Boltzmann Equation For Hard-sphere Molecules Research Articles

Variational derivation of thermal slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules and Cercignani–Lampis boundary conditions: Comparison with experimental results

In the present paper, a variational method is applied to solve the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules and the Cercignani–Lampis boundary conditions. This technique allows us to obtain an explicit relation between the first- and second-order thermal slip coefficients and the tangential momentum and normal energy accommodation coefficients, defined in the frame of the Cercignani–Lampis scattering kernel. Comparing the theoretical results with the experimental data from the work of Yamaguchi et al. [“Mass flow rate measurement of thermal creep flow from transitional to slip flow regime,” J. Fluid Mech. 795, 690 (2016)], a pair of accommodation coefficients has been extracted for each noble gas considered in the experiments. Then, these values have been used to compute, by means of our variational technique, the temperature-driven mass flow rates, and the outputs have been compared with the measurements for helium, neon, and argon. Good agreement has been obtained between the theoretical and the experimental data, within the range of validity of the proposed second-order slip model. For all the gases analyzed, the tangential accommodation coefficient is found to be much larger than the normal energy coefficient. The general trend, according to which, by increasing the molecular weight of the different gases, the values of both accommodation coefficients also increase, is confirmed in this study.

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.

Hagen–Poiseuille and thermal transpiration flows of a highly rarefied gas through a circular pipe

Hagen–Poiseuille and thermal transpiration flows of a highly rarefied gas through a circular pipe are investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. A numerical computation based on an iterative approximation method with an explicit convergence estimate is performed to construct a database of the mass fluxes for an arbitrary value of large Knudsen numbers. The singular behavior of the velocity distribution functions in that rarefied regime is also clarified.

Variational derivation of second-order slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules

The objective of the present paper is to provide an analytic expression for the first- and second-order velocity slip coefficients. Therefore, gas flow rates in microchannels have been rigorously evaluated in the near-continuum limit by means of a variational technique which applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator. The diffuse-specular reflection condition of Maxwell’s type has been considered in order to take into account the influence of the accommodation coefficient on the slip parameters. The polynomial form of the Knudsen number obtained for the Poiseuille mass flow rate and the values of the velocity slip coefficients, found on the basis of our variational solution of the linearized Boltzmann equation for hard-sphere molecules, are analyzed in the frame of potential applications of classical continuum numerical tools in simulations of microscale flows.

Vapor Flows Along a Plane Condensed Phase with Weak Condensation in the Presence of a Noncondensable Gas

A steady flow of a vapor in a half space condensing at incidence onto a plane condensed phase is considered in the case where another gas that does not condense (the noncondensable gas) is present near the condensed phase. A systematic asymptotic analysis of the Boltzmann equation for hard-sphere molecules is performed in the case where condensation is weak, and the relation among the parameters of the vapor flow at infinity, those associated with the plane condensed phase, and the amount of the noncondensable gas is derived in an analytical form. The result supplements the numerical result for the relation for arbitrarily strong condensation obtained on the basis of a model Boltzmann equation and under the restriction that the vapor molecules are mechanically identical with the noncondensable-gas molecules [Taguchi et al., Phys. Fluids 15: 689 (2003)].

Temperature, pressure, and concentration jumps for a binary mixture of vapors on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation

The half-space problem of the temperature, pressure, and concentration jumps for a binary mixture of vapors is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the complete condensation condition. First, the problem is shown to be reduced to three elemental ones: the problem of the jumps caused by the net evaporation or condensation, that caused by the gradient of temperature, and that caused by the gradient of concentration. Then, the latter two are investigated numerically in the present contribution because the first problem has already been studied [Yasuda, Takata, and Aoki, Phys. Fluids 17, 047105 (2005)]. The numerical method is a finite-difference one, in which the complicated collision integrals are computed by the extension of the method proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)] to the case of a gas mixture. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic quantities but also at the level of the velocity distribution function. In addition, accurate formulas of the temperature, pressure, and concentration jumps are constructed for arbitrary values of the concentration of the background reference state by the use of the Chebyshev polynomial approximation. The solution of the corresponding problem of a vapor-gas mixture and that of the temperature-jump problem on a simple solid wall are also obtained as special cases of the present problem.

Evaporation and condensation of a binary mixture of vapors on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation

Half-space problem of evaporation and condensation of a binary mixture of vapors is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the complete condensation condition. The problem is analyzed numerically by a finite-difference method, in which the complicated collision integrals are computed by the extension of the method proposed by Y. Sone, T. Ohwada, and K. Aoki [“Temperature jump and Knudsen layer in a rarefied gas over a plane wall: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules,” Phys. Fluids A 1, 363 (1989)] to the case of a gas mixture. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic quantities but also at the level of the velocity distribution function. In addition, accurate formulas of the temperature, pressure, and concentration jumps caused by the evaporation and condensation are constructed for arbitrary values of the concentration of the background reference state by the use of the Chebyshev polynomial approximation.

