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

This work studies the motion of an expanding or contracting bubble pinned a t 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 a nd drop methods for measuring dynamic surface tension, those of uniform sur factant concentration and of purely radial flow. Asymptotic solutions are o btained in the limit of the capillary number Ca --> 0 with the Reynolds num ber Re = o(Ca-1), non-zero Gibbs elasticity (G), and arbitrary Bond number (Bo). (Ca = mu Q/a(2)sigma(c),, where mu is the liquid viscosity, a is the tube radius, and sigma(c) is the clean surface tension.) This limit is rele vant to dynamic-tension experiments, and gives M --> infinity, 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 distributio n 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 isolate d translating bubble. These conclusions are confirmed by a boundary integra l 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.