Bubbly flow has great importance in diverse industrial processes: chemistry, environmental, metallurgical, petrochemical, pharmaceutical, biochemical, foods, etc., motivating numerous investigators to direct their attention to the study of these systems. The key problem in bubbly flow is to understand how the two phases interact locally. A general strategy may consist in distinguishing (1) the effect of the liquid on the bubbles, (2) the effect of the bubbles on the liquid, and (3) the role of the interactions between bubbles (Ellingsen and Risso, 2001). In the high and intermediate regimes of Reynolds numbers the dynamics of a pair of bubbles rising in line in a quiescent liquid differs from the single bubble. During the free-rise of gas bubbles through a liquid, an amount of liquid known as the wake is carried up behind the leader bubble. In the case of a pure liquid of impurities the effect of attraction of the wake on the trailing bubble culminates with coalescence between both bubbles at the time of the contact.

The interaction study of two or more bubbles in a flow is a necessary step in the derivation of averaging momentum equations for multiphase flows in bubbling columns (Yuan and Prosperetti, 1994) and the collisions between bubbles have a large effect in the bubble size distribution. In the last two decades main objectives of the investigators has been to obtain analytical expressions for the forces acting on bubbles.

In this paper a semianalytical expression is obtained for the motion of two bubbles rising in line under conditions of intermediate Reynolds Re~100 and small Weber numbers We<<1. Starting from the motion equation for a single bubble and considering the motion of the trailing bubble as quasi-steady state but taking into account the attraction effect of the wake we obtain an expression different to those obtained in previous works. The resulting equation is compared with experimental measurements of Katz and Meneveau (1995) and it is observed a better prediction that other models reported in the literature including added mass and Boussinesq-Basset forces. Finally, the obtained expression is substituted in a semianalytical expression of the drag force ratio for two particles obtained by Zhang and Fan (2002) and the resulting equation is validated with experimental data of the drag force ratio obtained by Zhu et al. (1994).

Keywords: Bubbly flow; Bubble rise velocity; Bubble-liquid interaction; Bubble–bubble interaction; Added mass force; Basset-Boussinesq force; Bubble drag force.

References:

Ellingsen, K., and Risso, F. (2001). On the rise of an ellipsoidal bubble in water: oscillatory paths and liquid-induced velocity. J. Fluid Mech. 440 235-268.

Katz, J., and Meneveau, C. (1996). Wake-induced relative motion of bubbles rising in line, Int. J. Multiphase flow, 22, 239-258.

Yuan, H., and Prosperetti, L. A. (1994). On the in-line of two spherical bubbles in a viscous fluid. J. Fluid. Mech. 278, 325-349.

Zhang, J., and Fan, L. -S. (2002). A semianalytical expression for the drag force of an interactive particle due to wake effect. Ind. Eng. Chem. Res. 41, 5094-5097.

Zhu, C., Liang, S. –C., and Fan, L. -S. (1994). Particle wake effects on the drag force of an interactive particle. Int. J. Multiphase flow, 20, 117-129.