Kinetic theory for a simple modeling of phase transition: Dynamics out of local equilibrium

This is a continuation of the previous work (Takata & Noguchi, J. Stat. Phys., 2018) that introduces the presumably simplest model of kinetic theory for phase transition. Here, main concern is to clarify the stability of uniform equilibrium states in the kinetic regime, rather than that in the continuum limit. It is found by the linear stability analysis that the linear neutral curve is invariant with respect to the Knudsen number, though the transition process is dependent on the Knudsen number. In addition, numerical computations of the (nonlinear) kinetic model are performed to investigate the transition processes in detail. Numerical results show that (unexpected) incomplete transitions may happen as well as clear phase transitions.Comment: 21 pages, 7 figure

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence

Sound waves propagating in a slightly rarefied gas over a smooth solid boundary

A time-evolution of a slightly rarefied gas from a uniform equilibrium state at rest is investigated on the basis of the linearized Boltzmann equation under the acoustic time scaling. By a systematic asymptotic analysis, linearized Euler sets of equations and acoustic-boundary-layer equations are derived, together with their slip and jump boundary conditions, as well as the correction formula in the Knudsen layer. Analysis is done up to the first order of the Knudsen number (Kn), with Kn¹/² being the small parameter. Several rarefaction effects, which are known as the effects of the second order in Kn in the diffusion scaling, are enhanced to be of the first order in Kn. This is because the variation of the macroscopic quantities along the normal direction is steep in the boundary layer and the compressibility of the gas is comparatively strong. The occurrence of secular terms associated with the Hilbert expansion is pointed out and a remedy for it is also given. Finally, as an application example, a sound propagation in a half space caused by a sinusoidal oscillation of flat boundary is examined on the basis of the Bhatnagar–Gross–Krook equation. The asymptotic solution agrees well with the direct numerical solution

Second-order Knudsen-layer analysis for the generalized slip-flow theory II: Curvature effects

Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists

Heat transfer in a dense gas between two parallel plates

Time-dependent heat transfer in a dense gas between two parallel plates, which is initiated by an abrupt change in temperature of one plate, is numerically investigated on the basis of the Enskog equation under the diffuse reflection boundary condition. Numerical computation is carried out by a finite-difference scheme combined with the Fourier spectral method for the efficient computation of the collision term of the Enskog equation. As a result, macroscopic quantities of the gas, such as heat flux and temperature, are obtained for various Knudsen numbers and ratios of the molecular diameter to the distance between plates. Compared to the case of an ideal gas, the heat flux in the stationary state is enhanced due to an effect of the finite size of molecules for not only small but also intermediate Knudsen numbers. The results imply that the finite-size effect also affects the propagation of disturbances in the initial stage, particularly for small Knudsen numbers

A revisit to the Cercignani–Lampis model: Langevin picture and its numerical simulation

Part of the Springer INdAM Series book series (SINDAMS, volume 48)The Workshop INdAM "Recent advances in kinetic equations and applications", which took place in Rome (Italy), from November 11th to November 15th, 2019.The Cercignani–Lampis (CL) model for the gas–surface interaction is revisited from the Langevin dynamics viewpoint. Starting from a time-independent Fokker–Planck formalism by Cercignani, its time-dependent extension and the corresponding Langevin description are introduced. The Langevin description sheds light on dynamical features of a stochastic process corresponding to the CL model. Numerical simulations on the basis of the Langevin description are performed as well to reproduce the scattering kernel and reflection intensity distribution numerically. Although the noise in the stochastic process is apparently simple, the Milstein scheme rather than the Euler–Maruyama scheme has to be adopted to achieve a satisfactory numerical convergence in time discretisation

On the entropic property of the Ellipsoidal Statistical model with the Prandtl number below 2/3

Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for $\mathrm{Pr}<2/3$, provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted

Modeling of gas transport in porous medium: Stochastic simulation of the Knudsen gas and a kinetic model with homogeneous scatterer

This paper is part of the Special Topic, Advances in Micro/Nano Fluid Flows: In Memory of Prof. Jason Reese.Mass transport of the Knudsen gas in a porous medium is investigated on the basis of the kinetic theory of gases. First, the mass flow conductance is computed numerically for various porosities and solid grain sizes by stochastic particle simulations (SPS). Then, a kinetic model with a homogeneous scatterer is introduced, which contains the reference Knudsen number as the sole parameter that characterizes the collision frequency of gas molecules with the micro-structural solid surface. With the aid of the standard asymptotic analyses for small and large Knudsen numbers combined with the percolation theory, the effective reference Knudsen number is identified to reproduce the SPS results for a wide range of porosities

Parabolic temperature profile and second-order temperature jump of a slightly rarefied gas in an unsteady two-surface problem

The behavior of a slightly rarefied monatomic gas between two parallel plates whose temperature grows slowly and linearly in time is investigated on the basis of the kinetic theory of gases. This problem is shown to be equivalent to a boundary-value problem of the steady linearized Boltzmann equation describing a rarefied gas subject to constant volumetric heating. The latter has been recently studied by Radtke, Hadjiconstantinou, Takata, and Aoki (RHTA) as a means of extracting the second-order temperature jump coefficient. This correspondence between the two problems gives a natural interpretation to the volumetric heating source and explains why the second-order temperature jump observed by RHTA is not covered by the general theory of slip flow for steady problems. A systematic asymptotic analysis of the time-dependent problem for small Knudsen numbers is carried out and the complete fluid-dynamic description, as well as the related half-space problems that determine the structure of the Knudsen layer and the coefficients of temperature jump, are obtained. Finally, a numerical solution is presented for both the Bhatnagar-Gross-Krook model and hard-sphere molecules. The jump coefficient is also calculated by the use of a symmetry relation; excellent agreement is found with the result of the numerical computation. The asymptotic solution and associated second-order jump coefficient obtained in the present paper agree well with the results by RHTA that are obtained by a low variance stochastic method

Bone erosions in rheumatoid arthritis can be repaired through reduction in disease activity with conventional disease-modifying antirheumatic drugs

We conducted the present study to determine whether repair of erosions occurs in patients with rheumatoid arthritis (RA) treated with conventional disease-modifying anti-rheumatic drugs (DMARDs) and to compare clinical characteristics between patients exhibiting and not exhibiting erosion repair. We included in the study a total of 122 RA patients who fulfilled the 1987 American College of Rheumatology criteria for RA; all patients had paired sequential radiographs of both hands and wrists showing erosive changes at baseline. Patients were classified into two groups according to the presence of repair of erosions at follow up, namely the 'repair observed' and 'repair not observed' groups. Clinical characteristics, disease activity, radiographic scores and treatment in the two groups were compared. Forty-four repairs were observed in 13 patients (10.7%). Compared with the repair not observed group, the functional class of the patients in the repair observed group was lower at baseline (P < 0.01) and the mean disease activity was lower at follow up (P < 0.005). The changes in radiographic scores per year (total radiographic score and erosion score) were lower (P < 0.05 and P < 0.01, respectively) in the repair observed group. No difference in treatment was observed. Repair of erosions was detected in 10.7% of RA patients treated with conventional DMARDs. Repairs were associated with low functional class at baseline and low disease activity at follow up. These observations support the importance of reduction in disease activity in RA patients. Because repair of erosions was detected in a substantial number of patients, assessment of erosion repair should be incorporated into the radiographic evaluation and scoring of RA