The collapse time of a closed cavity

Abstract

The collapse time of a closed cavity that is initially at rest in an incompressible, inviscid fluid of density ρ and ambient pressurep∞has the form\[ t_1 = \{\rho/(p_{\infty} - p_c)\}^{\frac{1}{2}}\ell, \]wherepcis the internal pressure, which is assumed to remain constant during collapse, and [ell ] is a length that depends only on the geometry of the cavity. A variational formulation of the dynamical problem is constructed from Jacobi's statement of the principle of least action. A single-degree-of-freedom approximation is developed from the similarity hypothesis that the cavity collapses through a family of similar surfaces with volume as the generalized co-ordinate. Two-degree-of-freedom approximations are given for both prolate and oblate spheroidal cavities and are used to obtain error estimates for the similarity approximation (approximately 2% for the limiting case of a needle-like, prolate spheroid and approximately ½ % for a disk-like, oblate spheroid). A perturbation analysis is developed for an approximately spherical cavity, which is found to have the same collapse time as a spherical cavity of equal volume within a factor 1 +O(e4), whereeis a representative eccentricity. A first-order correction for surface tension is obtained.

Bubble Dynamics and Cavitation

“…Let us observe here that, in the absence of surface-tension effects, one can compute from Equation (3.6) the time to required for complete collapse of the cavity : which is the result of Rayleigh (1917). An extension to the collapse of a closed cavity of arbitrary shape has been given by Miles (1966), who, for oblate and prolate spheroids, finds a correction to (3.7) of the order of the fourth power of the eccentricity (the quantity Ri is defined as the radius of a sphere of equivalent volume for this problem).…”

Observations of the dynamics and acoustics of travelling bubble cavitation

“…This formula adequately predicts the collapse time of spherical and quasi-spherical bubbles generated is still fluid (Miles (1966)) or over an axisymmetric body (Plesset (1949)) even though the analysis predicts an infinite bubble wall velocity at zero radius.…”

An analytical model for the microjet at the tip of a collapsing cavity

“…An expression is given for the collapse time that differs by a numerical factor from that obtained by MlLES [4] for very elongated bubbles, using a variational method. In Section 3 a better approximation is introduced, retaining in Bernoulli's equation the kineticenergy term obtained in first approximation; the results allow to explain the formation of microjets at the tips of the bubble.…”

“…As it will be shown later, equation (11) is expected to be valid for the whole collapse process if~<l'i 1. If equation (4) is used to calculate the kinetic energy, equation (8) …”

“…(**) Profesor Titular de Universidad. [2] and SHIMA and NAKAJIMA [3] ; however, they do not appear in the analysis of MILES [4]. A microjet also appears typically in situations such as in the bubble collapse in the presence of boundaries and in translating bubbles, as it is shown by BLAKE and GIBSON [5] and PLESSET and PROSPERETTI [6].…”