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Singularity Formation in Thin Jets with Surface Tension

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

### Abstract

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.