The oseenlet as a model for separated flow in a rotating viscous liquid

Abstract

The disturbance induced in the uniform flow of a viscous, rotating liquid by an axial point force −D is studied under the restrictions that the Ekman number, E = 2Ων/U2, be small and that D = O(1/logE) as E → 0. The method of matched asymptotic expansions is invoked to obtain inner and outer (with reference to the dimensionless axial co-ordinate x refered to the length U/(2Ω)) approximations to the solution of the Oseen equations as E → 0. The outer approximation, E → 0 with Ex fixed, is also an outer approximation to the solution of the Navier–Stokes equations. The mass flow across any transverse plane, which is equal to D/U for an oseenlet in a non-rotating flow, vanishes in this approximation. The corresponding inner limit yields a non-uniform, cylindrical flow far upstream of the force in the inviscid limit, E → 0, if and only if D ∝ 1/(log E + const.). This cylindrical flow is a one-term, inner approximation to the solution of the Navier–Stokes equations and suffices to show that separation implies the failure of Long's hypothesis of no upstream influence for inviscid, rotating flow past a finite body. A two-term inner representation of the solution is related to Stewartson's solution of the Oseen equations for a moving source in an inviscid, rotating fluid.