Singularity formation in vortex sheets and interfaces
Résumé
One of the paradigms of nonlinear science is that patterns result from instability and bifurcation. However, another pathway is possible: self-similar evolution, singularity formation, and form. One example of this process is the formation of spherical drops through the pinch off of a cylindrical thread of liquid. Other example is given by the evolution of a vortex sheet, which from an initial regular shape, develops a finite time singularity of the curvature, resulting in the generation of a spiraling vortex. We investigate some simple systems, a stretched vortex sheet, the free surface of a perfect fluid driven by a vortex dipole, and the splash produced by a convergent capillary wave, in order to illustrate some specific scenarios to the appearance of a “form'' through a singularity
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