An Analytical Study of the Early Stages of Unsteady Free Convective Flow from a Differentially Heated Rotating Sphere at Large Grashof Numbers

This research investigates the unsteady free convective flow of a viscous incompressible fluid from a differentially heated rotating sphere. The flow is assumed to remain laminar and to possess equatorial and azimuthal symmetry. The governing Navier-Stokes and energy equations are posed in terms of a scaled stream function - vorticity formulation and are solved subject to no-slip and specified surface temperature conditions. At t = 0 an impulsive heat flux is applied in the form of a jump in surface temperature. An asymptotic solution valid for large Grashof numbers and small times following the impulsive startup is constructed. Two small parameters have been identified and based on this the flow variables are expanded in a double series in powers of these parameters. The non-zero leading-order terms in the asymptotic expansions have been determined analytically and the corresponding heat transfer coefficient has been found. Future work will involve obtaining numerical solutions.