Collapse in conical viscous flows

Abstract

A class of steady conically similar axisymmetrical flows of viscous incompressible fluid is studied. The motion is driven by a vortex half-line or conical vortex in the presence of a rigid conical wall or in free space. The dependence of the solutions on parameters (say, the vortex circulation) is analysed. At some finite parameter values the solutions lose existence, i.e. a flow collapse takes place. This is correlated with the appearance of a sink singularity at the symmetry axis. Analytical estimates of the critical parameter values are performed, together with numerical calculations of sounds on the solution existence region in parameter space. Asymptotic analysis of the near-critical regimes shows that a strong axial jet develops. The jet momentum becomes infinite at the critical parameter value and a singularity occurs. These paradoxical features seem to be typical of conically similar viscous flows. Reasons for the paradox and ways of overcoming it are discussed. Solution non-uniqueness and a hysteresis phenomenon are found in the Serrin problem. Possible applications of the results to model some geophysical and asrophysical phenomena are outlined.