The Kramers problem: Velocity slip and defect for a hard sphere gas with arbitrary accommodation

The half-space problem of rarefied gas flow (the Kramers problem) is considered for the linearized Boltzmann equation and arbitrary gas-surface interaction. Accurate numerical results for the velocity slip coefficient and velocity defect are obtained for the rigid sphere interaction and Maxwellian boundary condition.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas

Plane Poiseuille flow of a highly rarefied gas that flows horizontally in the presence of weak gravitation is studied based on the Boltzmann equation for a hard sphere molecular gas and the diffuse reflection boundary condition. The behavior of the solution in the regime of large mean free path and small strength of gravity is studied numerically based on the one-dimensional Boltzmann equation derived by means of the asymptotic analysis for a slow variation in the flow direction. It is clarified that the effect of weak gravity on the flow is not negligible when the gas is so rarefied that the mean free path is comparable to the maximum range that the molecules travel along the parabolic path within the channel. When the mean free path is much larger than this range, the effect of gravity that makes the molecules fall plays the dominant role in determining the distribution function, and thus the over-concentration in the distribution function as well as the flow velocity does not increase further even if the mean free path is increased. The upper bound of the flow velocity and the mass flow rate of the gas are obtained as a function of the gravitational acceleration.     

Nonisothermal gas flow in a long channel analyzed on the basis of the kinetic <Emphasis Type="Italic">S</Emphasis>-Model

The nonisothermal steady rarefied gas flow driven by a given pressure gradient (Poiseuille flow) or a temperature gradient (thermal creep) in a long channel (pipe) of an arbitrary cross section is studied on the basis of the linearized kinetic S-model. The solution is constructed using a high-order accurate conservative method. The numerical computations are performed for a circular pipe and for a cross section in the form of a regular polygon inscribed in a circle. The basic characteristic of interest is the gas flow rate through the channel. The solutions are compared with previously known results. The flow rates computed for various cross sections are also compared with the corresponding results for a circular pipe.     

Heat transfer in a gas mixture between parallel plates

The one-dimensional steady-state heat flux and the temperature distribution in a rarefied gas mixture between two parallel plates with different temperatures are studied using the kinetic theory. The Boltzmann equation is solved by the projection method assuming that the gas consists of elastic hard spheres and the reflection from the surfaces is diffuse. The flow features are analyzed for a wide range of the Knudsen number. The molecular numerical densities of the components, the total temperature of the mixture, and the mixture heat flux are obtained. The behavior of the distribution functions for the components is discussed. A comparison with other authors’ results shows that the accuracy of the given method is good.     

The linearized Boltzmann equation: Sound-wave propagation in a rarefied gas

An analytical version of the discrete-ordinates method (the ADO method) is used to establish concise and particularly accurate solutions to the problem of sound-wave propagation in a rarefied gas. The analysis and the numerical work are based on a rigorous form of the linearized Boltzmann equation (for rigid-sphere interactions), and in contrast to many other works formulated (for an infinite medium) without a boundary condition, the solution reported here satisfies a boundary condition that models a diffusely-reflecting vibrating plate. In addition and in order to investigate the effect of kinetic models, solutions are developed for the BGK model, the S model, the Gross-Jackson model, as well as for the (newly defined) MRS model and the CES model. While the developed numerical results are compared to available experimental data, emphasis in this work is placed on the solutions of the problem of sound-wave propagation as described by the linearized Boltzmann equation and the five considered kinetic models. Received: November 22, 2004; revised: February 24, 2005     

Numerical modelling of rarefied gas flow through a slit at arbitrary pressure ratio based on the kinetic equation

A rarefied gas flow through a thin slit at an arbitrary gas pressure ratio is calculated on the basis of the kinetic model equations (BGK and S-model) applying the discrete velocity method. The calculations are carried out for the whole range of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate and distributions of density, bulk velocity and temperature are reported. Comparisons of the present results with those based on the direct simulation Monte Carlo method and on the linearized BGK kinetic equation are performed. The conditions of applicability of the linearized theory are discussed.     

Efficient method for computing rarefied gas flow in a long finite plane channel

The linearized kinetic S-model is used to study the nonisothermal steady rarefied gas flow driven by differences in pressure and temperature in a plane channel between long finite parallel plates joining two tanks of infinite volume. An efficient composite (asymptotic) method is developed: a one-dimensional asymptotic solution corresponding to an infinitely long channel is constructed in the middle part of the computational domain, while a solution of the two-dimensional kinetic equation matched with the middle-part asymptotic solution is constructed near the ends of the channel. The latter solution is found numerically by a high-order accurate conservative method. The basic quantity to be computed is the gas flow rate through the channel. Characteristic flow features are also investigated. The resulting solutions are compared with previously known results.     

Kinetic model of the Boltzmann equation for a power-law intermolecular interaction potential

A model kinetic equation approximating the Boltzmann equation with a linearized collision integral is constructed to describe rarefied gas flows at moderate and low Knudsen numbers. The kinetic model describes gas flows with a power-law intermolecular interaction potential and involves five relaxation parameters. The structure of a shock wave is computed, and the results are compared with an experiment for argon.     

Numerical analysis of oscillatory Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for a hard sphere molecular gas

The time-dependent motion of a rarefied gas between two parallel planes caused by an oscillatory motion of the plane is studied based on the linearized Boltzmann equation for a hard sphere molecular gas. With the aid of a deterministic numerical method, an accurate numerical analysis is carried out for a wide range of gas rarfaction and oscillatory frequency. The detailed data of the shear stress acting on the planes is provided in a complete form for a wide range of the parameters. The transition of the solution from low to high frequencies under various degrees of gas rarfaction is discussed.     

