Inertial shocks in surface waves and collapsing bubbles

Abstract

There exists a class of analytic, free-surface flows which describes qualitatively the highly-accelerated re-entrant jets seen in bubble generation and collapse. In semi-Langrangian coordinates such a flow is given by

$$ x + iy = ({F_1}\sin \omega - i{G_1}\cos {\rm{ }}\omega ) + {1 \over 2}({F_2}\sin 2\omega - i{\rm{ }}{G_2}{\rm{ }}\cos {\rm{ }}2\omega ) $$

where ? is a complex parameter, real at the free surface. The coefficients F ₁, G ₁, F ₂ and G ₂ are real functions of the time t only, related by a set of nonlinear, ordinary differential equations, so as to satisfy appropriate boundary conditions and conditions of regularity in the fluid. A wide variety of such flows exists, depending on the initial conditions. One remarkable feature of many of the flows is the occurrence of a sudden “inertial shock” at a critical time t = t _o. This usually happens when the surface locally has the form of a rectangular hyperbola — as in the previously known “Dirichlet hyperbola”. The phenomenon is illustrated by the behaviour of the inwards acceleration y _tt at the point ? = 0 on the free surface (the tip of the jet). The acceleration becomes infinite like (t — t _o)^−4/3, near the instant t _o.

Solutions also exist with two jets directed inwards from opposite sides of the bubble — or indeed any number of such jets per bubble.