Collapse of a Gas-Filled Spherical Cavity

Abstract

The PLK coordinate perturbation technique [10] is used to obtain a solution to the problem of collapse of gas-filled spherical cavity in an infinite compressible liquid. The gas is assumed to undergo quasi-static adiabatic compression and the liquid equation of state is taken to follow the modified Tate form [11]. The approach was first outlined by Benjamin [9] and in the present paper expressions for cavity wall history and fluid pressure, density, and particle velocity are carried out in complete detail for the first three terms in the expansions. It is found that the solutions for the variables can all be written as products of functions which depend on only one of the perturbed coordinates. For the coordinate corresponding to outward traveling characteristics (first used by Whitham [12]), only two functions are required; they are associated with cavity wall position and with velocity and satisfy second-order ordinary differential equations which are readily solved by digital computer. For the remaining coordinate (perturbed radius) the functions are all polynomials. A numerical example is presented and curves of cavity wall position, pressure, and velocity histories are given for the period associated with collapse and rebound of the cavity. Results are compared with earlier work based on the Gilmore adaptation of the Kirkwood-Bethe formulation [8], and good agreement is found.