The Partial Drift Volume Induced By a Translating Spherical Bubble at Finite Reynolds Number

The Lagrangian displacement, or drift, of fluid elements caused by moving bodies is important to pool boiling, dispersion in fixed beds, and oceanic bio-mixing. Previous theoretical calculations of drift have been conducted at the limits of inviscid flow and Stokes flow. Here, the drift caused by the translation of a bubble through a quiescent fluid is quantified at finite Reynolds number (Re). We consider a spherical, non-deformable bubble of radius a translating at a constant speed U through an unbounded Newtonian fluid. The bubble starts at the origin and moves in a straight line; axisymmetry of the flow about the direction of translation is assumed. The flow around the bubble is obtained for 0.01 < Re < 100 by numerically integrating the Navier-Stokes equations using a spectral element method. Then, the distortion of a circular material sheet of fluid elements of radius R » a centered on and perpendicular to the bubble's axis of translation is computed. This is accomplished by integrating the trajectories of individual fluid elements on the material sheet over time. The sheet is positioned at an initial distance L from the bubble at time t = 0. The volume swept out by this material sheet after a time t = t_f defines the partial drift volume V_D. We quantify V_D with respect to the parameters Re, L, R, and t_f. At finite Re, the displacement of fluid elements on the sheet, and hence V_D, grow unbounded as t_f → ∞. This is the result of the slow decay of the velocity field in the wake of the bubble, which acts as a point source of momentum at large distances. Specifically, we find that V_D ~ ln(t_f) as t_f → ∞. However, the coefficient of this scaling depends on R/L. We also find that increasing Re decreases V_D relative to that of the bubble. Notably, V_D ~ Re^-0.5 for Re » 1, which is the same as the scaling for the drag on the bubble at large Re. Physically, fluid elements are entrained by the viscous wake behind the bubble, which decreases in extent as Re increases.