The history force on a rapidly shrinking bubble rising at finite Reynolds number

Abstract

Using an optical device and a travelling system, we determine precisely the evolution of the radius and rising speed of a single CO2 or CO2–air bubble rising at finite Reynolds number in an aqueous NaOH solution and rapidly dissolving in it. We willingly introduce a slight amount of pentanol in the aqueous solution to force the bubble surface to obey a no-slip condition. The measurements allow us to evaluate directly the instantaneous magnitude of the buoyancy, added-mass, and quasisteady drag forces experienced by the bubble. We then deduce the magnitude of the so-called history force from the total force balance. It turns out that this force can be up to one-half of the buoyancy force during a certain stage of the motion. Using the argument developed by Magnaudet and Legendre [Phys. Fluids 10, 550 (1998)], we derive the counterpart of the Basset–Boussinesq history force for the case of a rigid sphere with a time-dependent radius, as well as the counterpart of some finite-Reynolds-number empirical extensions of this expression. We use these results to predict the time evolution of the history force in our experiments. Owing to finite-Reynolds-number effects, the zero-Reynolds-number expression yields totally unrealistic results. In contrast we find that, once properly generalized to a sphere of time-varying radius, the finite-Reynolds-number expression proposed by Kim, Elghobashi, and Sirignano [J. Fluid Mech. 367, 221 (1998)] provides an accurate estimate of the evolution of the history force and hence allows the bubble velocity to be precisely predicted all along the dissolution process.