Growing bubbles rising in line

John F Harper

doi:10.1155/s1173912601000062

Abstract

Over many years the author and others have given theories for bubbles rising in line in a liquid. Theory has usually suggested that the bubbles will tend towards a stable distance apart, but experiments have often showed them pairing off and sometimes coalescing. However, existing theory seems not to deal adequately with the case of bubbles growing as they rise, which they do if the liquid is boiling, or is a supersaturated solution of a gas, or simply because the pressure decreases with height. That omission is now addressed, for spherical bubbles rising at high Reynolds numbers. As the flow is then nearly irrotational, Lagrange's equations can be used with Rayleigh's dissipation function. The theory also works for bubbles shrinking as they rise because they dissolve.

Highlights

Consider bubbles rising in line in a liquid, with surface tension assumed large enough to keep them spherical, and surface activity small enough to ignore
Computational fluid dynamics has proved useful in this problem: Yuan & Prosperetti [21] dealt with two spherical bubbles at various Reynolds numbers up to 200, and Zinchenko et al [22] with two deformable bubbles or drops in Stokes flow
Bubbles that grow or shrink as they rise were studied in Stokes flow by Magnaudet & Legendre [15] and in inviscid flow by Chincholle [4], though the methods needed are much older [1], [2], [3], [8]

Summary

Introduction

Consider bubbles rising in line in a liquid, with surface tension assumed large enough to keep them spherical, and surface activity small enough to ignore. If they remain the same constant size as they rise, their motion presents a problem simple enough to solve analytically to leading order, with computation needed only for such things as quadrature and solution of linear matrix equations, in two special cases: several bubbles in Stokes flow (Reynolds number vanishingly small) [6], and two bubbles at high Reynolds numbers [5], [7]. Some of the theory is extended to more than two bubbles

Lagrangian Theory

Irrotational Theory for Two Bubbles

Other Effects

Conclusions