 5 ) e L 6 8 (1 + L 6 )))) 1 )((e L 4 L 4 +e L 3 L 5 )( ( ( e L 3 L 3 L 4 )( L 3 +L 5 )+( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) (( e L 3 +e L 6 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 6 )) + (e L 4 L 5 )( ( e L 3 L 3 L 4 )( L 3 +L 6 ) +( L 3 +L 4 )( e L 3 L 3 +e L 6 L 6 )))) 1 3 =( ( e L 3 )( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 )( L 3 +L 4 )) (e L 4 L 3 e L 3 L 5 ) 1 ((( e L 3 +e L 6 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 6 )) ( ( ( e L 3 ) ( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 )( L 3 +L 4 )) ( ( e L 3 L 3 L 4 )( L 3 +L 5 ) +( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) + (e L 4 L 5 )( ( ) ( e L 3 L 3 L 4 )+( L 3 +L 4 )(e L 3 5 (1 + L 3 ) 6 (1 + L 4 )+e L 5 7 (1 + L 5 )+e L 6 8 (1 + L 6 ))))) ((e L 4 L 5 )( ( ( e L 3 L 3 L 4 )( L 3 +L 5 )+( L 3 +L 4 )

15 Mathematical Problems in Engineering 15 ( e L 3 L 3 +e L 5 L 5 )) (( e L 3 +e L 6 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 6 )) + (e L 4 L 4 +e L 3 L 5 )( ( e L 3 L 3 L 4 )( L 3 +L 6 )+( L 3 +L 4 )( e L 3 L 3 +e L 6 L 6 )))) 1 4 =( ( ( e L 3 )( )+( e L 3 5 e L 4 6 e L 5 7 e L 6 8 )( L 3 +L 4 )) ( ( e L 3 L 3 5 = 6 = 7 = 8 = L 4 )( L 3 +L 5 )+( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) + (e L 4 L 5 )( ( )( e L 3 L 3 L 4 )+( L 3 +L 4 )( e L 3 5 (1 + L 3 ) e L 4 6 (1 + L 4 ) e L 5 7 (1 + L 5 ) e L 6 8 (1 + L 6 )))) ( ( ( e L 3 L 3 L 4 )( L 3 +L 5 )+( L 3 +L 4 )( e L 3 L 3 +e L 5 L 5 )) (( e L 3 +e L 6 )( L 3 +L 4 ) ( e L 3 )( L 3 +L 6 )) + (e L 4 L 5 )( ( e L 3 L 3 L 4 )( L 3 +L 6 ) +( L 3 +L 4 )( e L 3 L 3 +e L 6 L 6 ))) 1 αrl 7 L 2 3 (L2 3 π2 ) (L 3 L 4 )(L 3 L 5 )(L 3 L 6 ) αrl 8 L 2 4 (L2 4 π2 ) (L 4 L 3 )(L 4 L 5 )(L 4 L 6 ) αrl 9 L 2 5 (L2 5 π2 ) (L 5 L 3 )(L 5 L 4 )(L 5 L 6 ) αr0 L 2 6 (L2 6 π2 ) (L 6 L 3 )(L 6 L 4 )(L 6 L 5 ) 9 = L 7L 3 π 2 Re (L2 3 RL 3 (π 2 +M 2 )) L 20 = L 8L 4 π 2 Re (L2 4 RL 4 (π 2 +M 2 )) L 21 = L 9L 5 π 2 Re (L2 5 RL 5 (π 2 +M 2 )) L 22 = 0L 6 π 2 Re (L2 5 RL 5 (π 2 +M 2 )) L 23 = L 24 = L 25 = L 26 = 1 π 2 Re ((L2 3 RL 3 (π 2 +M 2 )) L (3L 2 3 2RL 3 (π 2 +M 2 )) 5 ) α(9 + α π 2 L 7L 2 3 )L 3 1 π 2 Re ((L2 4 RL 4 (π 2 +M 2 )) L (3L 2 4 2RL 4 (π 2 +M 2 )) 6 ) α(l 20 + α π 2 L 8L 2 4 )L 4 1 π 2 Re ((L2 5 RL 5 (π 2 +M 2 )) L (3L 2 5 2RL 5 (π 2 +M 2 )) 7 ) α(l 21 + α π 2 L 9L 2 5 )L 5 1 π 2 Re ((L2 6 RL 6 (π 2 +M 2 )) L (3L 2 6 2RL 6 (π 2 +M 2 )) 8 ) α(l 22 + α π 2 0L 2 6 )L 6 L 27 = 5L 3 π 2 Re (L2 3 RL 3 (π 2 +M 2 )) L 28 = 6L 4 π 2 Re (L2 4 RL 4 (π 2 +M 2 )) L 29 = 7L 5 π 2 Re (L2 5 RL 5 (π 2 +M 2 )) L 30 = 8L 6 π 2 Re (L2 5 RL 5 (π 2 +M 2 ))

