Analysis of Stagnation Point flow of an Incompressible Viscous Fluid between Porous Plates with Velocity Slip

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1 8, Issue 1 (018) 0- Journal of Advanced esearc in Fluid Mecanics and Termal Sciences Journal omepage: ISSN: Analysis of Stagnation Point flow of an Incompressible Viscous Fluid between Porous Plates wit Velocity Slip Open Access Aswini Bat 1,, Nagaraj N Katagi 1 1 Department of Matematics, Manipal Institute of Tecnology, Manipal Academy of Higer Education, Udupi Karkala oad, Manipal, Karnataka-610, India ATICLE INFO Article istory: eceived 8 April 018 eceived in revised form 1 June 018 Accepted 1 July 018 Available online 1 August 018 Keywords: Stagnation point flow, slip velocity, computer extended series, omotopy analysis metod, Domb-Sykes plot. ABSTACT Te present study investigates te effects of slip velocity on te stagnation point of an incompressible viscous fluid between porous plates. Te appropriate slip boundary conditions ave been introduced in place of no-slip condition. Te governing equations of motion is solved by Homotopy analysis and Computer extended series metod. Padé approximants are furter used to increase te domain and rate of convergence of te series so generated. Te above metods admits a desired accuracy wose validity increases up to a sufficiently large values of, eynolds number wit different slip coefficients. Te approximate analytical results ave been verified to be very accurate wen compared wit te previous solutions observed by Capman in te absence of slip coefficient. Copyrigt 018 PENEBIT AKADEMIA BAU - All rigts reserved 1. Introduction Te study of flow between te porous or non-porous disk as significant importance due to its applications in bot scientific and industry. Tese types of flows ave applications in bio-mecanics, semiconductor manufacturing process wit rotating wafers, ydrodynamical macines, etc. Hiemenz [1] was te first researcer to propose te basic two dimensional stagnation flow towards plate. Later, tis study was extended to tree dimensional case Howart [] and Davey[]. Axisymmetric stagnation flow on a cylinder was solved by Wang []. Many researcers ave investigated te problem on fluid flow between porous plates/ disks wit suction or blowing[, 6-10]. asmussen[11] numerically analysed te problem of flow between two porous co-axial disks. Capman and Bauer[1] presented te asymptotic and numerical solution for stagnation point viscous flow between porous plates wit uniform blowing. Later, te problem of steady stagnation point flow of an incompressible micro-polar fluid between two porous disks wit uniform blowing was analyzed by Agarwal and Danpal [1]. Tey ave used sooting tecniques for numerical solutions. Elcrat [1] obtained te teorem of existence and uniqueness for non-rotational fluid motion between fixed Corresponding autor. address: Aswini.bat@manipal.edu (Aswini Bat) 0

2 porous disks wit arbitrary suction or blowing. Bujurke et al., [1] discussed te solution of viscous flow between two parallel porous plates by computer extended series analysis followed by Euler transformation to increase te validity of series. A brief review of works on stagnation point flow can be found in paper by Wang [16,1]. Maapatra and Gupta [18] studied te laminar steady stagnationpoint flow of a viscoelastic fluid over a stretcing surface; tey studied flow wen te stretcing velocity of te surface is more (less) tan te free stream velocity. Following tis work, an extensive work as been carried out by many experts on te stagnation-point flow of viscoelastic fluid past a stretcing surface. In all te above analysis of flow wit porous boundaries, a zero slip condition was assumed, wic caracterizes flow wit te solid boundary walls. Howeve, te effect of slip was not considered by tem. Beavers and Josep [19] proved te existence of slip velocity at a porous surface troug teoretical explanations and experimental observations. Te istorical background to Beavers - Josep conditions at te interface of porous media and clear fluid were reported by Neild [0]. Aswini et al., [1,] ave implemented successfully tese Beavers - Josep conditions in te analysis of flow in cannels and pipes. It is clear from te literature tat no attempts ave been made to analyse te influence of slip velocity on te stagnation point flow of an incompressible viscous fluid between porous plates. Te current analysis as developed a model for te same by taking into consideration te velocity slip effects and filled tis gap. Te obtained solutions are in well agreement wit tat of Capman and BauerError! eference source not found. for smaller and large values of eynolds number wen slip effect is zero. Te resulting governing equations wit slip boundary conditions are solved by two novel semi-analytical tecniques for different values of slip coefficient at different eynolds number. Te influence of slip coefficient on pressure gradient, variations in dimensionless axial velocity, dimensionless axial velocity derivative in te presence of velocity slip ave been analysed.. Matematical Formulation Consider a steady, axially symmetric, laminar flow of a viscous incompressible fluid between two parallel porous disks separated by a distance L (Figure 1). Te fluid wit uniform velocity aving magnitude V is injected troug bot porous plates, continuously wic flows radially towards middle plane Z = 0. Under te assumed conditions, te relevent continuity and momentum equations wic governs te flow field and pressure distribution are [9], 1 ( rv r r ) z r v z = 0 (1) Fig. 1. Geometry of axi-symmetric flow between porous plates wit uniform injection velocities 1

