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.

On physical mechanisms in chemical reaction-driven tip-streaming

“…However, accounting for nonlinear inertia effects, which are present in our case since We w ϭ10 4 , is straightforward with the only difference that the solution can be found only numerically: one needs only to remark that for high enough Weber numbers one can expect nonexistence of a self-similar solution in view of generation of swirl such as in the Taylor cone problem. 34,35 Under these conditions, the self-similar solution is defined by…”

Numerical Solutions of Serrin’s Equations by Double Exponential Transformation

“…Therefore we can expect that P k 2 = 7.6478 is the critical value where Serrin's equations lose the uniqueness of the solution. Note that Goldshtik & Shtern [3] claimed numerically that the critical value for the lack of the uniqueness of solutions was P k 2 = 7.6447. Our result P k 2 = 7.6478 is very close to their result.…”

“…He also obtained some properties of functions f , Ω and G, and computed solutions numerically by successive iterations. Goldshtik & Shtern [3] studied some equations which were equivalent to Serrin's equations. They derived the asymptotic expansion of solutions and numerically computed solutions by the shooting method.…”

A study of similarity solutions for laminar swirling axisymmetric flows with both buoyancy and initial momentum flux

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