Implicit Finite Difference Solution of MHD Boundary Layer Heat Transfer over a Moving plate

Transcription

1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn:39-765x. Volume 9, Issue 3 (Nov. Dec. 03), PP 8-3.iosrournals.org Implicit Finite Difference Solution of MHD Boundary Layer Heat Transfer over a Moving plate Satish V Desale, V H Pradhan (R.C.Patel Institute of Technology, Shirpur, 45405) (S.V.National Institute of Technology, Surat, ) Abstract: In this paper, magneto hydrodynamics boundary layer heat transfer over a moving flat plate is discussed, using similarity transformation momentum and energy equations that are reduced in to nonlinear ordinary differential equations. The nonlinear differential equations are solved using implicit finite difference Keller box method. Graphical results of fluid velocity and temperature profile are presented and discussed for various parameters. Keyords: MHD boundary layer, moving surface, heat transfer, Keller box method. I. INTRODUCTION Magnetohydrodynamics (MHD) is the study of the collaboration of conducting fluids ith electromagnetic phenomena. The flo of an electrically conducting fluid in the occurrence of a magnetic field is important in various areas of technology and engineering such as MHD poer generation, MHD flo meters, MHD pumps, etc. Rosso [], as discussed MHD boundary layer flo past a flat plate. He discussed to cases namely, magnetic field is fixed relative to play and magnetic fluid is fixed relative to fluid. Heat transfer of MHD boundary layer flo past a flat plate has been discussed by Rosso [], Carrier and Green Span [], and Afzal [3]. Rosso[] has studied transverse magnetic field hereas Carrier et al. [] have considered the effect of a longitudinal magnetic field effect on the velocity and the temperature profile distribution. Sakiadis [4] studied the problem of forced convection along an isothermal moving plate. Erickson et al. [5] discussed the heat and mass transfer on a moving continuous flat plate ith suction or inection. Tsou et al. [6] considered flo and heat transfer in the boundary layer on a continuously moving surface. In this paper, MHD boundary layer heat transfer flo over a continuously moving plate is considered and solved ith the help of implicit finite difference Keller box method and various results are discussed graphically. II. GOVERNING EQUATIONS Consider the steady flo of an electrically conducting, viscous, incompressible fluid past a continuously moving flat plate ith even velocity and surface temperature in the presence of uniform transverse magnetic field. The magnetic Reynolds number of the flo is taken to be very small so that the induced magnetic field can be ignored. The fluid properties are assumed to be isotropic and constant. Therefore, under the usual boundary layer approximations, the governing equations of motion are [7] u v 0 x y () u u u B0 u u v x y y () T T u u B0 u v x y y Cp y Cp u (3) here u and v are velocity components in x and y directions respectively, is the kinematic viscosity, is the electrical conductivity, T is the temperature, is the thermal diffusivity of the fluid and heat at constant pressure. C is the specific p The boundary conditions are y 0; u U, v 0, T T (4).iosrournals.org 8 Page

2 y ; u U, T T here U and U are constants and denote the free stream and sheet velocities, respectively. Define the stream function ( x, y) such that u ; v y x Using the folloing similarity transformation (5) ( x, y) xu f ( ) T T ( ) T T U here y and U= U x U is reference velocity. The momentum and energy equation reduces to f ff Mf 0 Pr f Pr Ec f M Pr Ec f 0 (6) (7) With the boundary conditions 0 : f 0, f r, U Where r U U Here : f r, 0 xb M 0 U is moving parameter (0 r ) and prime denotes differentiation ith respect to. is the magnetic parameter. Pr is Prandtl number. U Ec is the Eckert Number. CpT T (8) III. KELLER BOX METHOD Equation (6)-(7) subect to the boundary conditions (8) is solved numerically using implicit finite difference method that is knon as Keller-box in combination ith the Neton s linearization techniques as described by Cebeci and Bradsha [8]. This method is completely stable and has second-order accuracy. In this method the transformed differential equation (6)-(7) are rites in terms of first order system, for that introduce ne dependent variable uv, such that f u (9) u v (0) () here prime denotes the differentiation.r.to. Equation (6)-(7) become.iosrournals.org 9 Page

