J. Basic. Appl. Sci. Res., 4()37-33, 4 4, TextRoad Publication ISSN 9-434 Journal of Basic and Applied Scientific Researc www.textroad.com Heat Transfer and MHD Boundary Layer Flow over a Rotating Disk * Dr. Farooq Amad, Dr. Sajjad Hussain Punjab Higer Education Department, Principal, Government Degree College Darya Kan, (Bakkar), Pakistan Centre for Advanced Studies in Pure and Applied Matematics, Baauddin Zakariya University, Multan, Pakistan Received: December 3 Accepted: January 4 ABSTRACT An analysis of eat transfer wit MHD boundary layer flow of an electrically conducting viscous fluid over a rotating disk in te presence of magnetic field as been carried out. Te governing partial differential equations ave been transformed in to ordinary differential form by using similarity transformations. Te resulting equations ave been solved numerically by Successive over relaxation (SOR) metod and Simpson s (/3) rule. Te numerical results ave been improved by Ricardson extrapolation to te limit. Te effects of magnetic parameter M, Prandtl number Pr and temperature index parameter n are investigated on te features of fluid flow and temperature profiles. AMS Subject Classification: 76M. KEY WORDS: Rotating Disk, Similarity transformations, Heat Transfer, Ricardson's extrapolation, INTRODUCTION Von Karaman [] was te premier to study te fluid flow past a rotating disk, e reduced te Navier-Stokes equations to self similar form. Te numerical solutions of tese equations were obtained and presented by Cocran []. Sparrow and Gregg [3] and Benton [4] investigated different aspects related to tis area. Watson and Wang [5] considered te deceleration of a rotating disk in a viscous fluid. Watanabe and Oyama [6] studied magnetoydrodynamic boundary layer flow over a rotating disk. Ariel [7] examined te steady, laminar flow of a second grade fluid near a rotating disk. Nazar et al. [8] analyzed unsteady boundary layer flow due to a rotating fluid. Ali and Al-Yousef [9] discussed laminar mixed convection boundary layer under te effect of linearly stretcing permeable surface. Sevcuk and Buscmann [] analyzed eat transfer and fluid flow over a single disk in a fluid rotating as a rigid body. Fang [] presented an exact solution for flow over a stretcable disk along wit a special case for rotating disk. Te influence of magnetic field on te flow of electrically conducting fluids is important because of its applications in engineering and researc. Pavlov [] obtained exact solution of te MHD boundary layer equations of a steady, electrically conducting fluid flow over a rotating two dimensional plane surface under te effect of a uniform magnetic field. Asir et al. [3] considered magnetoydrodynamic electrically conducting fluid over a rotating disk wit suction. Attia [4] investigated eat transfer in an electrically conducting fluid flow due to a rotating, non conducting porous disk in te presence of magnetic field. Unsteady tree dimensional MHD flow and eat transfer in boundary layer was considered by Xu et al. [5]. Hussain and Kamal [6] studied te boundary layer flow of an electrically conducting micropolar fluid on a rotating disk in te presence of magnetic field. In te present work, MHD boundary layer flow over a rotating disk as been examined wit eat convection. Te numerical solution of te problem is found using a very easy, straigtforward and efficient numerical sceme involving finite differences. Te results ave been obtained and presented for several values of te parameters involved in te problem. MATHEMATICAL ANALYSIS An electrically conducting fluid flow over an infinite rotating disk is considered to be incompressible, steady, laminar and axisymmetic. Cylindrical polar coordinates (r,, z) are used, r being te radial distance from te axis, te polar angle and z te normal distance from te disk. Te Navier-Stokes equations of motion for maganetoydrodynamic boundary layer flow as given in ref [6] and te energy equation are transformed by using te following similarity functions: * Corresponding Autor: Dr. Farooq Amad, Punjab Higer Education, Department, Principal, Government Degree College Darya Kan, (Bakkar), Pakistan. Email: farooqgujar@gmail.com 37
Amad and Hussain 4 u rf( ), v rg( ), w H ( ), p P( ) and ( ) ( T T ) /( Tw T ), were z is te dimensionless variable, is density and is angular velocity, being kinematics viscosity, T w is temperature at wall and T is temperature n of outer edge of te boundary layer and Tw ( x) T cx. Here c and n are constants. A set of governing ordinary differential equations is obtained as: F H () F G HF F MF () FG HG G MG, (3) Pr( H n F ) (4) B were primes denote differentiation wit respect to and M is Magnetic interaction parameter, is carge density, B is te magnetic field strengt applied perpendicular to te plane of te disk, Pr is Prandtl number and is termal conductivity. Te boundary conditions are : F, G, H, : F, G, (5) In order to obtain te numerical solution of te differential equations () to (4), we approximate te derivatives involved using central differences at a typical point n of te interval [,), to obtain ( H ) F ( H ) F G ( M F ) F (6) n n n n n n n ( H ) G ( H ) G ( M F ) G (7) n n n n n n ( Pr H n ) (4 P r n n F n ) n ( P r H n ) n (8) were denotes a grid size and Fn F( n ), Gn G( n ) and H n H( n ). Wile equation () is integrated numerically. For