The growth or dissolution of small gas bubbles (R0<15 microm) by rectified diffusion in nearly saturated liquids, subject to low frequencies (20 kHz

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 Péclet 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 behaviour is reminiscent of the so-called rectified diffusion problem, leading to slow growth of a gas bubble oscillating in a sound field. A rigorous method formerly proposed by Fyrillas & Szeri (J. Fluid Mech. vol. 277, 1994, p. 381) to solve the latter problem is used to solve the present one. It is based on splitting the problem into a smooth part and an oscillatory part. The smooth part is solved by a multiple scales method and yields the slowly varying average concentration field everywhere in the liquid. The oscillatory part is obtained by matched asymptotic expansions in terms of the small parameter pe−1/2: the inner solution is required to satisfy the oscillatory balance between pressure diffusion and Fick diffusion at the bubble wall, while the outer solution is required to be zero. Matching both solutions yields a unique splitting of the problem. The final analytical solution, truncated to leading order, compares successfully to direct numerical simulation of the full convection–diffusion equation. The analytical expressions for both smooth and oscillatory parts are calculated for various sets of bubble parameters: driving pressure, frequency and ambient radius. The smooth problem always yields an average depletion of the heaviest species at the bubble wall, only noticeable for large molecules or nano-particles. 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 the time of 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.

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Abstract. A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schrödinger equation with a quasi-exactly-solvable potential

The determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy, namely the Wronskian of two solutions of the Schrödinger equation which are regular respectively at the origin and at infinity. We show in this paper how to evaluate that Wronskian exactly, except for numerical rounding errors. The procedure is illustrated by application to the gx 2 + x 2N (N a positive integer) oscillator.