Fluid kinetic energy asymptotic expansion for two variable radii moving spherical bubbles at small separation distance

TL;DR

This paper derives an exact series expression for the fluid kinetic energy of two moving spherical bubbles with variable radii, enabling asymptotic analysis at small separation distances to study bubble approach and coalescence.

Contribution

It introduces a new series expansion for the fluid kinetic energy in bispherical coordinates, allowing asymptotic analysis at small bubble separation distances.

Findings

Exact series for kinetic energy quadratic form coefficients

Asymptotic expansions at small separation distances

Convergence analysis of the series

Abstract

Two spherical bubbles with changing radii are considered to be moving in ideal fluid along their center-line. The exact expression for the fluid kinetic energy is obtained. The Stokes stream function is expanded in Gegenbauer polynomials in bispherical coordinates. This expansion is used to obtain the exact series for the fluid kinetic energy quadratic form coefficients. The new series are confirmed to be correct by comparison with the known ones. The main advantage of the new kinetic energy form is the possibility to obtain asymptotic expansions at small separation distance between the bubbles. These expansions are obtained and their convergence is analyzed. The results of this work can be used to describe the bubbles approach before the contact and their coalescence in acoustic field.