On Oscillation Properties of Self-Adjoint Boundary Value Problems of Fourth Order

The connection between the number of internal zeros of nontrivial solutions to fourth-order self-adjoint boundary value problems and the inertia index of these problems is studied. We specify the types of problems for which such a connection can be established. In addition, we specify the types of problems for which a connection between the inertia index and the number of internal zeros of the derivatives of nontrivial solutions can be established. Examples demonstrating the effectiveness of the proposed new approach to an oscillatory problem are considered.

Dissolution or growth of soluble spherical oscillating bubbles: the effect of surfactants

A new theoretical formulation is developed for the effects of surfactants on mass transport across the dynamic interface of a bubble which undergoes spherically symmetric volume oscillations. Owing to the presence of surfactants, the Henry's law boundary condition is no longer applicable; it is replaced by a flux boundary condition that features an interfacial resistance that depends on the concentration of surfactant molecules on the interface. The driving force is the disequilibrium partitioning of the gas between free and dissolved states across the interface. As in the clean surface problem analysed recently (Fyrillas & Szeri 1994), the transport problem is split into two parts: the smooth problem and the oscillatory problem. The smooth problem is treated using the method of multiple scales. An asymptotic solution to the oscillatory problem, valid in the limit of large Péclet number, is developed using the method of matched asymptotic expansions. By requiring that the outer limit of the inner approximation match zero, the splitting into smooth and oscillatory problems is determined unambiguously in successive powers of [weierp ]−1/2, where [weierp ] is the Péclet number. To leading order, the clean surface solution is recovered. Continuing to higher order it is shown that the concentration field depends on RI[weierp ]−1/2, where RI is the (dimensionless) interfacial resistance associated with the presence of surfactants. Although the influence of surfactants appears at higher order in the small parameter, surfactants are shown to have a very significant effect on bubble growth rates owing to the fact that the magnitude of RI is approximately the same as the magnitude of [weierp ]1/2 at conditions of practical interest. Hence the higher-order ‘corrections’ happen numerically to be of the same magnitude as the leading-order, clean surface problem. This is the fundamental reason for major increases in the bubble growth rates associated with the addition of surfactants. This is in contrast to the case of a still, surfactant-covered bubble, in which the first-order correction to the growth rate is of order RI[weierp ]−1 and presents a [weierp ]−1/2 correction. Finally, although existing experimental results have shown only enhancement of bubble growth in the presence of a surfactant the present theory suggests that it is possible for a surfactant, characterized by weak dependence of interfacial resistance on surface concentration, to inhibit rather than enhance the growth of bubbles by rectified diffusion.

Dissolution or growth of soluble spherical oscillating bubbles

A new theoretical formulation is presented for mass transport across the dynamic interface associated with a spherical bubble undergoing volume oscillations. As a consequence of the changing internal pressure of the bubble that accompanies volume oscillations, the concentration of the dissolved gas in the liquid at the interface undergoes large-amplitude oscillations. The convection-diffusion equations governing transport of dissolved gas in the liquid are written in Lagrangian coordinates to account for the moving domain. The Henry's law boundary condition is split into a constant and an oscillating part, yielding the smooth and the oscillatory problems respectively. The solution of the oscillatory problem is valid everywhere in the liquid but differs from zero only in a thin layer of the liquid in the neighbourhood of the bubble surface. The solution to the smooth problem is also valid everywhere in the liquid; it evolves via convection-enhanced diffusion on a slow timescale controlled by the Péclet number, assumed to be large. Both the oscillatory and smooth problems are treated by singular perturbation methods: the oscillatory problem by boundary-layer analysis, and the smooth problem by the method of multiple scales in time. Using this new formulation, expressions are developed for the concentration field outside a bubble undergoing arbitrary nonlinear periodic volume oscillations. In addition, the rate of growth or dissolution of the bubble is determined and compared with available experimental results. Finally, a new technique is described for computing periodically driven nonlinear bubble oscillations that depend on one or more physical parameters. This work extends a large body of previous work on rectified diffusion that has been restricted to the assumptions of infinitesimal bubble oscillations or of threshold conditions, or both. The new formulation represents the first self-consistent, analytical treatment of the depletion layer that accompanies nonlinear oscillating bubbles that grow via rectified diffusion.