The unsteady matched Stokes-Oseen solution for the flow past a sphere

Abstract

The authors present the matched asymptotic expansion type of solution for the unsteady viscous incompressible flow past a sphere. Most of the analysis is developed under the assumption that a constant rectilinear velocity is suddenly imparted to a sphere in an otherwise quiescent infinite body of fluid. The Reynolds number based on that velocity is taken to be small, and the analysis is then extended to other transient flows that satisfy this requirement. Evidently in the unsteady cases under discussion one can recognize inner and outer regimes. The leading terms in the expansions representing the flow in these are governed by the unsteady Stokes and the hitherto unreported unsteady Oseen equations. The streamline patterns calculated show the ‘birth’ of a ring vortex close to the equator and its gradual migration downstream and outwards. This result is also verified qualitatively by a crude experiment.