Equilibrium shape of a bubble in a liquid-filled horizontal capillary

Abstract

Bubbles in liquid-filled capillaries feature in many practical applications. Here we describe the equilibrium shape of a bubble in a horizontal capillary of circular cross section, as computed numerically using energy minimization. The data is presented in terms of empirical scaling laws for bubble extension along and perpendicular to the tube axis, over a large range of the ratio of cylinder to bubble diameter and Bond numbers. We also consider the limiting case of a bubble underneath a flat plate, for which approximate analytical solutions are available. Furthermore, we deduce an equation for the mean curvature of the gas–liquid interface and the variation of contact areas, the latter being of relevance also for estimating the wall drag forces of flowing bubbles.