3 r-momentum and z-momentum equations. vr vr p 1 ( rvr ) vr ρ ( vr vz ) = µ ( ( ) ) (1) r z r r r r z vz vz p 1 vz vz ρ( vr vz ) = µ ( ( r ) ) ρg r z z r r r z () It is assumed tat v θ = 0. Te boundary conditions are, vz = V at z = L v = v at z = L r slip v z = 0 at z = 0 () v r = 0 at z = 0 (6) z and P= P at r = 0, z = 0 () 0 Te boundary conditions (6) and () are due to planar symmetry, tat is we consider only upper alf of te flow field and Equation () is te slip boundary condition by Beavers and Josep [16]. Te slip velocity at porous surface is being proportional to sear rate at te porous boundary, we ave vr k v r = vslip =, were = is te slip coefficient. αl Because of te symmetrical geometrical properties and uniform boundary conditions, it allows to assume, v r = rϕ ( z) (8) () () v So tat, z = ( z) z (9) P Substituting (8) and (9) into (1) and () concludes tat te quantity, [ / r] is a constant. Equation r (1) becomes, ρ dvz vz d vz ( ) ρ ( dz dz µ d vz ) ( dz P ) =[ / r] r (11) Now, wit dimensionless quantities defined as, z v =, θ = z, LVρ e= and L V µ D p P L = [ ] (10) r ρrv Equation Error! eference source not found. reduces to θ ( ) θ ( ) θ ( ) ( θ ( )) = D p (11) e

4 or θ ( ) = e θ ( ) θ ( ) (1) Te boundary conditions ()-() reduces to, θ = 1 at = 1 (1) θ = 0 at = 0 (1) θ = 0 at = 0 (1) k and θ = ϕθ at = 1 were ϕ =, slip coefficient. (16) αl Also, from (), () and (10) te solution for pressure gradient can be rewritten as, P µ V ρv Dpr P0 [ θθ θ ] d. L (1) 0 8L = From (8) and (9) te radial component of velocity is given by, V v r = rθ ( ) (18) L Tus Equation (1) along wit boundary conditions (1)-(16) describes te entire flow situation and expressions for θ (, ), D p () provides solution to te problem. Solution of Equation (1) is usually solved by direct integration wic frequently involves more tan one integration process because of two point nature of boundary conditions. Tus te use of proposed series metod provides an attractive alternative approac. Also, te terms in te series metod are capable of providing results to any desired accuracy.. Metod of Solution.1 Series Solution Metod We seek te solution of equation (1) for small values of can be expressed in te form of power series as, 0 n n=1 n θ( ) = θ ( ) θ ( ) (19) substituting equation (19) into equation (1) and comparing like powers of on bot sides, we get, n1 θ n = θ n 1 rθ r, n =1,,,K (0) r=0 te boundary conditions are, θn(0) =0, θ n (0) =0, n 0 (1)

