Singularity Formation in Thin Jets with Surface Tension

with M. J. Shelley, CPAM 51(1998)733-795.


We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be incompressible, inviscid, irrotational, and density matched. One such thin jet model is a coupled system of PDEs with nonlocal terms -- Hilbert transforms -- that result from expansion of a Biot-Savart integral. In order to make the asymptotic model well-posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of finite-time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension, and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz instability with surface tension, and in other related studies. Dropping the nonlocal terms of the model, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we find that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model.

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M.P. was supported by an NSF post-doctoral fellowship while at Courant and by the Ambrose Monell Foundation while at the Institute for Advanced Study. M.J.S. acknowledges support from Department of Energy grant DE-FG02-88ER25053, National Science Foundation grants DMS-9396403 (PYI) and DMS-9404554, and the Exxon Educational Foundation.