The viscous vortex induced by a sink on the axis of a circulating fluid in the presence of a plane free surface

Abstract

The flow under discussion represents an idealization of the bath-tub vortex; distortions of the free surface, finite sink size, and all rigid boundaries have been eliminated from the problem in order to isolate the effect of the non-uniform stretching of vortex lines produced by the sink flow. A boundary-layer type of approximation is made about the axis, which requires that the meridional Reynolds number (N) be large, and since the problem is still intractable, an expansion is made in powers of K = R²/N (where R is the swirl Reynolds number), which measures the strength of the interaction between the swirl and meridional velocity fields. In the limit of zero K the flow is a modified Burgers vortex whose radius decreases to zero at the sink. For non-zero K, the interaction is not restricted to the vortex core, because the presence of the vortex modifies the outer irrotational flow, inducing a radial mass flux into the core, whose dependence on the axial co-ordinate is calculated to the first order in K. The structure of the core is obtained, again to the first order in K, from two co-ordinate expansions, one near the stagnation point on the axis, and the other near the sink, although only the first few terms of the latter can be determined explicitly. It is shown how the methods can be extended not only to higher orders in K, but also to any other narrow viscous vortex in which the vortex lines are stretched non-uniformly away from an internal stagnation point.