Laminar Boundary Layer Flow of Non-Newtonian Power Law Fluid past a Porous Flat Plate

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Volume 1, No. 10, October 2013 Journal of Global Research in Mathematical Archives RESEARCH PAPER Available online at http://www.jgrma.

1 Volume 1, No. 10, October 2013 Journal of Global Research in Mathematical Archives RESEARCH PAPER Available online at Laminar Boundary Layer Flow of Non-Newtonian Power Law Fluid past a Porous Flat Plate Dr. B.P. Jadhav, DEPARTMENT OF MATHEMATICS, THE NEW COLLEGE, KOLHAPUR bpj.jadhav@gmail.com.Abstract:- Laminar boundary layer flow of non-newtonian Power law fluid past a Porous flat plate has been considered. The governing equations of continuity and momentum are transformed into ordinary differential equations using similarity transformations. The equation is solved by using method of successive approximations starting with zeroth approximation. For n=1 the results tallies with Corresponding results for Newtonian fluids. Velocity profiles are drawn for different values of parameter n, suction/injection parameter shows the behavior of power law fluids. Keywords: - Laminar boundary layer, Power law fluids, successive approximations, velocity profiles, suction /injection. Introduction:-Boundary layer flows of viscous, incompressible fluid past semi infinite flat plate were studied by Blassius [1], Haworth [2], Acrious et al. [3], Schowalter[4]. Jadhav & Wagmode [5] have extended the problem to laminar boundary layer flow of power law fluid Past a flat plate and obtained a approximate solution to the problem for n=2,3.. T.G.Mayer[6] used integral method to find skin friction coefficient and boundary layer thickness for various values of n. Guedda and Hammouch[7] have obtained similarity solution to the power law fluid flow past semi infinite plate.the boundary layer flow of non-newtonian fluids with suction/injection were reported in literature by Patni and Atolia[8],Mahapatra[9], Rajvanshi[10], Mishra and Das[11],Mathur[12],Datta[13]. These studies have not included the flow of Non-Newtonian Power law fluid past a Porous flat plate with suction /injection. Abdel and Khider et al.[14] have obtained the boundary layer parameters for the flow of electrically conducting power law fluid past a porous plate. In this paper we have used the method of successive approximations starting with zeroth approximation to find the solution of the problem. The boundary layer parameters for have been calculated different values of parameter n, suction/injection parameter. Velocity profiles are drawn for different values of flow index parameter n, suction/injection parameter power law fluids. It is found that the method employed gives good agreement with the existing results. shows the behavior of MATHEMATICAL ANALYSIS: Consider a steady, two dimensional flow of an electrically conducting, non-newtonian power law fluid past a semi infinite porous flat plate. The boundary layer equations are = γ (1) + = (2) Where γ = k/ρ, is the coefficient of viscosity. The boundary conditions are u=, v= at y=0 and u ---- (3) JGRMA 2013, All Rights Reserved 46

2 THE SOLUTION OF THE BASIC EQUATIONS: To solve the equations, introduce a stream function (x,y) such that, (4) Let η= y (, ( η) (5) We obtain the velocity components and their derivatives as, η f ) (6) With these values, the equation (1) reduces to (7) The Boundary conditions are (8) Where is suction/injection parameter is modified Reynold number. METHOD OF SOLUTION: To solve the non-linear differential equation (7) under the boundary conditions (8), we use method of successive approximations starting with zeroth approximation. For zeroth approximation, we assume η) = (9) Where, β is arbitrary constant to be determined such that for the first approximation. i. e. β is real root of the equation = (10) The different successive approximations can be obtained from =, (11) For the first approximation, we have, (12) Integration of equation (12) under the boundary conditions (8) leads to η) = (13) JGRMA 2013, All Rights Reserved 47

3 η) = + ] (14) Where,,,, = (15) BOUNDARY LAYER PARAMETERS: 1) Displacement Thickness : ( ) It is defined by = = (16) 2) Momentum Thickness : ( It is defined by = = (17) Skin Friction; ( ) The Sharing stress at the wall is given by at y=0 = K ( [ The local skin friction coefficient is given by = 2 [ = ½ [ ] = [ = (18) DISCUSSIONS: The equation (9) is solved for various values of flow index n and suction /injection parameter. The values of β are obtained by trial method.the boundary layer parameters are calculated for given values of and n. These results have been tabulated in JGRMA 2013, All Rights Reserved 48