5 θ (1) = 1, θ n (1) =0, n 1 () 0 θ (1) = ϕθ (1) 0 () n n n Te solution of above system of equations up to term in are, θ ( ) 0 θ ( ) 1 6 = ( 1) = ( ) ( 1) Te solution for = 0 is given by Capman and Bauer [9]. () Computer extended perturbation solution: As te series (0) is slowly converging, it is not possible to analyze te problem accurately wit just two terms [,]. We need sufficiently large number of universal polynomial coefficients wic reveal te true nature of te solution represented by series (19) [,6]. Manually evaluating te coefficients beyond second order terms is very difficult as one proceeds to iger approximations te algebra becomes cumbersome. Towards tis goal, we proposed recurrence relations along wit Matematica, wic efficiently generates iger order terms of te series. Te axial velocity component is directly obtained as θ ( ) and dimensionless axial velocity derivative is, n n θ ( ) = θ0 ( ) θn ( ) = an () n=1 n=0 Te dimensionless pressure gradient D p is represented by te series, D p = θ (1) θ (1) (6) Coefficients of te series (19) representing θ ( ) and pressure gradient (6) are decreasing in magnitude and ave no fixed sign pattern. Domb-Sykes plot [] is drawn to find te nature of nearest singularities wic restricts convergence of te series.. Homotopy Analysis Metod To compare te solution obtained by extended series metod, we also solve te governing equations wit boundary conditions by anoter useful semi analytical tecnique called Homotopy analysis metod (HAM). As HAM does not depend on a small parameter like oter series metods and allows to transfer a non-linear problem into an infinite number of linear sub-problems, along wit Padé sum it guarantees convergence of te solution in any case.

6 Zerot-order deformation problem: We seek solution of Equation (1) by using HAM and coose te base function to express θ ( ) [8,9]. Te initial guess wic satisfies te boundary conditions is θ 0( ) = () ( 1) 1 ( 1) and auxiliary linear operator is given by, L [ θ ] = θ (8) Te above linear operator wic satisfies te following property, L C C C 6 [ 1 C ] = 0 were C 1, C, C and C are constants to be determined. If q [0,1] ten te zerot order deformation problem can be constructed as, (1 q) L[ θ( n, q) θ0 ( )]= qh( ) N[ θ(, q)] (9) wit relevant boundary conditions, θ (0, q) θ (1, q) θ (1, q) θ (0, q) = 0 = 1 = θ (1, q) = 0 (0) were 0 q 1 is an embedding parameter, and H are non-zero auxiliary parameter and auxiliary function respectively. Furter, N is a non-linear differential operator and is defined as, θ(, q) θ(, q) N [ θ(, q)] = θ (, q) (1) For q = 0 and q = 1, Equation (9) as solution, θ(,0) = θ0( ) θ(,1) = θ( ) () As q varies from 0 to 1, θ (, q) varies from initial guess θ 0( ) to exact solution θ ( ) By Taylor's teorem, Equation () can be expressed as

7 0( ) θm( m () m=1 θ(, q ) = θ )q m 1 θ were, θ m( ) = m q= 0. Convergence of te above series () depends on te convergence m! q control parameter, wic is cosen in suc a way tat () is convergent at q = 1. Ten we ave, 0 m m=1 θ( ) = θ ( ) θ ( ) () m t -order deformation problem: Differentiating te zerot order deformation problem equation-(9) 'm' times wit respect to q t and lastly setting q = 0. Te resulting m order deformation problem becomes, L[ θ ( ) χ θ 1( )]= H( ) ( ) () m m m and te omogeneous boundary conditions are, (0) =0, θm(1)=0, θm (1)= θ m (1), θm (0)=0 m m θ (6) were m1 n n=0 ( ) = 1 ' 1' m θm θnθm () and 0, m 1 χ m = (8) 1, m > 1 We systematically utilized Matematical software, Matematica to obtain te solution for system of linear equations () wit appropriate omogeneous boundary conditions (6). Te solutions up to second order approximations are sown is Eq. (1). Convergence of HAM: Te proposed series () contains te auxiliary parameter wic influences te convergence region and rate of approximations for te HAM solutions. Tis parameter is known as convergence control parameter. To ensure tis series converges, we need to coose a suitable value for. To obtain te permissible ranges of te parameter, -curves are plotted (Fig.6). Figure 6 sows curve for te series D for corresponding values of and slip coefficient at 10t order approximation. p 6

