On the Stability of Fluid Flows with Spherical Symmetry

The conditions for the stability or instability of the interface between two immiscible incompressible fluids in radial motion are deduced. The stability conditions derived by Taylor for the interface of two fluids in plane motion do not apply to spherical flows without significant modifications

A Nonsteady Heat Diffusion Problem with Spherical Symmetry

A solution in successive approximations is presented for the heat diffusion across a spherical boundary with radial motion. The approximation procedure converges rapidly provided the temperature variations are appreciable only in a thin layer adjacent to the spherical boundary. An explicit solution for the temperature field is given in the zero order when the temperature at infinity and the temperature gradient at the spherical boundary are specified. The first-order correction for the temperature field may also be found. It may be noted that the requirements for rapid convergence of the approximate solution are satisfied for the particular problem of the growth or collapse of a spherical vapor bubble in a liquid when the translational motion of the bubble is neglected

On the Classical Model of Nuclear Fission

The first experiments on neutron bombardment of various elements carried out by Fermi and his collaborators included the study of the group of activities observed in uranium which were at that time ascribed to transuranic elements. The great number of studies following this first work led finally to the results of Hahn and Strassmann which showed clearly that many of the activities ascribed to transuranic elements came, instead, from nuclei of approximately half the mass of uranium. The startling conclusion that these activities must arise from the splitting of the uranium nucleus under neutron bombardment into two fragment nuclei was pointed out by Meitner and Frisch, and was quickly confirmed by subsequent experiments. In the first theoretical discussion of this new type of nuclear reaction, Meitner and Frisch proposed the name fission for the process, and compared it with the splitting that may take place in a liquid drop in oscillation. This model was supported by Bohr who correlated it with other nuclear properties and, at the same time, emphasized how far the phenomenon of nuclear fission may be described classically. A very complete theoretical discussion of both the classical and quantum aspects of fission was given by Bohr and Wheeler, and it is proposed here to describe some of the classical theory of fission developed by these authors

The Dynamics of Cavitation Bubbles

Three regimes of liquid flow over a body are defined, namely: (a) noncavitating flow; (b) cavitating flow with a relatively small number of cavitation bubbles in the field of flow; and (c) cavitating flow with a single large cavity about the body. The assumption is made that, for the second regime of flow, the pressure coefficient in the flow field is no different from that in the noncavitating flow. On this basis, the equation of motion for the growth and collapse of a cavitation bubble containing vapor is derived and applied to experimental observations on such bubbles. The limitations of this equation of motion are pointed out, and include the effect of the finite rate of evaporation and condensation, and compressibility of vapor and liquid. A brief discussion of the role of "nuclei" in the liquid in the rate of formation of cavitation bubbles is also given

On the Dynamics of Small Vapor Bubbles in Liquids

When a vapor bubble in a liquid changes size, evaporation or condensation of the vapor takes place at the surface of the bubble. Because of the latent heat requirement of evaporation, a change in bubble size must therefore be accompanied by a heat transfer across the bubble wall, such as to cool the surrounding liquid when the bubble grows (or heat it when the bubble becomes smaller). Since the vapor pressure at the bubble wall is determined by the temperature there, the result of a cooling of the liquid is a decrease of the vapor pressure, and this causes a decrease in the rate of bubble growth. A similar effect occurs during the collapse of a bubble which tends to slow down the collapse. In order to obtain a satisfactory theory of the behavior of a vapor bubble in a liquid, these heat transfer effects must be taken into account. In this paper, the equations of motion for a spherical vapor bubble will be derived and applied to the case of a bubble expanding in superheated liquid and a bubble collapsing in liquid below its boiling point. Because of the inclusion of the heat transfer effects, the equations are nonlinear, integro-differential equations. In the case of the collapsing bubble, large temperature variations occur; therefore, tabulated vapor pressure data were used, and the equations of motion were integrated numerically. Analytic solutions are obtainable for the case of the expanding bubble if the period of growth is subdivided into several regimes and the simplifications possible in each regime are utilized. The growth is considered here only during the time that the bubble is small. An asymptotic solution of the equations of motion, valid when the bubble becomes large (i.e. observable), has been presented previously, together with experimental verification. We shall be specifically concerned in the following discussion with the dynamics of vapor bubbles in water. This restriction was made for convenience only, since the theory is applicable without modification to many other liquids

