Bubbles rising in line: why is the first approximation so bad?

Abstract

An analytical theory is given for the viscous wake behind a spherical bubble rising steadily in a pure liquid at high Reynolds number, and for that wake's effect on the motion of a second bubble rising underneath the first. Previous theoretical work on this subject consists of just two papers: a first approximation ignoring wake vorticity diffusion between the bubbles, and a full numerical solution avoiding simplifying approximations (apart from that of spherical shape of the bubbles). A second approximation is now found; it removes much of the discrepancy between the first approximation and the full solution. The leading-order calculation of wake vorticity diffusion uses a transformation of the independent variables which appears to be new. Experimental work to date has disagreed with all the theoretical work, but it addresses a somewhat different problem: a line of many bubbles.