The Nonlinear Evolution of Vortex Sheets with Surface Tension in Axisymmetric Flows

The presence of surface tension for interfacial flows usually leads to severe stability constraints for explicit time integration methods. Moreover, the nonlocality and nonlinearity of the high-order terms make the application of implicit methods difficult. In this paper, a computational strategy is presented for computing the motion of fluid interfaces with surface tension in axisymmetric flows using boundary integral techniques. This method is based on adaptive quadratures for the principal-value integrals and a small-scale decomposition for the treatment of surface tension through a vector-potential formulation. A study of the method is conducted in the context of vortex sheet evolution with surface tension in axisymmetric flows. The method is found to be accurate, efficient, and robust. Numerical simulations indicate that the dynamics of vortex sheets with surface tension frequently result in topological singularities (i.e., self-intersection). Away from the axis of symmetry, these singularities are similar to those found in the two-dimensional flows. Singularities occurring near the axis of symmetry take a different form.