Numerical analysis of the shear flow of a binary mixture of hard-sphere gases over a plane wall

The shear flow of a binary mixture of rarefied gases over a plane wall is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. This fundamental problem in rarefied gas dynamics is analyzed numerically by a finite-difference method, in which the complicated collision integrals are computed by the extension to the case of a gas mixture of the method proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)]. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic variables but also at the level of the velocity distribution function. In addition, an accurate formula of the shear-slip (viscous-slip) coefficient for arbitrary values of the concentration of a component gas is constructed by the use of the Chebyshev polynomial approximation.

平面凝縮相における2成分混合蒸気の蒸発・凝縮 : 線形化ボルツマン方程式の数値解析(S27-2 熱流体現象に対する分子気体力学および分子動力学法の応用(2),S27 熱流体現象に対する分子気体力学および分子動力学法の応用)

Half-space problem of evaporation and condensation of a binary mixture of vapors is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the complete condensation condition. The problem is analyzed numerically by a finite-difference method, in which the complicated collision integrals are computed by the extension of the method proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)] to the case of a gas mixture. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic quantities but also at the level of the velocity distribution function. In addition, accurate formulas of the temperature, pressure, and concentration jumps caused by the evaporation or condensation are constructed for arbitrary values of the concentration of a component gas by the use of the Chebyshev polynomial approximation.

Numerical analysis of thermal-slip and diffusion-slip flows of a binary mixture of hard-sphere molecular gases

The thermal-slip (thermal-creep) and the diffusion-slip problems for a binary mixture of gases are investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. The problems are analyzed numerically by the finite-difference method incorporated with the numerical kernel method, which was first proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)] for a single-component gas. As a result, the behavior of the mixture is clarified accurately not only at the level of the macroscopic variables but also at the level of the velocity distribution function. In addition, accurate formulas of the thermal-slip and the diffusion-slip coefficients for arbitrary values of the concentration of a component gas are constructed by the use of the Chebyshev polynomial approximation.

Contributions of steady heat conduction to the rate of chemical reaction

We have derived the effect of steady heat flux on the rate of chemical reaction based on the line-of-centers model using the explicit velocity distribution function of the steady-state Boltzmann equation for hard-sphere molecules to second order. It is found that the second-order velocity distribution function plays an essential role for the calculation of it. We have also compared our result with those from the steady-state Bhatnagar–Gross–Krook (BGK) equation and information theory, and found no qualitative differences among them.

Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann equation for hard-sphere molecules

The structure of a normal shock wave for a binary mixture of hard-sphere gases is analyzed numerically on the basis of the Boltzmann equation by a finite-difference method. In the analysis, the complicated collision integrals are computed efficiently as well as accurately by means of the numerical kernel method, which is the generalization to the case of a binary mixture of the method devised by Ohwada in 1993 in the shock-structure analysis for a single-component gas. The transition from the upstream to the downstream uniform state is clarified not only for the macroscopic quantities but also for the velocity distribution functions.

The velocity distribution function in an infinitely strong shock wave

The structure of an infinitely strong shock wave (i.e., a shock wave with infinitely large upstream Mach number) is investigated on the basis of the Boltzmann equation for hard-sphere molecules. The velocity distribution function is expressed as a sum of a multiple of the Dirac delta function, centered at the upstream bulk velocity, and the remainder. The equation for the latter, which contains the linear collision term linearized around the delta function and the nonlinear collision term, is solved numerically by an accurate finite-difference method after the nonlinear collision term is replaced by the BGK collision model. The result not only confirms the singularity in the remainder observed in the previous Monte Carlo simulation [Cercignani et al., Phys. Fluids 11, 2757 (1999)] but also provides more detailed information about the structure of the singularity.

Higher order time integration of spatially nonhomogeneous Boltzmann equation: Deterministic and stochastic computations

High accuracy of the time-differencing method for the spatially non-homogeneous Boltzmann equation, recently developed in [T. Ohwada, Journal of Compt. Phys., 139, 1 (1998)] as the second order approximation of the integral form of the equation along its characteristic line, is demonstrated numerically in both deterministic and stochastic computations of the standard Boltzmann equation for hard-sphere molecules. Comparisons are made with the results by the conventional splitting method and those by the splitting method with second order accurate collision step. Although the error of splitting method is reduced by the improvement of accuracy of the collision step, however, the accuracy is still first order because of the intrinsic error of the method. The intrinsic error is observed clearly in the stochastic computation of an initial value problem with a steep initial function, where the modified direct simulation Monte-Carlo proposed in the previous study is shown to work well.