3 v fv Mu 0 () f Ecv M Ecu Pr Pr Pr 0 ith ne independent boundary conditions are f (0) 0, u(0) r, (0) u( ) r, ( ) 0 (4) No rite the finite difference approximations of the ordinary differential equations (9)-() for the midpoint n, x of the segment using centered difference derivatives, this is called centering about n, x f f u u h u u v v h h Ordinary differential equations ()-(3) is approximated by the centering about the mid-point the rectangle. (3). n This can be done in to steps. In first step, approximate equation at x, ithout specifying. v fv M u v fv M u n n (5) (6) (7) n x, Pr Pr Ec n Pr Ec n Pr Pr Ec n f v M u f v M Pr Ecu n In second step approximate equations at (for simplicity, remove n) v v f f v v u u u u v v M fv M u h h n n of (8) f f Pr v v Pr Ec v v u u u u M Pr Ec h n n n Pr f Pr Ecv M Pr Ecu h No linearize the nonlinear system of equations (5)-(9) using the Neton s quasi-linearization method [5] For that use, (9).iosrournals.org 0 Page

4 f f f n u u u n n n Equation (5)-(9) reduces to h f f u u r h u u v v r h r 3 ( a ) v ( a ) v ( a ) f ( a ) f ( a ) u ( a ) u r ( b ) ( b ) ( b ) f ( b ) f ( b ) u ( b ) u ( b ) v ( b ) v r The linearized difference equation of the system (0) has a block tridiagonal structure. In a vector matrix form, it can be ritten as A C B A C B A C [ ] [ r] [ ] [ r] [ ] [ r3] : : : : BJ AJ C J : : BJ AJ [ J] [ rj] This block tridiagonal structure can be solved using LU method explained by Na [9] IV. RESULTS AND DISCUSSION Graphically, effects of magnetic parameter (M), moving parameter (r) and Eckert Number (Ec) on velocity profile as ell as on temperature profile are shon in folloing figures. Fig. and Fig shos, the fluid velocity profile decreases as the magnetic parameter increases for constant value of moving parameter r =0 and r =0. respectively. Fig 3 presents velocity profile for different values of moving parameter. In Fig 4 and Fig 5, as the magnetic parameter M and Eckert number (Ec) increases the temperature profile increases. While in Fig 6, temperature profile decreases as Prandtl number increases. (0).iosrournals.org Page

5 Figure Effect of magnetic parameter on velocity profile hen r=0, Pr=0.7,Ec=0.. Figure Effect of magnetic parameter on velocity profile hen r=0., Pr=0.7,Ec=0.. Figure 3 Effect of moving parameter (r) on velocity profile hen M=0., Pr=0.7,Ec=0.. Figure 4 Effect of magnetic parameter (M) on temperature profile hen r=0., Pr=0.7,Ec=0...iosrournals.org Page

6 Figure 5 Effect of Eckert Number (Ec) on temperature profile hen r=0., Pr=0.7, M=0.. Figure 6 Effect of Prandtl Number (Pr) on temperature profile hen r=0., EC=0., M=0.. REFERENCES [] Rosso VJ, On the flo of electrically conducting fluid over a flat plate in the presence of transverse magnetic field.-maca, Rept.(958),358 [] Carrier GF and Greenspan HP, J. Fluid Mech. 67, 959, 77. [3] Afzal N, Int. J. Heat Mass Trans. 5, 97, 863. [4] Sakiadis B C, Boundary layer behaviour on continuous moving solid surfaces: I. Boundary layer equations for to dimensional and axi-symmetric flo, II. Boundary layer on continuous flat surface, III. Boundary layer on a continuous cylindrical surface. AICHE Journal, 7, 96, 6-8, -5, [5] Erickson L E, Fan L T and Fox V G, Heat and mass transfer on a moving continuous flat plate ith suction or inection, Ind. Engg. Chem. Fundam.5, 966, 9-5. [6] Tsou F K, Sparro E M and Goldstein R J, Flo and heat transfer in the boundary layer on a continuous moving surface, Int J Heat Mass Transfer, 0, 967, [7] R. N. Jat, Abhishek Neemaat,Dinesh Raotia, MHD Boundary Layer Flo and Heat Transfer over a Continuously Moving Flat Plate, International Journal of Statistika and Mathematika,3(3),0,0-08. [8] Cebeci T., Bradsha P. Physical and computational Aspects of Convective Heat Transfer, (Ne York: Springer, 988), [9] Na T.Y., Computational Methods in Engineering Boundary Value Problem, (Ne York: Academic Press 979), -8..iosrournals.org 3 Page