Analytical and numerical analyses have been made of the physical behaviour of a collapsing bubble in a liquid. The mathematical formulation takes into account the effects of compressibility of the liquid, non-equilibrium condensation of the vapour, heat conduction and the temperature discontinuity at the phase interface. Numerical solutions for the collapse of the bubble are obtained beyond the time when the bubble reaches its minimum radius up to the stage when a pressure wave forms and propagates outward into the liquid. The numerical results indicate that evaporation and condensation strongly influence the dynamical behaviour of the bubble.In addition, the propagation of the stress wave, both in a solid and a liquid, due to the collapse of the bubble has been observed by means of the dynamic photoelasticity. It is clearly demonstrated that the stress wave in a photoelastic specimen is caused by impact of the pressure wave radiated from the bubble.

The evaporation and condensation at an interface of vapor and its condensed phase is considered. The validity of kinetic boundary condition for the Boltzmann equation, which prescribes the velocity distribution function of molecules outgoing from the interface, is investigated by the numerical method of molecular dynamics for argon. From the simulations of evaporation into vacuum, the spontaneous-evaporation flux determined by the temperature of condensed phase is discovered. Condensation coefficient in equilibrium states can then be determined without any ambiguity. It is found that the condensation coefficient is close to unity below the triple-point temperature and decreases gradually as the temperature rises. The velocity distribution of spontaneously evaporating molecules is found to be nearly a half-Maxwellian at a low temperature. This fact supports the kinetic boundary condition widely used so far. At high temperatures, on the other hand, the velocity distribution deviates from the half-Maxwellian.

The structural change of a phospholipid bilayer in water under the action of a shock wave is numerically studied with unsteady nonequilibrium molecular dynamics simulations. The action of shock waves is modeled by the momentum change of water molecules, and thereby we demonstrate that the resulting collapse and rebound of the bilayer are followed by the penetration of water molecules into the hydrophobic region of the bilayer. The high-speed phenomenon that occurs during the collapse and rebound of the bilayer is analyzed in detail, particularly focusing on the change of bilayer thickness, the acyl chain bend angles, the lateral fluidity of lipid molecules, and the penetration rate of water molecules. The result shows that the high-speed phenomenon can be divided into two stages: in the first stage the thickness of bilayer and the order parameter are rapidly reduced, and then in the second stage they are recovered relatively slowly. It is in the second stage that water molecules are steadily introduced into the hydrophobic region. The penetration of water molecules is enhanced by the shock wave impulse and this qualitatively agrees with a recent experimental result.

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The kinetic boundary condition for the Boltzmann equation at an interface between a polyatomic vapor and its liquid phase is investigated by the numerical method of molecular dynamics, with particular emphasis on the functional form of the evaporation part of the boundary condition, including the evaporation coefficient. The present study is an extension of a previous one for argon [Ishiyama, Yano, and Fujikawa, Phys. Fluids 16, 2899 (2004)] to water and methanol, typical examples of polyatomic molecules. As in the previous study, molecular dynamics simulations of vapor-liquid equilibrium states and those of evaporation from liquid into a virtual vacuum are carried out for water and methanol. In spite of the formation of molecular clusters in the vapor phase and the presence of the preferential orientation of molecules at the interface, essentially the same results as in the previous study are obtained. When the bulk liquid temperature is relatively low, the evaporation part is the product of the half range Maxwellian for the translational velocity of molecules of saturated vapor at the temperature of the bulk liquid phase, the equilibrium distribution of rotational energy of molecules at the temperature, and the evaporation coefficient (or the condensation coefficient in the equilibrium state). The evaporation coefficients of water and methanol are determined without any ambiguity as decreasing functions of the temperature, and are found to approach unity with the decrease of the temperature.

We propose a systematic derivation method of the Korteweg-de Vries-Burgers (KdVB) equation and nonlinear Schrödinger (NLS) equation for nonlinear waves in bubbly liquids on the basis of appropriate choices of scaling relations of physical parameters. The basic equations are composed of a set of conservation equations for mass and momentum and the equation of bubble dynamics in a two-fluid model. The scaling of parameters is related to the wavelength, frequency, propagation speed, and amplitude of waves concerned. With the help of the method of multiple scales, appropriate choices of the parameter scaling allow us to derive various nonlinear wave equations systematically from a set of basic equations. The result shows that the one-dimensional nonlinear propagation of a long wave with a low frequency is described by the KdVB equation, and that of an envelope of a carrier wave with a high frequency by the NLS equation. Thus, we establish a unified theory of derivation of nonlinear wave equations in bubbly liquids.

By molecular dynamics simulations, the boundary condition for the Boltzmann equation at a vaporliquid interface is found to be the product of three one-dimensional Maxwellian distributions for the three velocity components of vapor molecules and a factor including a well-defined condensation coefficient. The Maxwellian distribution for the velocity component normal to the interface is characterized by the liquid temperature, as in a conventional model boundary condition, while those for the tangential components are prescribed by a different temperature, which is a linear function of energy flux across the interface. The condensation coefficient is found to be constant and equal to the evaporation coefficient determined by the liquid temperature only.

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