The Growth of Vapor Bubbles in Superheated Liquids

The growth of a vapor bubble in a superheated liquid is controlled by three factors: the inertia of the liquid, the surface tension, and the vapor pressure. As the bubble grows, evaporation takes place at the bubble boundary, and the temperature and vapor pressure in the bubble are thereby decreased. The heat inflow requirement of evaporation, however, depends on the rate of bubble growth, so that the dynamic problem is linked with a heat diffusion problem. Since the heat diffusion problem has been solved, a quantitative formulation of the dynamic problem can be given. A solution for the radius of the vapor bubble as a function of time is obtained which is valid for sufficiently large radius. This asymptotic solution covers the range of physical interest since the radius at which it becomes valid is near the lower limit of experimental observation. It shows the strong effect of heat diffusion on the rate of bubble growth. Comparison of the predicted radius-time behavior is made with experimental observations in superheated water, and very good agreement is found

On the Stability of Gas Bubbles in Liquid-Gas Solutions

With the neglect of the translational motion of the bubble, approximate solutions may be found for the rate of solution by diffusion of a gas bubble in an undersaturated liquid-gas solution; approximate solutions are also presented for the rate of growth of a bubble in an oversaturated liquid-gas solution. The effect of surface tension on the diffusion process is also considered

Scattering and Absorption of Gamma-Rays

A formulation is presented of the scattering and absorption of gamma-rays in different materials. The range of gamma-ray energies considered is from 1 to 10 mc^2. Results are given for the transmission of gamma-rays through air and lead

Phase Diagrams for Sonoluminescing Bubbles

Sound driven gas bubbles in water can emit light pulses. This phenomenon is called sonoluminescence (SL). Two different phases of single bubble SL have been proposed: diffusively stable and diffusively unstable SL. We present phase diagrams in the gas concentration vs forcing pressure state space and also in the ambient radius vs gas concentration and vs forcing pressure state spaces. These phase diagrams are based on the thresholds for energy focusing in the bubble and two kinds of instabilities, namely (i) shape instabilities and (ii) diffusive instabilities. Stable SL only occurs in a tiny parameter window of large forcing pressure amplitude $P_a \sim 1.2 - 1.5$atm and low gas concentration of less than $0.4\%$ of the saturation. The upper concentration threshold becomes smaller with increasing forcing. Our results quantitatively agree with experimental results of Putterman's UCLA group on argon, but not on air. However, air bubbles and other gas mixtures can also successfully be treated in this approach if in addition (iii) chemical instabilities are considered. -- All statements are based on the Rayleigh-Plesset ODE approximation of the bubble dynamics, extended in an adiabatic approximation to include mass diffusion effects. This approximation is the only way to explore considerable portions of parameter space, as solving the full PDEs is numerically too expensive. Therefore, we checked the adiabatic approximation by comparison with the full numerical solution of the advection diffusion PDE and find good agreement.Comment: Phys. Fluids, in press; latex; 46 pages, 16 eps-figures, small figures tarred and gzipped and uuencoded; large ones replaced by dummies; full version can by obtained from: http://staff-www.uni-marburg.de/~lohse

Analytical Approximations for the Collapse of an Empty Spherical Bubble

The Rayleigh equation 3/2 R'+RR"+p/rho=0 with initial conditions R(0)=Rmax, R'(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density rho. The solution for r=R/Rmax as a function of time t=T/Tcollapse, where R(Tcollapse)=0, is independent of Rmax, p, and rho. While no closed-form expression for r(t) is known we find that s(t)=(1-t^2)^(2/5) approximates r(t) with an error below 1%. A systematic development in orders of t^2 further yields the 0.001%-approximation r*(t)=s(t)[1-a Li(2.21,t^2)], where a=-0.01832099 is a constant and Li is the polylogarithm. The usefulness of these approximations is demonstrated by comparison to high-precision cavitation data obtained in microgravity.Comment: 5 pages, 2 figure