Kelvin方程的一种理论推导

从液滴平衡条件推导出严格意义的Kelvin方程,验证了其在宏观尺度可以转化为经典形式.利用Tolman方程,在考虑表面张力与曲率半径关系的条件下,给出在液体压缩性可忽略时,饱和蒸气压、蒸气密度、蒸气摩尔体积和曲率半径等关系;液体压缩性不可忽略时,得出以等温压缩系数和Tolman长度表示的饱和蒸气压与液滴半径的关系.

Comparative study using small-angle x-ray scattering and nitrogen adsorption in the characterization of silica xerogels and aerogels

A comparative study using small-angle x-ray scattering (SAXS) and nitrogen adsorption has been carried out in the structural characterization of silica xerogels and aerogels, obtained from tetraethoxysilane sonohydrolysis. The specific surface and the mean pore size as measured by both the techniques were found to be in notable agreement in all cases for aerogels and xerogels. According to the SAXS data, aerogels at 500 °C exhibit a mass fractal structure with fractal dimension D∼2.4 in the range between the correlation length ξ∼5.3 nm and a∼0.75 nm. An experimental method to probe the mass fractal structure of aerogels from exclusively nitrogen adsorption isotherms has been presented. For aerogels at 500 °C, we have found D∼2.4 in the range between the pore width 2rξ∼33 nm and 2ra∼4.5 nm, which is in notable agreement with the SAXS results (D ∼2.4, ξ∼5.3 nm, a∼0.75 nm) if we assign the pore width 2r probed by the Kelvin equation in the adsorption method to the Bragg distance 2π/q associated to the correlation length 1/q probed by SAXS.

Exact solution for the energy spectrum of Kelvin-wave turbulence in superfluids

We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at near-zero temperatures. In this paper, we show analytically that the solution proposed by [L’vov and Nazarenko, JETP Lett. 91, 428 (2010)] enjoys existence, uniqueness, and regularity of the prefactor. Furthermore, we present numerical results of the dynamical equation that describes to leading order the nonlocal regime of the Kelvin-wave dynamics. We compare our findings with the analytical results from the proposed local and nonlocal theories for Kelvin-wave dynamics and show an agreement with the nonlocal predictions. Accordingly, the spectrum proposed by L’vov and Nazarenko should be used in future theories of quantum turbulence. Finally, for weaker wave forcing we observe an intermittent behavior of the wave spectrum with a fluctuating dissipative scale, which we interpreted as a finite-size effect characteristic of mesoscopic wave turbulence.

Kelvin-wave cascade in the vortex filament model

The small-scale energy-transfer mechanism in zero-temperature superfluid turbulence of helium-4 is still a widely debated topic. Currently, the main hypothesis is that weakly nonlinear interacting Kelvin waves (KWs) transfer energy to sufficiently small scales such that energy is dissipated as heat via phonon excitations. Theoretically, there are at least two proposed theories for Kelvin-wave interactions. We perform the most comprehensive numerical simulation of weakly nonlinear interacting KWs to date and show, using a specially designed numerical algorithm incorporating the full Biot-Savart equation, that our results are consistent with the nonlocal six-wave KW interactions as proposed by L'vov and Nazarenko.

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

Kelvin-Helmholtz instability in solar chromospheric jets: theory and observation

Using data obtained by the high-resolution CRisp Imaging SpectroPolarimeter instrument on the Swedish 1 m Solar Telescope, we investigate the dynamics and stability of quiet-Sun chromospheric jets observed at the disk center. Small-scale features, such as rapid redshifted and blueshifted excursions, appearing as high-peed jets in the wings of the Hα line, are characterized by short lifetimes and rapid fading without any descending behavior. To study the theoretical aspects of their stability without considering their formation mechanism, we model chromospheric jets as twisted magnetic flux tubes moving along their axis, and use the ideal linear incompressible magnetohydrodynamic approximation to derive the governing dispersion equation. Analytical solutions of the dispersion equation indicate that this type of jet is unstable to Kelvin–Helmholtz instability (KHI), with a very short (few seconds) instability growth time at high upflow speeds. The generated vortices and unresolved turbulent flows associated with the KHI could be observed as a broadening of chromospheric spectral lines. Analysis of the Hα line profiles shows that the detected structures have enhanced line widths with respect to the background. We also investigate the stability of a larger-scale Hα jet that was ejected along the line of sight. Vortex-like features, rapidly developing around the jet’s boundary, are considered as evidence of the KHI. The analysis of the energy equation in the partially ionized plasma shows that ion–neutral collisions may lead to fast heating of the KH vortices over timescales comparable to the lifetime of chromospheric jets.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Exploring first-year academic achievement through structural equation modelling

The purpose of this research was to develop and test a multicausal model of the individual characteristics associated with academic success in first-year Australian university students. This model comprised the constructs of: previous academic performance, achievement motivation, self-regulatory learning strategies, and personality traits, with end-of-semester grades the dependent variable of interest. The study involved the distribution of a questionnaire, which assessed motivation, self-regulatory learning strategies and personality traits, to 1193 students at the start of their first year at university. Students' academic records were accessed at the end of their first year of study to ascertain their first and second semester grades. This study established that previous high academic performance, use of self-regulatory learning strategies, and being introverted and agreeable, were indicators of academic success in the first semester of university study. Achievement motivation and the personality trait of conscientiousness were indirectly related to first semester grades, through the influence they had on the students' use of self-regulatory learning strategies. First semester grades were predictive of second semester grades. This research provides valuable information for both educators and students about the factors intrinsic to the individual that are associated with successful performance in the first year at university.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.