16 16 Mathematical Problems in Engineering L 31 = (L 21 +L 22 )e +L 2 L 32 = (L 21 +L 22 )e +L 2 e e L 2 L 33 = αrl 2 2 ( )(L2 ) L 34 = αrl 2 1 ( )(L2 ) L 35 = R R2 +4(π 2 +M 2 ) 2 L 36 = R+ R2 +4(π 2 +M 2 ) 2 L 37 = L 38 = 1 e L 35 e L 36 (L 39 (e L 36 e L 2+L 3 ) L 40 (e L 36 e +L 3 )+L 41 (e L 36 e L 2+L 4 ) L 42 (e L 36 e L 2+L 4 )+L 43 (e L 36 e L 2+L 5 ) L 44 (e L 36 e +L 5 )+L 45 (e L 36 e L 2+L 6 ) L 46 (e L 36 e +L 6 )) 1 e L 36 e L 35 (L 39 (e L 35 e L 2+L 3 ) L 40 (e L 35 e +L 3 )+L 41 (e L 35 e L 2+L 4 ) L 42 (e L 35 e L 2+L 4 )+L 43 (e L 35 e L 2+L 5 ) L 44 (e L 35 e +L 5 )+L 45 (e L 35 e L 2+L 6 ) L 46 (e L 35 e +L 6 )) L 39 = Re L 7L 2 e ( )L47 L 40 = Re L 7 e L 2 ( )L48 L 41 = Re L 8L 2 e ( )L49 L 42 = Re L 8 e L 2 ( )L50 L 43 = Re L 9L 2 e ( )L51 L 44 = Re L 9 e L 2 ( )L52 L 45 = Re 0L 2 e ( )L53 L 46 = Re 0 e L 2 ( )L54 L 47 =(L 2 +L 3 L 35 )(L 2 +L 3 L 36 ) L 48 =( +L 3 L 35 )( +L 3 L 36 ) =(L 2 +L 4 L 35 )(L 2 +L 4 L 36 ) =( +L 4 L 35 )( +L 4 L 36 )

17 Mathematical Problems in Engineering 17 =(L 2 +L 5 L 35 )(L 2 +L 5 L 36 ) L 52 =( +L 5 L 35 )( +L 5 L 36 ) L 53 =(L 2 +L 6 L 35 )( +L 6 L 36 ) =( +L 6 L 35 )( +L 6 L 36 ) L 55 = 1 e L ( L 57 e L L e L 36 + L 59 (e L 36 e L 2+L 3 )+ L 60 (e L 36 e +L 3 )+ L 61 (e L 36 e L 2+L 4 ) 35 e L 36 L 36 L 35 L 35 L 36 L 47 L 48 + L 62 (e L 36 e +L 4 )+ L 63 (e L 36 e L 2+L 5 )+ L 64 L 52 (e L 36 e +L 5 )+ L 65 L 53 (e L 36 e L 2+L 6 )+ L 66 (e L 36 e +L 6 ) L 67 ((1 2(L 2 +L 3 ) R )e L 2+L 3 + 2(L 2 +L 3 ) R e L 36 ) L 68 ((1 2( +L 3 ) R )e +L 3 L 47 L 47 L 48 L 48 L ( +L 3 ) R L 48 e L 36 ) L 69 ((1 2(L 2 +L 4 ) R 2( +L 4 ) R )e +L 4 + 2( +L 4 ) R e L 36 ) L 71 L 52 )e L 2+L 4 + 2(L 2 +L 4 ) R e L 36 ) L 70 ((1 ((1 2(L 2 +L 5 ) R )e L 2+L 5 + 2(L 2 +L 5 ) R e L 36 ) L 72 ((1 2( +L 5 ) R )e +L 5 + 2( +L 5 ) R e L 36 ) L 73 ((1 2(L 2 +L 6 ) R )e L 2+L 6 L 52 L 52 L 53 L 53 L 56 = + 2(L 2 +L 6 ) R L 53 e L 36 ) L 74 ((1 2( +L 6 ) R )e +L 6 + 2( +L 6 ) R e L 36 )) 1 e L ( L 57 e L L e L 36 + L 59 (e L 35 e L 2+L 3 )+ L 60 (e L 35 e +L 3 )+ L 61 (e L 35 e L 2+L 4 ) 36 e L 35 L 36 L 35 L 35 L 36 L 47 L 48 + L 62 (e L 35 e +L 4 )+ L 63 (e L 35 e L 2+L 5 )+ L 64 L 52 (e L 35 e +L 5 )+ L 65 L 53 (e L 35 e L 2+L 6 )+ L 66 (e L 35 e +L 6 ) L 67 ((1 2(L 2 +L 3 ) R )e L 2+L 3 + 2(L 2 +L 3 ) R e L 35 ) L 68 ((1 2( +L 3 ) R )e +L 3 L 47 L 47 L 48 L 48 L ( +L 3 ) R L 48 e L 35 ) L 69 ((1 2(L 2 +L 4 ) R 2( +L 4 ) R )e +L 4 + 2( +L 4 ) R e L 35 ) L 71 L 52 )e L 2+L 4 + 2(L 2 +L 4 ) R e L 35 ) L 70 ((1 ((1 2(L 2 +L 5 ) R )e L 2+L 5 + 2(L 2 +L 5 ) R e L 35 ) L 72 ((1 2( +L 5 ) R )e +L 5 + 2( +L 5 ) R e L 35 ) L 73 ((1 2(L 2 +L 6 ) R )e L 2+L 6 L 52 L 52 L 53 L (L 2 +L 6 ) R L 53 e L 35 ) L 74 L 57 =α 2 RL 37 L 2 35 L 58 =α 2 RL 38 L 2 36 ((1 2( +L 6 ) R L 59 = Re (L 7 L 32 L 2 +α 2 L 39 (L 2 +L 3 ) 2 +e (L 7 L 33 + L 60 = Re (L 7 L 31 α 2 L 40 ( +L 3 ) 2 +e L 2 (L 7 L 34 L 61 = Re (L 8 L 32 L 2 +α 2 L 41 (L 2 +L 4 ) 2 +e (L 8 L 33 + )e +L 6 + 2( +L 6 ) R e L 35 )) L 2 (2αL 7 L L 3 ))) (2αL L 3 ))) L 2 (2αL 8 L L 20 L 4 )))

