Terminal velocity of a bubble in a vertically vibrated liquid

Abstract

We rigorously derive a formula for the terminal velocity of a small bubble in a vertically vibrated viscous incompressible liquid starting from the full Navier-Stokes equations and the exact boundary conditions at the bubble surface. This formula is derived using a perturbation analysis in which the small parameter is the nondimensional amplitude of the pressure oscillation. The analysis does not assume that the bubble remains spherical but does assume that the bubble is axisymmetric. It is shown that the bubble terminal velocity can be computed to second order while computing the full solution only to first order by applying a compatibility condition on the first-order solution. To second order, the bubble terminal velocity is shown to be the net value from an upward steady term and a rectified term that can be downward or upward. The perturbation formula depends on the vibration frequency nondimensionalized by the bubble radius and the liquid kinematic viscosity. We show that our perturbation formula links two heuristically developed formulas for the rectified component, which we denote the velocity-averaged and force-averaged formulas. Our perturbation formula reproduces the velocity-averaged formula for low frequencies and the forced-averaged formula for high frequencies and varies monotonically between these limits for intermediate frequencies. We furthermore develop a high-resolution spectral code specifically to simulate this type of bubble motion. Results from this code verify that the perturbation formula is correct for infinitesimal oscillating pressure amplitudes and suggest that it provides an upper bound for finite amplitudes of the pressure oscillation.