Evaporation from or condensation onto a sphere: Numerical analysis of the Boltzmann equation for hard-sphere molecules

A steady weak evaporation from or condensation onto a spherical condensed phase in its vapor gas is investigated, mainly numerically, on the basis of the Boltzmann equation for hard-sphere molecules. The numerical method is the combination of the hybrid-difference-scheme method, which can describe the discontinuity of the velocity distribution function in the gas, and the numerical kernel method [1]. The velocity distribution function, macroscopic variables (pressure, temperature, flow velocity, heat flow), as well as mass and heat fluxes from the sphere into the gas, are obtained for the whole range of the Knudsen number. In addition, the solution of the heat transfer problem of a nonvolatile sphere in a rarefied gas is obtained as a special case of the present problem. The problem is also discussed from the standpoint of nonequilibrium thermodynamics.

Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules

Heat flow and temperature and density distributions in a rarefied gas between two parallel plates at rest with different uniform temperatures are analyzed numerically on the basis of the full nonlinear Boltzmann equation for hard-sphere molecules and the Maxwell-type boundary condition by a finite difference method where the collision term is computed direct numerically. The accurate results are presented for the case in the density measurement by Teagan and Springer [Phys. Fluids 11, 497 (1968)], where the temperature ratio is 1.326, the value of the accommodation coefficient is 0.826, and the ratio of mean free path to plate spacing (Knudsen number Kn) is 0.0658≤Kn≤0.7582. It is found that there is a considerable difference between the present density distribution and the experimental data. The reason for this discrepancy is also discussed. The accurate numerical results of the linearized problem are also presented for comparison.

Measurement of the thermophoretic force by electrodynamic levitation: Microspheres in air

The thermophoretic force on single microspheres has been measured over a wide range of Knudsen numbers and particle thermal conductivities for solid and liquid spheres in air. The microspheres were dioctyl phthalate (DOP) droplets, metallic nickel, polystyrene latex (PSL), and glass. This covers a thermal conductivity ratio (gas to particle) range of 2 × 10 −4−0.17. In a companion paper, results are reported for monatomic (helium) and polyatomic gases (carbon dioxide and air) to examine the effects of gas properties. The measurements were accomplished by levitating the particle between heated and cooled plates mounted in a vacuum chamber. Light-scattering phase functions were used to determine the size of all materials except the nickel spheres, and for the highly absorbing metallic spheres the size was obtained by measuring the marginal stability limit of the particle. The apparatus developed for the research and the experimental techniques are discussed, and the experimental results in the Knudsen number range 0.05–20 are compared with previously proposed theories. For particle Knudsen numbers greater than about 10, the effects of the temperature jumps (thermal slip) at the heated and cooled plates become significant. The data in the transition regime overlap Loyalka's (1992, J. Aerosol Sci. 23, 291–300) solution of the linearized Boltzmann equation for hard-sphere molecules, and the data in the slip regime agree with Brock's (1962, J. Colloid Sci. 17, 768–780) theory. The effects of particle thermal conductivity are shown to be nearly negligible in the Knudsen regime, for the data for particles having greatly different thermal conductivities overlap.

Numerical Analysis of a Uniform Flow of a Rarefied Gas Past a Sphere on the Basis of the Boltzmann Equation for Hard-Sphere Molecules

Numerical Analysis of a Uniform Flow of a Rarefied Gas Past a Sphere on the Basis of the Boltzmann Equation for Hard-Sphere Molecules

Numerical analysis of a rarefied gas flow past a volatile particle using the Boltzmann equation for hard-sphere molecules

A spherical condensed phase with a uniform surface temperature that is placed in a slow uniform flow of its vapor gas is considered. The steady behavior of the gas accompanied by evaporation and condensation on the sphere is investigated mainly numerically on the basis of the Boltzmann equation for hard-sphere molecules. The numerical method is a combination of the hybrid-difference-scheme method, capable of describing the discontinuity of the velocity distribution function in the gas, and the numerical kernel method [Phys. Fluids A 5, 716 (1993)]. The velocity distribution function of the gas molecules, the macroscopic variables such as the density, velocity, and temperature of the gas, and the force (drag) acting on the sphere are obtained precisely for the whole range of the Knudsen number (the mean free path of the uniform flow divided by the radius of the condensed phase). In particular, the behavior of the discontinuity of the velocity distribution function in the gas is described accurately.

Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules

A slow uniform flow of a rarefied gas past a sphere is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules and the diffuse reflection condition. With the aid of a similarity solution, the Boltzmann equation is reduced to two simultaneous integrodifferential equations with three independent variables, which are solved numerically. The collision integral is computed efficiently by the use of a numerical collision kernel [Phys. Fluids A 1, 363 (1989)]. The velocity distribution function of the gas molecules, which has discontinuity in the gas, the density, flow velocity, and temperature fields of the gas, and the drag on the sphere are obtained accurately for the whole range of the Knudsen number. In spite of slow flow, the temperature is nonuniform (thermal polarization). From the behavior of the velocity distribution function, the kinetic transition region is clearly seen to separate into the Knudsen layer and the S layer for small Knudsen numbers.