8 ) (6 1) 800( 1 = ) ( ) ( 1) 60( 1 = ) ( r θ θ (9). esults Te equation of motion for steady stagnation point flow between two parallel plates is governed by nonlinear differential equation (1) togeter wit boundary conditions (1)-(16) are solved by computer extended series metod and Homotopy Analysis Metod. We study te effect of slip coefficient on velocity profiles and pressure gradient at different eynolds number. Te results for velocity profiles and pressure gradient ave been presented troug figures and tables. Te proposed series expansion sceme using recurrence relation and Matematica software we generate large number 0) = (n of universal polynomial functions ) ( θ n for different slip coefficients. Te series representing velocity profiles ) ( θ, ) ( θ and pressure gradient p D are analyzed using Padé approximants for larger eynolds number for different slip coefficients. Domb-Sykes plot given in Figure sows te singularity restricting convergence of te series representing velocity profiles, wic gives te nature and location of nearest singularity. After extrapolation, using rational approximation yields te radius of convergence of series () to be 9.01, 9.10 and 9.0 for = 0, 0.1 ϕ and 0. respectively. Te influence of slip coefficient on te velocity profiles are sown in Figure wic are found to be identical wit HAM curves. It sows tat velocity profiles are decreasing wit increasing value of. It is also observed tat te sape of axial velocity profiles does not depend very strongly on eynolds number. Figure sows te variation of axial velocity derivative profiles for different values of eynolds number. It is noted tat larger values of eynolds number results in linear profiles. Influence of slip coefficient on pressure gradient is explained by Figure. It is seen tat pressure gradient, D p decreases wit te influence of and attains a constant value Dp =,. and. for = 0, 0.1 and 0. respectively. To ceck validity of te metods, results were compared for = 0 wit tat of Capman and Bauer Error! eference source not found. are given in Table 1. Te agreement wit earlier findings is an excellent lending support to te metods proposed.

9 Fig.. Domb-Syke plot for velocity profiles Fig.. Variation in dimensionless axial velocity for different eynolds number 8

10 Fig.. Variation in dimensionless axial velocity derivative for different eynolds number Fig.. Dimensionless pressure gradient as a function of eynolds number for different slip coefficient = 0,0.1,0. 9

11 To compare and prove efficiency of te results obtained by Computer extended Series metod, te problem is additionally analyzed by omotopy analysis metod (HAM) alongside Padé sum to accelerate convergence of te series. We plot -curves to discover te convergence range and furtermore te rate of approximations for te series representing θ (0) and D wen p = 0.1, = 0 t respectively from 10 order HAM approximations. Te range for admissible values of for different values of and is different. From te figures 6 and it is observed tat series representing θ (0) and D are convergent wen. 0.1 and.8 0. respectively. p Fig. 6. -curves for (0) -10t order approximations Fig.. -curves for pressure gradient, -10t order approximations 0