Numerical analysis of oscillatory Couette flow of a rarefied gas on the basis of the linearized Boltzmann equation for a hard sphere molecular gas

The time-dependent motion of a rarefied gas between two parallel planes caused by an oscillatory motion of the plane is studied based on the linearized Boltzmann equation for a hard sphere molecular gas. With the aid of a deterministic numerical method, an accurate numerical analysis is carried out for a wide range of gas rarfaction and oscillatory frequency. The detailed data of the shear stress acting on the planes is provided in a complete form for a wide range of the parameters. The transition of the solution from low to high frequencies under various degrees of gas rarfaction is discussed.     

Models of a linearized Boltzmann collision integral

For rarefied gas flows at moderate and low Knudsen numbers, model equations are derived that approximate the Boltzmann equation with a linearized collision integral. The new kinetic models generalize and refine the S-model kinetic equation.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

On the behavior of a slightly rarefied gas mixture over plane boundaries

Summary The behavior of a slightly rarefied gas mixture bounded by plane boundaries is investigated on the basis of the linearized Boltzmann equation ofB-G-K type for gas mixtures under the diffusive boundary condition. A useful result of the present analysis is that the macroscopic equations and the appropriate boundary conditions in terms of slip and jump are obtained together with the Knudsen-layer corrections near the boundaries. This system of equations makes possible the treatment at fluid dynamic level for various problems of gas mixtures with plane geometry which require kinetic theory consideration. As an application of this system, some basic flow problems of a slightly rarefied gas mixture, namely, Couette flow, thermal slip flow and diffusion slip flow between two plates are taken up. The total velocity distributions of these concrete problems are explicitly obtained for the first time, and their dependence on the properties and concentration of the component gases in the mixture are clarified in some detail.

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Zusammenfassung Das Verhalten einer verdünnten Gasmischung, bei kleiner aber nicht vernachlässigbarer Knudsen Zahl, zwischen zwei parallelen Platten wird analytisch untersucht. Die linearisierten Boltzmann Gleichungen desB-G-K Typs für Gasmischungen mit diffusiven Randbedingungen werden angewendet. Aus der vorliegenden Untersuchung resultieren brauchbare makroskopische Gleichungen mit den zugehörigen Randbedingungen, — mit Gleitgeschwindigkeit und Temperatursprung formuliert, — sowie die Korrekturen für die wandnahe Knudsen-Schicht. Verschiedene Strömungen von Gasmischungen, bei denen gas-kinetische Effekte eine Rolle spielen, entlang ebener Begrenzungen können mit den Gleichungssystemen auf strömungsmechanischem Niveau behandelt werden. Das System wird auf die Couette Strömung und die thermal-slip und diffusion-slip Strömungen angewendet. Zum ersten Mal werden die Geschwindigkeitsprofile dieser elementaren Strömungen explizit berechnet. Der Einfluß der Gaseigenschaften und Konzentrationen auf diese Profile werden weitgehend erklärt.

     

Numerical study of the radiometric phenomenon exhibited by a rotating Crookes radiometer

The two-dimensional rarefied gas flow around a rotating Crookes radiometer and the arising radiometric forces are studied by numerically solving the Boltzmann kinetic equation. The computations are performed in a noninertial frame of reference rotating together with the radiometer. The collision integral is directly evaluated using a projection method, while second- and third-order accurate TVD schemes are used to solve the advection equation and the equation for inertia-induced transport in the velocity space, respectively. The radiometric forces are found as functions of the rotation frequency.     

A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows

In this paper, an alternative approach to the traditional continuum analysis of flow problems is presented. The traditional methods, that have been popular with the CFD community in recent times, include potential flow, Euler and Navier–Stokes solvers. The method presented here involves solving the governing equation of the molecular gas dynamics that underlies the macroscopic behaviour described by the macroscopic governing equations. The equation solved is the Boltzmann kinetic equation in its simplified collisionless and BGK forms. The algorithm used is a discontinuous Taylor–Galerkin type and it is applied to the 2D problems of a highly rarefied gas expanding into a vacuum, flow over a vertical plate, rarefied hypersonic flow over a double ellipse, and subsonic and transonic flow over an aerofoil. The benefit of this type of solver is that it is not restricted to continuum regime (low Knudsen number) problems. However, it is a computationally expensive technique.     

Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels

Unsteady rarefied gas flows in narrow channels accompanied by shock wave formation and propagation were studied by solving the Boltzmann kinetic equation. The formation of a shock wave from an initial discontinuity of gas parameters, its propagation, damping, and reflection from the channel end face were analyzed. The Boltzmann equation was solved using finite differences. The collision integral was calculated on a fixed velocity grid by a conservative projection method. A detector of shock wave position was developed to keep track of the wave front. Parallel computations were implemented on a cluster of computers with the use of the MPI technology. Plots of shock wave damping and detailed flow fields are presented.     

Nonlinear nonequilibrium kinetic model of the boltzmann equation for monatomic gases

A model kinetic equation approximating the Boltzmann equation in a wide range of nonequilibrium gas states was constructed to describe rarefied gas flows. The kinetic model was based on a distribution function depending on the absolute velocity of the gas particles. Highly efficient in numerical computations, the model kinetic equation was used to compute a shock wave structure. The numerical results were compared with experimental data for argon.