Segregation of a liquid mixture by a radially oscillating bubble
A theoretical formulation is proposed for forced mass transport by pressure gradients in a liquid binary mixture around a spherical bubble undergoing volume oscillations in a sound field. Assuming the impermeability of the bubble wall to both species, diffusion driven by pressure gradients and classical Fick-diffusion must cancel at the bubble wall, so that an oscillatory concentration gradient arises in the vicinity of the bubble. The Peclet number Pe is generally high in typical situations and Fick diffusion cannot restore equilibrium immediately, so that an asymptotic average concentration profile may progressively build up in the liquid over large times. Such a behavior is reminiscent of the so-called rectified diffusion problem, leading to slow growth of gas bubble oscillating in a sound field. A rigorous method formerly proposed by Fyrillas & Szeri (1994) to solve the latter problem is used in this paper to solve the present one. It is based on splitting the problem into a smooth part and an oscillatory part. [...] For driving pressures sufficiently high to yield inertial oscillations of the bubble, the oscillatory problem predicts a periodic peak excess concentration of the heaviest species at the bubble wall at each collapse, lingering on several tens of time the characteristic duration of the bubble rebound. The two effects may compete for large molecules and practical implications of this segregation phenomenon are proposed for various processes involving acoustic cavitation.