Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip

Abstract

This work studies the motion of an expanding or contracting bubble pinned at a submerged tube tip and covered with an insoluble Volmer surfactant. The motion is driven by constant flow rate Q into or out of the tube tip. The purpose is to examine two central assumptions commonly made in the bubble and drop methods for measuring dynamic surface tension, those of uniform surfactant concentration and of purely radial flow. Asymptotic solutions are obtained in the limit of the capillary number Ca→0 with the Reynolds number Re=o(Ca⁻¹, non-zero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca=μQ/a²σ_c, where μ is the liquid viscosity, a is the tube radius, and σ_c is the clean surface tension.) This limit is relevant to dynamic-tension experiments, and gives M→∞, where M=G/Ca is the Marangoni number. We find that in this limit the deforming bubble at each instant in time takes the static shape. The surfactant distribution is uniform, but its value varies with time as the bubble area changes. To maintain a uniform distribution at all times, a tangential flow is induced, the magnitude of which is more than twice that in the clean case. This is in contrast to the surface-immobilizing effect of surfactant on an isolated translating bubble. These conclusions are confirmed by a boundary integral solution of Stokes flow valid for arbitrary Ca, G and Bo. The uniformity in surfactant distribution validates the first assumption in the bubble and drop methods, but the enhanced tangential flow contradicts the second.