On the impulsively started rotating sphere

Abstract

The velocity field generated in a fluid of viscosity, v, by impulsively starting at time t = 0, a sphere of radius a spinning with angular velocity Ω about a diameter is described using a new expansion variable 2 √vt/r. It is first shown how the standard time-dependent boundary-layer equations can be modified to give series solutions satisfying all the boundary conditions. Next, that these new solutions are relevant when the Reynolds number R = a²Ω/v goes to infinity in such a way that $R^{\frac{1}{3}} \Omega t$ is large. Lastly, solutions are given, applicable at small times for non-zero Reynolds numbers. These last expansions show that the velocity components decay algebraically rather than exponentially at large distances.