4 table no.1.for fixed n, increase in suction leads to decrease in boundary layer thicknesses and increase in the skin friction, while increase in the injection leads to increase in boundary layer thicknesses and fall in the skin friction. For fixed values of suction parameter also., increase in flow index n decrease in boundary layer thicknesses and skin friction For fixed values of suction parameter, decrease in flow index n increase in boundary layer thicknesses and skin friction also. The calculated values of skin friction for zero suction and various values of flow index n are closure to the values obtained by Acrious et al.[3]. For n=1,the results obtained agrees with the results obtained by Blassious[1] for Newtonian fluid. The velocity profiles for drawn for different values of flow index parameter n, suction/injection parameter are shown in figs 1-4. It has been observed that for fixed n, increase in suction leads to increase in velocity, while reverse nature occurs with injection. For fixed values of suction/injection parameter, increase in flow index n, decreases the velocity. The velocity profiles show the typical behavior of non- Newtonian fluids. CONCLUSIONS: The method employed gives good agreement with the results obtained for zero suction for Newtonian and non- Newtonian fluids. The effect of suction is to decrease in boundary layer thicknesses and increase in the skin friction. Table no.-1 Calculation of Boundary layer Parameters n JGRMA 2013, All Rights Reserved 49

5 REFERENCES: [1]Blassius (1908):- Grenzschichten in Flussigkeiten mit kleiner Reibung. Z.Math.u.phys., 56,1-36. [2] Howarth,L(1949):- On the solution of the laminar boundary layer equations, Proc. Roy.Soc.London,A 164, [3] Acrious, A,at. el. (1960):- Momentum and Heat Transfer in laminar boundary layer flow of non- Newtonian fluids past external surfaces.a.i.ch.e.j.6, 312. [4] Schowlter,W.R(1960):-The application of power law Pseudo plastic fluids ; similar solutions; A.I.Ch.E.J.6,24. [5]Jadhav& Wagmode(1988):- Approximate solution of laminar boundary layer flow of power law fluid past flat plate. Jour. Of Ravishankar University, Raipur,vol-1,pp [6] T.G.Mayers(2011):- The approximate method for boundary layer flow of power law fluid over flat plate. Int. Jour of Heat and Mass transfer [7] Mohamed Guedda, Zakia Hammouch(2008):- Similarity Flow Solutions of a Non-Newtonian Power-law Fluid. Vol.6, No.3, pp [8] Atolia(2004):- A simple solution of boundary layer flow of power law fluid past a semi infinite plate. [9] Mahapatra G(1966):-Flow of second order fluid past a porous plate. Ind. Jour. of Pure and Appl.Phys., 4(40, [10] Rajvanshi S.C.(1968);-Steady Laminar flow of Visco-elastic fluid through parallel and uniform walls of different permeability. Ind. Jour. of Pure and Appl.Phys., 6(9),512. A.I.Ch.E.J. [11] Mishra and Das(1973);- Laminar flow of Second order fluid between rotating porous discs. Ind. Jour. of Phys.,47, [12] Mathur M.N. et al.(1975):-genral Plane Coutte flow of polar fluid with suction or injection. Rheologica Acta,14, [13] Datta S.K.(1961):-.Laminar flow of non-newtonian fluid in porous walls. Bull. of Cal. Maths. Soc., 53,1. [14] Abdel and Khider et al.(1980):-on the solution of boundary layer equations for non-newtonian electrically conducting power law fluid past a semi infinite porous plate. Bull. of Cal. Maths. Soc. 72, JGRMA 2013, All Rights Reserved 50

the lines of the solution obtained in for the twodimensional for an incompressible secondorder

the lines of the solution obtained in for the twodimensional for an incompressible secondorder Flow of an Incompressible Second-Order Fluid past a Body of Revolution M.S.Saroa Department of Mathematics, M.M.E.C., Maharishi Markandeshwar University, Mullana (Ambala), Haryana, India ABSTRACT- The