A note on a source in a rotating liquid

Extract

Consider an inviscid incompressible fluid confined within an infinite tube of radius b and assume that its motion initially consists of a uniform translation with velocity U and a uniform rotation, with angular velocity ω, about the tube axis. Suppose now that at time t = 0 a unit source of fluid situated at a point O of the axis begins to emit fluid at a uniform rate and that ultimately the flow pattern so modified becomes steady (at a fixed point as t → ∞). Under the further assumption that far upstream flow conditions are undisturbed by the source the stream function ψ of the disturbance satisfies

where z denotes distance along the axis from O, r distance from the axis and k = 2ω/U.