Propagation and stopping of air bubbles in Carbopol solutions

We present results of an experimental study of the motion of air bubbles rising in a column of Carbopol solution under gravity. In Dubash and Frigaard [N. Dubash, I.A. Frigaard, Conditions for static bubbles in viscoplastic fluids, Phys. Fluids 16 (12) (2004) 4319-4330] we have predicted conditions under which sufficiently small bubbles will not propagate in a yield stress fluid. The bubbles in our experiments all exceed the bounds in Dubash and Frigaard (2004), which show themselves to be quite conservative. For smaller bubbles we observe a rapid increase in terminal bubble velocity as the ratio of bubble to column radius, R/Rc increases. For R/Rc ≥ 0.7 the terminal velocity approaches a maximum. By considering a simple control volume model for the energy dissipation rate of a steadily rising bubble, we are able to show that all our data collapse onto a linear relation between the work done by buoyancy and the sum of the contributions from inertia, yield stress, viscous dissipation and surface tension. By extrapolating this linear relation to the case of zero bubble velocity, we derive a new empirical prediction of the stopping condition. This condition is compatible with those previously derived, but is much less conservative.