18 18 Mathematical Problems in Engineering L 62 = Re (L 8 L 31 α 2 L 42 ( +L 3 ) 2 +e L 2 (L 8 L 34 L 63 = Re (L 9 L 32 L 2 +α 2 L 43 (L 2 +L 5 ) 2 +e (L 9 L 33 + L 64 = Re (L 9 L 31 α 2 L 44 ( +L 5 ) 2 +e L 2 (L 9 L 34 L 65 = Re (0 L 32 L 2 +α 2 L 45 (L 2 +L 6 ) 2 +e (0 L 33 + L 66 = Re (0 L 31 α 2 L 46 ( +L 6 ) 2 +e L 2 (0 L 34 L 67 = Re L 2 (L 7 L 33 + L 68 = Re (L 7 L 34 L 69 = Re L 2 (L 8 L 33 + L 70 = Re (L 8 L 34 L 71 = Re L 2 (L 9 L 33 + L 72 = Re (L 9 L 34 L 73 = Re L 2 (0 L )e 5 )e L 2 6 )e 6 )e L 2 7 )e 7 )e L 2 8 )e (2αL L 20 L 4 ))) L 2 (2αL 9 L L 21 L 5 ))) (2αL L 21 L 5 ))) L 2 (2α0 L L 22 L 6 ))) (2α0 +4 +L 22 L 6 ))) L 74 = Re (0 L 34 8 )e L 2 (A.1) Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper. References [1] G. N. V. Lachmann Boundar Laer and Flow Control Its Principles and Applicationvol.1-2PergamonPress1961. [2]K.GerstenandJ.F.Gross Flowandheattransferalonga plane wall with periodic suction Zeitschrift für angewandte Mathematik und Phsikvol.25no.3pp [3] K. D. Singh Three-dimensional Couette flow with transpiration cooling Zeitschrift fur Angewandte Mathematik und Phsikvol.50no.4pp [4] R. C. Chaudhar M. C. Goal and U. Gupta Threedimensional Couette flow with transpiration cooling between twohorizontalparallelporousplates J. Mech. Cont. & Maths. Scivol.2no.1pp [5] M. Guria and R. N. Jana Three-dimensional fluctuating Couette flow through the porous plates with heat transfer International Journal of Mathematics and Mathematical Sciencesvol Article ID pages [6] B. K. Sharma M. Agarwal and R. C. Chaudhar Radiation effect on temperature distribution in three-dimensional Couette flow with suction or injection Applied Mathematics and Mechanics (English Edition) vol. 28 no. 3 pp [7] D. S. Chauhan and V. Kumar Heat transfer effects in a couette flow through a composite channel partl filled b a porous medium with a transverse sinusoidal injection velocit and heat source Thermal Science vol. 15 supplement 2 pp. S175 S [8] H. Schlichting Boundar Laer TheorMcGraw-Hill [9]P.SinghJ.K.SharmaMisraandK.A.Naraan Threedimensional convective flow and heat transfer in a porous medium Indian Journal of Pure and Applied Mathematicsvol. 19 no. 11 pp [10] N. Ahmed and D. Sarma Three dimensional free convective flow and heat transfer through a porous medium Indian Journal of Pure and Applied Mathematics vol.28no.10pp [11] M. Guria and R. N. Jana Hdrodnamic effect on the threedimensional flow past a vertical porous plate International Journal of Mathematics and Mathematical Sciences vol.2005 no. 20 pp [12] S. S. Das Effect of constant suction and injection on MHD three dimensional couette flow and heat transfer through a