computational purposes, we sall replace te interval [,) by [, b], were b is sufficiently large. Results and Discussion Finite difference equations (6) and (7) are solved by using SOR metod Smit [7] and te equation () is integrated using Simpson's (/3) rule Gerald [8] wit te formula given by Milne [9] subject to te appropriate boundary conditions. Te numerical solutions of F, G and are of order of accuracy O ( ) due to second order finite differences approximations to te derivatives. However, te solution of H is accurate to te order of accuracy O( 5 ) and te iger order accuracy O ( 6 ) for te solution of F, G and is acieved by using Ricardson's Extrapolation Burden []. Numerical results ave been computed for several values of te parameters namely magnetic parameter M, Prandtl number Pr and temperature index parameter n. Te calculations are made on tree different grid sizes namely, and 4 to ceck te accuracy of te numerical results. Te results are presented in tabular form in table and table for representative values of te parameters. 38
J. Basic. Appl. Sci. Res., 4()37-33, 4 Te effect of magnetic parameter M on velocity components and temperature distribution is presented in figs. to 4. Te increasing values of te magnetic field reduce tese pysical quantities. It is noticed tat te effect of M is strong. Similarly, te increasing values of te prandtl number Pr and temperature index parameter n bot decrease te temperature distribution as sown in fig.5 and fig.6. Table : Numerical results using SOR Metod and Simpson s Rule for M =.5, Pr=.7and.5 F G H.......344.4443 -.6877.68636..5588.557 -.3976.44887 3..688.55654 -.43988.3495 4..765.733 -.43888.985 5... -.438377. n =.5..5........347.449 -.6883.68457..558.534 -.39368.4866 3..683.5564 -.4476.37595 4..76.737 -.43968.99778 5... -.438454........3486.443 -.68855.686..558.55 -.3944.4633 3..678.55657 -.445.3583 4..758.7335 -.43987.98894 5... -.438469. Table : Numerical results using Ricardson extrapolation metod for M=.5, Pr=.7and n =.5. =.5 =. 5 =. Extrapolated F F F F.......344.347.3486.349..5588.558.558.558 3..688.683.678.676 4..765.76.758.757 5....... =.5 =. 5 =. Extrapolated G G G G.......373.37.384.376..5344.5346.53389.53383 3..57.4.3.8 4..663.658.654.65 5......8 F.6.4 M=.5,,, 3, 5 G.5 M=.5,,, 3, 5. 3 4 5 3 4 5 Fig.: Grap of F for different values of M wen Pr=.7and n =.5. Fig.: Grap of G for different values of M wen Pr=.7and n =.5. 39
Amad and Hussain 4 -..8 -. H -.3 M=.5,,, 3, 5.6.4 Pr =.,.7,., 5., 7. -.4. -.5 3 4 5 Fig.3: Grap of H for different values of M wen Pr=.7and n =.5. 3 4 5 Fig.5: Grap of for different values of Pr, M=.8.6.4. M=,, 3, 4, 5 3 4 5.8.6 n= -3, -,, 3, 5.4. 3 4 5 Fig.4: Grap of for different values of M wen Pr=.7and n =.5. Fig.6: Grap of for different values of n, M= REFERENCES [] Von, T., 9. Karaman, Uberlaminare und Turbulent Reibung, Z. Agnew Mat. Mec. : 33-5. [] Cocran, W. G., 934. Te flow due to a rotating disk, Proceedings of te Cambridge Pilosopical Society. 3: 365-375. [3] Sparrow, E. M. and J. L. Gregg, 96. Flow about an unsteady rotating disk, Journal of Aeronautical Sciences. 7(4): 5-57. [4] Benton, E. R., 966. On te flow due to a rotating disk, Journal of Fluid Mecanics. 4 (4): 78-8. [5] Watson, L. T. and C. Y. Wang, 979. Deceleration of a rotating disk in a viscous fluid, Pysics of Fluids. (): 67-69. [6] Watanabe, T. and T. Oyama, 99. Magnetoydrodynamic Boundary layer flow over a rotating disk, Z. Agnew Mat. Mec. 7(): 5-54. [7] Ariel, P. D., 997. Computation of flow of a second grade fluid near a rotating disk, International journal of Engineering Science. 3: 335-357. 33
Amad and Hussain 4 [8] Nazar, R., N. Amin and I. Pop, 4. Unsteady boundary layer flow due to a rotating fluid, Mecanics Researc Communications. 3: -8. [9] Ali, M. E. and F. Al-Yousef,. Laminar mixed convection boundary layer induced by linearly stretcing permeable surface, Heat Mass Transfer. 45: 4-45. [] Sevcuk, I. V. and M. H. Buscmann, 4. Heat transfer and fluid flow over a single disk in a fluid rotating as a rigid body, J. Termal Science. 3 (3): 79-8. [] Fang, T., 7. Flow over a stretcable disk, Pysics of fluids. 9: -4. [] Pavlov, K. B., 994. Magnetoydrodynamic flow of an incompressible viscous fluid caused by te deformation of a plane surface, Magnitnaya Gidrodinamika. 4: 46-47. [3] Asir, B. E., A. Mansour, Bataine and R. A. Arar, 6. A new ybrid ananlytical analysis of te magnetoydrodynamic flow over a rotating disk under uniform suction, Journal of applied Science. 6 (5): 59-65. [4] Attia, H. A., 9. Steady flow over a rotating disk in a porous medium wit eat transfer, Nonlinear Analysis: Modelling and Control. 4 (): -6. [5 ] Xu, H., S. J. Liao and I. Pop, 7. Series Solution of Unsteady tree dimensional MHD flow and eat transfer in boundary layer over an impulsively stretcing plate, European Journal of Mecanics. 5 (): 5-7. [6] Hussain, S. and Kamal,. Magnetoydrodynamic Boundary Layer micropolar fluid flow over a rotating disk, International Journal of Computational and Applied Mecanics. 7 (3): 3-33. [7] Smit, G. D., 979. Numerical Solution of Partial Differential Equation, Clarendon Press, Oxford. [8] Gerald, C. F., 989. Applied Numerical Analysis, Addison-Wesley Pub. NY. [9] Milne, W. E., 97. Numerical Solution of Differential Equation, Dover Pub. [] Burden, R. L., 985. Numerical Analysis, Prindle, Weber & Scmidt, Boston. 33