The long-time motion of vortex sheets with surface tension

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Abstract
We study numerically the simplest model of two incompressible, immiscible fluids shearing past one another. The fluids are two-dimensional, inviscid, irrotational, density matched, and separated by a sharp interface under a surface tension. The nonlinear growth and evolution of this interface is governed by only the competing effects of the Kelvin–Helmholtz instability and the dispersion due to surface tension. We have developed new and highly accurate numerical methods designed to treat the difficulties associated with the presence of surface tension. This allows us to accurately simulate the evolution of the interface over much longer times than has been done previously. A surprisingly rich variety of behavior is found. For small Weber numbers, where there are no unstable length-scales, the flow is dispersively dominated and oscillatory behavior is observed. For intermediate Weber numbers, where there are only a few unstable length-scales, the interface forms elongating and interpenetrating fingers of fluid. At larger Weber numbers, where there are many unstable scales, the interface rolls-up into a “Kelvin-Helmholtz” spiral with its late evolution terminated by the collision of the interface with itself, forming at that instant bubbles of fluid at the core of the spiral. Using locally refined grids, this singular event (a “topological” or “pinching” singularity) is studied carefully. Our computations suggest at least a partial conformance to a local self-similar scaling. For fixed initial data, the pinching singularity times decrease as the surface tension is reduced, apparently towards the singularity time associated with the zero surface tension problem, as studied by Moore and others. Simulations from more complicated, multi-modal initial data show the evolution as a combination of these fingers, spirals, and pinches.