The rising velocity of a slowly pulsating bubble in a shear-thinning fluid

Abstract

We study the rising motion of small bubbles that undergo contraction, expansion, or oscillation in a shear-thinning fluid. We model the non-Newtonian response of the fluid using the Carreau-Yasuda constitutive equation, under the assumptions that the inertia of the fluid and the bubble is negligible and that the bubble remains spherical. These assumptions imply that the rising velocity of the bubble is instantaneously proportional to the buoyancy force, with the proportionality constant given by the inverse of the friction coefficient. Instead of computing the rising velocity for a particular radial dynamics of the bubble, we evaluate its friction coefficient as a function of the rheological parameters and of the instantaneous expansion/contraction rate. To compute the friction coefficient, we impose a translational motion and we linearize the governing equations around the expansion/contraction dynamics of the bubble, which we solve using a perturbation expansion along with the finite element method. Our results show that the radial motion of the bubble reduces the viscosity of the surrounding fluid and may thus markedly decrease the friction coefficient of the bubble. We use the friction coefficient to compute the average rise velocity of a bubble with periodic variations of its radius, which we find to be strongly increased by the radial pulsations. Finally, we compare our predictions with the experiments performed by Iwata et al. [“Pressure-oscillation defoaming for viscoelastic fluid,” J. Non-Newtonian Fluid Mech. 151(1-3), 30–37 (2008)], who found that the rise velocity of bubbles that undergo radial pulsations is increased by orders of magnitude compared to the case of bubbles that do not pulsate. Our results shed light on the mechanism responsible for enhanced bubble release in shear-thinning fluids, which has implications for bubble removal from complex fluids.