No abstract

The stability and bifurcations associated with the loss of azimuthal symmetry of planar flows of a viscous incompressible fluid, such as vortex-source and Jeffery–Hamel flows, are studied by employing linear, weakly nonlinear and fully nonlinear analyses, and features of new solutions are explained. We address here steady self-similar solutions of the Navier–Stokes equations and their stability to spatially developing disturbances. By considering bifurcations of a potential vortex-source flow, we find secondary solutions. They include asymmetric vortices which are generalizations of the classical point vortex to vortical flows with non-axisymmetric vorticity distributions. Another class of solutions we report relates to transition trajectories that connect new bifurcation-produced solutions with the primary ones. Such solutions provide far-field asymptotes for a number of jet-like flows. In particular, we consider a flow which is a combination of a jet and a sink, a tripolar jet, a jet emerging from a slit in a plane wall, a jet emerging from a plane channel and the reattachment phenomenon in the Jeffery–Hamel flow in divergent channels.

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.

Axisymmetric vortex breakdown in a steady, inviscid, incompressible ow in a semi-inÿnite circular pipe is considered analytically. We suggest a new perception of vortex breakdown and compare ours with other approaches. In our view, vortex breakdown occurs due to solution nonuniqueness in some range of in ow parameters when the entire steady ow experiences a jump to another metastable steady state with the same boundary conditions. These co-existing solutions are smooth along the pipe length; they have the same mechanical energy but, in general, di erent ow forces. Vortex breakdown necessarily occurs by a continuous change in ow parameters (usually the swirl number) when the solution fails to exist (locally) because of fold or similar catastrophe, but spontaneous jumps (in some range of parameters) between di erent metastable solutions (not on a fold) can also be caused by large ow perturbations. The folds can appear due to transcritical bifurcation, which is destroyed (in the case considered here) by the injection of azimuthal vorticity into the vortex core at the pipe entrance. A high level of the entrance swirl leads to separation zones (even for solid-body in ow!) where the steady ow is undetermined. We ÿnd that the nonuniqueness interval in parameter space is connected with the ow pattern inside the separation zone. We consider models for dealing with two such ow patterns: the traditional analytic continuation (leading to a recirculation zone) and a new stagnant separation zone model. We reveal serious defects of the analytic continuation approach. The stagnation zone model is superior in that solutions always exist and, for large enough in ow swirl, exhibit nonuniqueness and folds, thus explaining the experimentally observed hysteretic jump transitions in vortex breakdown. We also predict some new phenomena, which deserve experimental investigation.