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VOLUME 61 (1995) | ISSUE 1 | PAGE 28
A general property of solutions of Eulerian hydrodynamic equations: the formation of discontinuities at a liquid surface
The evolution of a boundary η is studied. A collapse is observed, consisting of the spontaneous formation of singularities on solutions of the Cauchy problem of a general type which are initially smooth everywhere. This collapse occurs after a finite time rsing. An important advance is that the analysis is based on equations which incorporate a realistic strong nonlinearity. (The weakly nonlinear approximation is valid when the singularities are far from a boundary. A strong nonlinearity is taken into account by carrying out a high-order expansion and analyzing the convergence of the results as the order of the expansion varies.) In addition, a singularity has been observed in valleys between surges or "between fingers." (The onset of singularities at points of acoustic catastrophes, corresponding to surges of the liquid, has been predicted previously on the basis of weakly nonlinear approximations which lead to equations analogous to gasdynamic equations with a negative pressure and an imaginary sound velocity. Liquid surges or jets are often called "fingers.") Details of the stages before and after the collapse are reported. © 1995 American Institute of Physics.