Traveling Waves from the Arclength Parameterization: Vortex Sheets With Surface Tension
We study traveling waves for the vortex sheet with surface tension. We use the angle-arclength description of the interface rather than Cartesian coordinates, and we utilize an arclength parameterization as well. In this setting, we make a new formulation of the traveling wave ansatz. For this problem, it should be possible for traveling waves to overturn, and notably, our formulation does allow for waves with multi-valued height. We prove that there exist traveling vortex sheets with surface tension bifurcating from equilibrium. We compute these waves by means of a quasi-Newton iteration in Fourier space; we find continua of traveling waves bifurcating from equilibrium and extending to include overturning waves, for a variety of values of the mean vortex sheet strength.
Interfaces and Free Boundaries
Akers, B. F., Ambrose, D. M., & Wright, J. D. (2013). Traveling waves from the arclength parameterization: Vortex sheets with surface tension. Interfaces and Free Boundaries, 15(3), 359–380. https://doi.org/10.4171/IFB/306