On the motion of vortex sheets with surface tension in the 3D Euler equations with vorticity

The motion of vortex sheets with surface tension has been analyzed in the setting of irrotational flows by Ambrose [1] and Ambrose & Masmoudi [2] in 2D, and by Ambrose & Masmoudi [3] in 3D. With irrotationality, the nonlinear Euler equations reduce to Poisson’s equation for the pressure function in the bulk, and the motion of the vortex sheet is decoupled from that of the fluid, thus allowing boundary integral methods to be employed. In a general flow with vorticity, the full two-phase Euler equations must be analyzed; in this instance, the motion of the two phases of fluid is coupled to the motion of the vortex sheet, and entirely new mathematical methods must be developed to obtain a well-posedness theory. In particular, a new class of approximation schemes must be employed which preserve the transport-type structure of the vorticity, an issue that, by definition, does not arrise in either the irrotational theory, or in the analysis of the Euler equations on fixed domains. In the general case with vorticity present in the fluid, the vortex sheet is a surface of discontinuity propogated by the fluid, representing the material interface between two incompressible inviscid fluids with densities ρ+ and ρ−, respectively. The tangential velocity of the fluid suffers a jump discontinuity along the material interface, leading to the well-known Kelvin-Helmholtz or Rayleigh-Taylor instabilities when surface tension is neglected. The velocity of the vortex sheet is the normal component of the fluid velocity, whose continuity across the material interface Γ(t) is enforced. In addition to incompressibility, the continuity of the normal, rather than tangential, component of velocity across Γ(t) is a fundamental difference between multi-D shock wave evolution, wherein the velocity of the surface of discontinuity is determined by the generalized Rankine-Hugoniot condition. Unlike multi-D shocks, the vortex sheet problem is characteristic; nevertheless, the