Laminar boundary layer on an impulsively started rotating sphere

Abstract

Viscous, incompressible, axially symmetric flow about an impulsively started rotating sphere is studied in terms of non-steady, partially linearized Navier-Stokes equations. The non-linear centripetal acceleration is included in full, but the other non-linear terms are neglected because of the restriction in interest to the case of large (but subcritical) Reynolds or Taylor numbers, a2Ω/v. Approximate closed-form solutions for u(r,θ,t), v(r,θ,t), w(r,θ,t) are found which satisfy all relevant boundary and initial conditions. The linearization approximation is checked for consistency and a restriction on Ωt is found. The velocity profiles, in the range of validity, are shown to be approximately similar in time, so their shapes may be qualitatively correct for larger values of Ωt. Some comparison with existing steady-state theories is given and the boundary-layer displacement thickness and viscous torque on the sphere are calculated.