A vectorized evolution equation for interfaces represented as vortex-entrainment sheets.

This work considers the motion of an interface between two inviscid fluids. The evolution equation for the vortex-entrainment sheet strength representing the interface is presented in a fully vectorized format. The sheet strenghs are treated as a vector quantity that naturally occurs within the jump in the Euler equation across the sheet. The tagnetial component of the equation evolves the vortex sheet strengh, while the normal component evolves the entrainment sheet strength, although they are coupled. Even in the case of a conventional vortex sheet (i.e. zero entrainment), the equation still produces tangential and normal components that are non-trivial. This approach is in contrast to works that derive the corresponding Bernoulli equation and then spatially differentiate it to obtain the evolution equation for the vortex-sheet strength. Specifically, the role of curvature and the evolution of the geometry are made clear. Moreover, this is done in the arclength domain in which the equations take a particularly useful form for free vortex sheets. This form is used to investigate some classical problems, such as singularity formation and roll-up.