International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 5, May 2013) 718 Numerical Solution For The Deceleration of A Rotating Disk in A Viscous Fluid Sajjad Hussain 1 , Dr. Farooq Ahmad 2 , M. Shafique 3 , Sifat Hussain 4 1, 4 Centre for Advanced Studies in Pure and Applied Mathematics, B. Z. Uni., Multan, Pakistan 2, a Corresponding author 2 Mathematics Department, Government Degree College Darya Khan (Bhakkar) 30000, Punjab, Pakistan 3 Department of Mathematics, Gomal University, D. I. Khan, Pakistan Abstract--The unsteady similarity equations have been obtained for a disk rotating in a viscous fluid that decelerates with angular velocity proportional to time. The resulting ordinary differential equations have been solved numerically using an easy numerical technique for range 0 100    s of the non-dimensional parameter s which measures unsteadiness. The calculations have been carried out using three different grid sizes to check the accuracy of the results. The results are compared with the previous work and found in good agreement. AMS Subject Classification: 76M20. Keywords-- Newtonian Fluids, Numerical Analysis, Rotating Disk I. INTRODUCTION Unsteady flows are of importance from the practical point of view and full unsteady Navier-Stokes equations with all the unsteady, nonlinear and viscous terms are difficult to solve whereas exact solutions are rare. However, similarity solution to the governing equations is of special interest in case of fluid flow along a rotating disk. The flow of an incompressible viscous fluid past an infinitely rotating disk was first studied by Von Karman [1] who reduced the necessary Navier-Stokes equations to self- similar form by means of some transformations, and derived approximate solutions. Cochran [2] at a later stage presented accurate numerical solutions to these equations. Different physical situations were studied in this area by Benton [3] and Sparrow & Gregg [4]. Pop [5] investigated the problem of unsteady flow past a wall which starts impulsively to stretch from rest. Nazar et al [6] investigated unsteady boundary layer flow due to a rotating fluid. Xu et al [7] considered unsteady three dimensional MHD flow and heat transfer in boundary layer over an impulsively stretching plate. The motion of an electrically conducting fluid film squeezed between two parallel disks in the presence of a transverse magnetic field was studied by Hamza [8]. Watson et al [9] considered the two dimensional channel flow symmetrically driven by accelerating walls. Ariel [10] studied the problem of steady laminar flow of a second grade fluid near a rotating disk. MHD flow due to non- coaxial rotation of an accelerated disk and a fluid at infinity was analyzed by Asghar et al [11]. The influence of an external uniform magnetic field on the flow due to a rotating disk was studied by Attia [12]. The steady flow of an incompressible viscous fluid above an infinite rotating disk in a porous medium with heat transfer was considered by Attia [13]. Watson and Wang [14] studied deceleration of a rotating disk in a viscous fluid for the range 0 20    s . In the present work, we have extended the numerical solutions up to the range 0 100    s by using SOR method, Richardson's extrapolation and Simpson's (1/3) rule. The numerical scheme used is very easy, straight forward and efficient. II. MATHEMATICAL ANALYSIS The fluid flow is unsteady and incompressible. w v u , , are velocity components in cylindrical polar coordinates (r,  , z). The z-axis is the axis of rotation of the disk, with z = 0 on the surface of the disk. The following similarity transformations are used: ) 1 ( 0 t r u     ) (  f  , ) 1 ( 0 t r v     g ( ), 2 / 1 2 / 1 0 ) 1 ( ) ( 2 t w       f ) (  and ) ( ) 1 ( 0     P t p     , (1)