Development of the Stokes flow field associated with a line vortex perpendicular to a plane wall

Abstract

A viscous incompressible fluid occupying the space θ≤π/2 and bounded by the wall θ=π/2 of a spherical polar coordinate system (r,θ,φ), is stirred by a line vortex along the line θ=0 which is switched on at time t=0. The line vortex is perpendicular to the wall. The development of the flow configuration is considered for the case where the poloidal flow is weak and does not affect the structure of the inducing azimuthal flow. The problem is formulated in terms of the similarity variable r/2(νt)1/2 and the polar angle θ, where ν is the kinematic viscosity of the fluid. An analytical solution is constructed for the azimuthal flow. At any given station r the steady azimuthal velocity field is, practically, reached within time r2/ν. The equations governing the poloidal flow are coupled partial differential equations of mixed elliptic–parabolic type which are transformed to equations that are elliptic throughout the solution domain. These equations are solved numerically using the methods of successive overrelaxation and fast Fourier transform. The results show that the poloidal flow in a meridional plane at time t forms closed loops about the point r≊1.58(νt)1/2, θ=π/4, where the velocity has only an azimuthal component. The case of a diffusing configuration from the steady state, due to switching off at t=0 of the agent generating the flow, is also considered. For this case the poloidal field consists of open streamlines and at t=2r2/ν its intensity is a very small fraction of that associated with the steady state.