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Symmetry and the hydrodynamic blow-up problem. (English) Zbl 1002.76095
Summary: We address the problem of whether a spontaneous singularity can occur in finite time in an incompressible inviscid fluid flow. As suggested by previous numerical simulations, candidate flows are restricted to be invariant under the octahedral group of symmetries and to have a compact vortex tube in the fundamental domain. It is shown that in such a flow the image vorticity contributes strongly to the axial strain rate in a way which is only weakly proportional to the curvalure of vortex lines. Analysis of a model flow shows that axial strain rate scales as the inverse square of the distance to the origin, and that the velocity field forms a topological trap in which the vortex tube is accelerated towards the origin – a degenerate critical point. Evidence from simulations supports these findings. These features suggest that linear strain rate/vorticity coupling can occur in a finite-time pointwise collapse of such symmetric flows.

MSC:
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
76M22 Spectral methods applied to problems in fluid mechanics
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