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 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

Oscillatory enhancement of the squeezing flow of yield stress fluids: A novel experimental result

The extrusion of a yield stress fluid from the space between two parallel plates is investigated experimentally. Oscillating the magnitude of the squeezing force about a mean value (F = f[1+αcos(ωt)]) was observed to significantly enhance the flow rate of yield stress fluids, while having no effect on the flow rate of Newtonian fluids. This is a novel result. The enhancement depends on the magnitude of the force, the oscillatory frequency and amplitude, the fluid being squeezed, and the thickness of the fluid layer. Non-dimensional results for the various flow quantities have been presented by using the flow predicted for the constant-force squeezing of a Herschel-Bulkley yield stress fluid as the reference. In the limit of constant-force squeezing, the present experimental results compare very well with those of our earlier theoretical model for this situation (Zwick, Ayyaswamy & Cohen 1996). The results presented in this paper have significance, among many applications, for injection moulding, in the adhesive bonding of microelectronic chips, and in surgical procedures employed in health care

Long quantum channels for high-quality entanglement transfer

High-quality quantum-state and entanglement transfer can be achieved in an unmodulated spin bus operating in the ballistic regime, which occurs when the endpoint qubits A and B are coupled to the chain by an exchange interaction $j_0$ comparable with the intrachain exchange. Indeed, the transition amplitude characterizing the transfer quality exhibits a maximum for a finite optimal value $j_0^{opt}(N)$, where $N$ is the channel length. We show that $j_0^{opt}(N)$ scales as $N^{-1/6}$ for large $N$ and that it ensures a high-quality entanglement transfer even in the limit of arbitrarily long channels, almost independently of the channel initialization. For instance, the average quantum-state transmission fidelity exceeds 90% for any chain length. We emphasize that, taking the reverse point of view, should $j_0$ be experimentally constrained, high-quality transfer can still be obtained by adjusting the channel length to its optimal value.Comment: 12 pages, 9 figure

Boundary effects on one-particle spectra of Luttinger liquids

We calculate one-particle spectra for a variety of models of Luttinger liquids with open boundary conditions. For the repulsive Hubbard model the spectral weight close to the boundary is enhanced in a large energy range around the chemical potential. A power law suppression, previously predicted by bosonization, only occurs after a crossover at energies very close to the chemical potential. Our comparison with exact spectra shows that the effects of boundaries can partly be understood within the Hartree-Fock approximation.Comment: 4 pages including 4 figures, revised version, to be published in Phys. Rev. B, January 200

Symmetric Strategy Improvement

Symmetry is inherent in the definition of most of the two-player zero-sum games, including parity, mean-payoff, and discounted-payoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann's traps, which shook the belief in the potential of classic strategy improvement to be polynomial

The level set method for the two-sided eigenproblem

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event systems, and show that the reconstruction of spectrum is pseudopolynomia