12 . Conclusions In tis article, we ave examined te steady stagnation point flow between two permeable plates by Computer Extended Series metod (CES) and Homotopy Analysis Metod (HAM). Te impact of non-zero tangential slip velocity on velocity field and pressure gradient are analysed. Te validity of series solution is extended to a large values of eynolds number by utilizing analytic continuation. Te examination affirms tat te proposed metods converges to te solution for very large values of eynolds number as compared to te earlier findings wen slip coefficient is reduced to zero. eferences [1] Hiemenz, Karl. "Die Grenzscict an einem in den gleicformigen Flussigkeitsstrom eingetaucten geraden Kreiszylinder." Dinglers Polytec. J. 6 (1911): 1-. [] Howart, L. "CXLIV. Te boundary layer in tree dimensional flow. Part II. Te flow near a stagnation point." Te London, Edinburg, and Dublin Pilosopical Magazine and Journal of Science, no. (191): [] Davey, A. "Boundary-layer flow at a saddle point of attacment." Journal of Fluid Mecanics 10, no. (1961): [] Wang, Cang-Yi. "Axisymmetric stagnation flow on a cylinder." Quarterly of Applied Matematics, no. (19): 0-1. [] Berman, Abraam S. "Laminar flow in cannels wit porous walls." Journal of Applied pysics, no. 9 (19): 1-1. [6] Sellars, Jon. "Laminar flow in cannels wit porous walls at ig suction eynolds numbers." Journal of Applied Pysics 6, no. (19): [] Yuan, S. W. "Furter investigation of laminar flow in cannels wit porous walls." Journal of Applied Pysics, no. (196): [8] Mamud, K.., M. M. aman, and A. K. Al Azad. "Numerical Simulation and Analysis of Incompressible Newtonian Fluid Flows using FreeFem." 6: [9] Amed. L. Abdulla, and Fuat Yilmaz. "Computational Analysis of Heat Transfer Enancement in a Circular Tube Fitted wit Different Inserts." Journal of Advanced researc in fluid mecanics and termal sciences. 0(01): [10] Soo Weng Beng, and Wan Mod. Arif Aziz Japar. "Numerical Analysis of Heat and Fluid Flow in Microcannel Heat Sink wit Triangular Cavities". Journal of Advanced researc in fluid mecanics and termal sciences. (01): [11] asmussen, Henning. "Steady viscous flow between two porous disks." Zeitscrift für angewandte Matematik und Pysik ZAMP 1, no. (190): [1] Capman, Tomas W., and Gerald L. Bauer. "Stagnation-point viscous flow of an incompressible fluid between porous plates wit uniform blowing." Applied Scientific esearc 1, no. (19): -9. [1] Agarwal,. S., and C. Danapal. "Stagnation point micropolar fluid flow between porous discs wit uniform blowing." International journal of engineering science 6, no. (1988): [1] Elcrat, Alan. "On te radial flow of a viscous fluid between porous disks." Arcive for ational Mecanics and Analysis61, no. 1 (196): [1] Bujurke, N. M., N. P. Pai, and P. K. Acar. "Semi-analytic approac to stagnation-point flow between porous plates wit mass transfer." Indian Journal of Pure and Applied Matematics 6 (199): -90. [16] Wang, C. Y. "Exact solutions of te steady-state Navier-Stokes equations." Annual eview of Fluid Mecanics, no. 1 (1991): [1] Wang, C. Y. "Similarity stagnation point solutions of te Navier Stokes equations review and extension." European Journal of Mecanics-B/Fluids, no. 6 (008): [18] Maapatra, T. oy, and A. S. Gupta. "Magnetoydrodynamic stagnation-point flow towards a stretcing seet." Acta Mecanica 1, no. 1- (001): [19] Beavers, Gordon S., and Daniel D. Josep. "Boundary conditions at a naturally permeable wall." Journal of fluid mecanics 0, no. 1 (196): [0] Nield, D. A. "Te Beavers Josep boundary condition and related matters: a istorical and critical note." Transport in porous media 8, no. (009):. [1] Bat, Aswini, N. N. Katagi, and Sesappa A. ai. "Analysis of Laminar Flow troug a Porous Cannel wit Velocity Slip." Malaysian Journal of Matematical Sciences 11, no.. 1

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14 Appendix Table 1: Values of, and at various eynolds number for =0. =0.1 =1 =10 =100 =1000 =10000 = z q q' q q' q q' q q' q q' q q' q q' Dp

15 Table : Values of, and at various eynolds number for =0.1. =0.1 =1 =10 =100 =1000 =10000 = z q q' q q' q q' q q' q q' q q' q q' Dp

16 Table : Values of, and at various eynolds number for =0.. =0.1 =1 =10 =100 =1000 =10000 = z q q' q q' q q' q q' q q' q q' q q' Dp

17 Appendix Table 1 Values of, and at various eynolds number for =0. =0.1 =1 =10 =100 =1000 =10000 = z q q' q q' q q' q q' q q' q q' q q' Dp

18 Table Values of, and at various eynolds number for =0.1. =0.1 =1 =10 =100 =1000 =10000 = z q q' q q' q q' q q' q q' q q' q q' Dp

19 Table Values of, and at various eynolds number for =0.. =0.1 =1 =10 =100 =1000 =10000 = z q q' q q' q q' q q' q q' q q' q q' Dp