Interaction between two spherical bubbles rising in a viscous liquid

The three-dimensional flow around two spherical bubbles moving in a viscous fluid is studied numerically by solving the full Navier-Stokes equations. The study considers the interaction between two bubbles for moderate Reynolds numbers (50 <= Re <= 500, Re being based on the bubble diameter) and for positions described by the separation S (2.5 <= S <= 10, S being the distance between the bubble centres normalised by the bubble radius) and the angle theta (0 degrees <= theta <= 90 degrees) formed between the centreline and the direction perpendicular to the direction of the motion. We provide a general description of the interaction extending the results obtained for two bubbles moving side by side (theta = 0 degrees) by Legendre, Magnaudet & Mougin (J. Fluid Mech., vol. 497, 2003, p. 133) and for two bubbles moving in line (theta degrees = 90 degrees) by Yuan & Prosperetti (J. Fluid Mech., vol. 278, 1994, p. 325). Simple models based on physical arguments are given for the drag and lift forces experienced by each bubble. The interaction is the combination of three effects: a potential effect, a viscous correction (Moore's correction) and a significant wake effect observed on both the drag and the transverse forces of the second bubble when located in the wake of the first one.