19 Mathematical Problems in Engineering 19 porous medium Journal of Naval Architecture and Marine Engineeringvol.6no.1pp [13] P. K. Sharma and R. C. Chaudhar Magnetohdrodnamics effect on three-dimensional viscous incompressible flow between two horizontal parallel porous plates and heat transfer with periodic injection/suction International Journal of Mathematics and Mathematical Sciencesvol.2004no.62pp [14] M. Goal and N. Narania MHD three dimensional free convection Couette flow with transpiration cooling International Journal of Science and Research (IJSR) Index Copernicus Value vol. 6 article [15] L. Sreekala and E. K. Redd Stead MHD Couette flow of an incompresseble viscous fluid through a porous medium between two infinite parallel plates under effect of inclined magnetic field The International Journal of Engineering and Science (IJES)vol.3no.9pp [16] A. Idowu and J. Olabode Unstead MHD poiseuille flow between two infinite parallel plates in an inclined magnetic field with heat transfer IOSR Journal of Mathematics vol.10no.3 pp [17] K. Sumathi T. Arunachalam and N. Radha Effect of magnetic field on three dimensional fluctuating couette slip flow past porous plates Applied Mathematical Sciences vol.8no pp [18] M. Shoaib M. A. Rana and A. M. Siddiqui Three dimensional flow of upper convected Maxwell fluid along an infinite plane wall with periodic suction Journal of Computational and Theoretical Nano sciencevol.13no.8pp [19] M. Shoaib M. A. Rana A. M. Siddiqui and M. Darus Threedimensional magnetohdrodnamics flow of upper convected Maxwell fluid along an infinite plane wall with periodic suction Journal of Computational and Theoretical Nanoscience vol. 13 no. 11 pp [20] M. Shoaib A. M. Siddiqui M. A. Rana and M. Imran Threedimensional flow of a second grade fluid along an infinite horizontal plane wall with periodic suction Research Journal for Engineering Technolog and Sciences (ASRJETS)vol.18no. 1pp [21] M. Shoaib M. A. Rana and A. M. Siddiqui The effect of slip condition on the three-dimensional flow of Jeffre fluid along a plane wall with periodic suction JournaloftheBrazilianSociet of Mechanical Sciences and Engineering2017. [22] A. M. Siddiqui M. Shoaib and M. A. Rana Three-dimensional flow of Jeffre fluid along an infinite plane wall with periodic suction Meccanica 2017.

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