Singular solutions and ill-posedness for the evolution of vortex sheets
SIAM Journal on Mathematical Analysis
Removing the stiffness from interfacial flows with surface tension
Journal of Computational Physics
A study of singularity formation in the Kelvin-Helmholtz instability with surface tension
SIAM Journal on Applied Mathematics
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Stable Methods for Vortex Sheet Motion in the Presence of Surface Tension
SIAM Journal on Scientific Computing
The nonlinear evolution of vortex sheets with surface tension in axisymmetric flows
Journal of Computational Physics
Numerical simulations of inviscid capillary pinchoff
Journal of Computational Physics
A Eulerian level set/vortex sheet method for two-phase interface dynamics
Journal of Computational Physics
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In a recent analytical study, the author has proved well-posedness of the vortex sheet with surface tension. This work included using a formulation of the problem introduced by Hou, Lowengrub, and Shelley for a numerical study of the same problem. The analytical study required identification of a term in the evolution equations which can be viewed as being responsible for the Kelvin-Helmholtz instability; this term is of lower order than the surface tension term. In the present work, the author introduces a simple model for the vortex sheet with surface tension which maintains the same dispersion relation and the same destabilizing force as in the vortex sheet with surface tension. For the model problem, it is found that finite-time singularities can form when the initial data is taken from a certain class. For the vortex sheet with surface tension, the only observed singularities thus far in numerical work have coincided with self-intersection of the fluid interface. There is no analogue of self-intersection in the model problem, and thus the singularities observed in the present work may well be related to a previously unobserved singularity for the full vortex sheet problem.