The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a simplified singular-perturbation method to first order in the bubble-wall Mach number. It is shown that, at this order, a one-parameter family of approximate equations for the bubble radius exists, which includes those previously derived by Herring and Keller as special cases. The relative merits of these and other equations of the family are judged by comparison with numerical results obtained from the complete partial-differential-equation formulation by the method of characteristics. It is concluded that an equation close to the Keller form, but written in terms of the enthalpy of the liquid at the bubble wall, rather than the pressure, is most accurate, at least for the cases considered of collapse in a constant-pressure field and collapse driven by a Gaussian pressure pulse. A physical discussion of the magnitude and nature of compressibility effects is also given.

The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a rigorous singular-perturbation method to second order in the bubble-wall Mach number. The results of Part 1 (Prosperetti & Lezzi, 1986) are recovered at orders zero and one. At second order the ordinary inner and outer structure of the solution proves inadequate to correctly describe the fields and it is necessary to introduce an intermediate region the characteristic length of which is the geometric mean of the inner and outer lengthscales. The degree of indeterminacy for the radial equation of motion found at first order is significantly increased by going to second order. As in Part 1 we examine several of the possible forms of this equation by comparison with results obtained from the numerical integration of the complete partial-differential-equation formulation. Expressions and results for the pressure and velocity fields in the liquid are also reported.

Spinodal decomposition of deeply quenched mixtures is studied experimentally, with particular emphasis on the domain growth rate during the late stage of coarsening. We provide some experimental evidence that at high Péclet number, the process is isotropic and the domain growth is linear in time, even at finite quenching rates. In fact, the quenching rate appears to influence the magnitude of the growth rate, but not its scaling law. In the second part of the work we analyze the effect of viscosity on the growth rate. As predicted by the diffuse interface model, we do not find any effect of viscosity on the growth rate of the nucleating drops, although, as expected, the viscosity of the continuous phase does influence the settling speed and thus the total separation time.

Spinodal decomposition can be driven by either diffusion or self-induced convection; the importance of convection relative to diffusion depends on the Péclet number, defined as the ratio between convective and diffusive mass fluxes. Diffusion is the dominating mechanism of phase segregation when the Péclet number is small - i.e., when viscosity and diffusivity are large - or when the domain characteristic size is small. For low-viscosity mixtures, convection is the dominating process and the segregation is very rapid as it takes a few seconds compared to the hours needed in the case of pure diffusion. In such cases, strong convective motion of the phase segregating domains is generated even in small-size systems and is almost independent of the temperature difference as long as it is below the transition value. We study experimentally the enhancement of heat transfer in a 1-mm -thick cell. A water-acetonitrile-toulene mixture is quenched into a two-phase region so as to induce convection-driven spinodal decomposition. The heat transfer rate is measured and compared to that obtained in the absence of convective motion. A substantial reduction in the cooling time obtains in the case of spinodal decomposition. The heat transfer enhancement induced by this self-induced, disordered but effectively convective effect may be exploited in the cooling or heating of small-scale systems whereby forced convection cannot be achieved because of the small sizes involved. A scaling analysis of the data based on the diffuse interface H model for a symmetric mixture near the equilibrium point yields very encouraging agreement and insights.

A linear analysis of the effects of compressibility on the stability of two superposed isentropic fluids is presented. The results of the analysis, which differ from those available in the literature for other unperturbed stratifications, are illustrated with several numerical examples. It is found that, in the present conditions, compressibility has a stabilizing effect at small wavelengths and a destabilizing effect at long wavelengths. The magnitude of these effects is, however, small in most circumstances. A physical basis for the interpretation of the results is also described in qualitative terms. This discussion sheds some light on the nature of the differences between the present results and those relative to the case of isothermal stratification.