Time-Dependent Stokes Flow Induced by Instantaneous Sources

Abstract

Analytical solutions are provided for Stokes flow fields induced by an instantaneous thermal source located (i) in an infinite fluid and (ii) at the center of a solid spherical surface. In both cases it is assumed that the flow is driven by a body force that arises from buoyancy effects in the fluid due to the gradient of the source’s temperature field. In accordance with Stokes approximation, convective acceleration terms are left out of the momentum equations but the local acceleration term is retained. The retention of the local acceleration term is crucial if all the salient features of the flow, especially in its early stages, are to be included in the solution for low Prandtl number. The evolution of the flow field in each of the two cases is demonstrated by plotting streamline patterns at various times. The influence of Prandtl number on flow field development is also discussed. Problem (i) is the fundamental one for this class of problems. From its solution, solutions to problems where the heat-generating rate is a general but known function of position and time to bounded and unbounded domains may be synthesised using well-known techniques. These results may be fruitfully applied to such practical problems as the motion of vessels in which nuclear wastes are stored, the fluid dynamics of weak point explosions, and the cooling of electrical and electronic equipments.