Singularity Formation for Vortex Sheets in Axi-symmetric Flows

 

Abstract

Motion of vortex sheets provides simple models for thin shear layers and interfaces between immiscible liquids. One of the signature behavior of the motion of vortex sheets in the absence of regularization for two-dimensional flows is the formation of curvature singularities in finite time. A open question at hand is whether vortex sheets in three-dimensional flows form singularities. Even with the simplification of axi-symmetry, it has been a challenge to study the singularity formation numerically due to lack of accurate numerical methods. In this talk, I will first discuss a numerical technique using adaptive strategies to compute the motion of vortex sheets in axi-symmetric flows, and study the singularity formation numerically. I'll also talk about a robust and efficient numerical technique to compute the vortex sheets in axi-symmetric flows with surface tension to investigate its regularization of the singularity.