This paper is concerned with calculations regarding a collection of small gas bubbles rising under buoyancy in a clear liquid. For dilute mixtures interactions can be restricted to those between two bubbles. In the analysis of binary interactions it is assumed that the Reynolds number for relative motion between bubbles and liquid is large, that surfactants, if present, do not modify the condition of zero tangential stress at the bubble-liquid interface, and that bubbles bounce at an encounter.
A two-bubble probability density is derived from the analysis, valid on a short timescale associated with the interaction. It is shown that on a long timescale, based on viscous dissipation, clustering together of pairs takes place, most likely even when triple encounters are allowed for. An analysis is given of the vertical motion of pairs, followed by a calculation of the mean vertical bubble velocity with help of the (short timescale) probability density function. The result is compared with experimental data.