On conical swirling flows in an infinite fluid

Abstract

The steady axisymmetric flow generated in an unbounded incompressible viscous fluid, of density ρ and kinematic viscosity v, by torque-producing singularities with constant line density c along the semi-infinite line θ = 0 of a spherical polar coordinate system (r, θ, ϕ) that was investigated by Paull & Pillow (1985b), is reconsidered. The numerical solution constructed revealed the following features. (i) For values of c up to about 46.9 there is only one solution where the axial component of the meridional flow is directed from θ = 0 to θ = π. This solution can be continued to all values of c. (ii) For c > 46.9 the system of equations allows bifurcation and two more solutions with a single separatrix are possible. (iii) For c = ∞ one of the two branches of the separatrix asymptotes to θ = ½π and the other to θ = π. The asymptotic solution for large c constructed by Paull & Pillow (1985b), where the meridional flow consists of two colliding flows, relates to the bifurcation solution where the separatrix asymptotes to ½π as c → ∞.