28TH INTERNATIONAL SYMPOSIUM ON RAREFIED GAS DYNAMICS 2012

Preface: 28th International Symposium on Rarefied Gas Dynamics
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 PLENARY  HAROLD GRAD LECTURE


Kinetic theory for active and granular particles
View Description Hide DescriptionA dilute suspension of hard sphere impurities in a host gas is described by the BoltzmanLorentz kinetic theory, modified to account for active particles. The case of heavy particles is discussed via the FokkerPlanck limit and the corresponding Langevin equations. Effects of activity on diffusion are calculated in a truncated cumulant approximation.

 PLENARY  LLOYD THOMAS LECTURE


Experiments with sizeselected clusters: From condensation in molecular beams to dust particles in the Enceladus plume
View Description Hide DescriptionThe size selection of clusters and typical applications are reviewed. These include the comparison of cluster distributions with simulations based on condensation theory, the presentation of the infrared spectroscopy of water cluster over a large size range, and the explanation of the icy grains and smaller clusters in the plume of the cryovolcanic Saturn moon Enceladus.

 PLENARY  GRAEME A. BIRD LECTURE


Computer experiments on the onset of turbulence
View Description Hide DescriptionWe are investigating if small amplitude distributed wall roughness, combined with fluctuations, could nucleate the onset of turbulence in bounded flows. Our direct numerical simulations of turbulent transition isolate the effects of the roughness since the only direct flow perturbations we consider are those due to natural hydrodynamic fluctuations. To properly resolve the range of length scales, we developed a conservative mesh refinement approach for the latticeBoltzmann method.

 KINETIC AND TRANSPORT THEORY, BOLTZMANN AND RELATED EQUATIONS


Analytical and numerical computations for high frequency MEMS
View Description Hide DescriptionWe consider a MicroElectroMechanical Systems (MEMS) device vibrating at high frequency. We present (theoretical and numerical) computations allowing to assess the damping of the gas trapped in the channel of the device. Semiexplicit solutions for the transient and permanent regimes associated to the linearized BGK equations with Maxwellian boundary conditions and a periodic forcing are established. Then, different numerical methods are briefly described for the treatment of these equations.

Moment method and nonequilibrium thermodynamics of rarefied gas mixture
View Description Hide DescriptionThe problem of the kinetic justification of the extended irreversible thermodynamics on the basis of Grad's moment method for multicomponent rarefied gas mixture is discussed. The examination is carried out of linearized kinetic Boltzmann equation which is used to obtain an infinite chain of linked equations (moment equations) for the coefficients of expansion of the distribution function in a system of tensor polynomials. Using these equations generalized expressions are obtained for the entropy density, entropy flux density and entropy production as functions of an arbitrary number of state variables, which allow different variants of the relations between fluxes and thermodynamic forces to be considered. For spatially homogeneous systems these relations correspond to the Onsager version of the linear nonequilibrium thermodynamics. In a more general case thermodynamic forces involved in the transport equations are significantly redefined to include, besides the usual gradients of the initial macroscopic parameters, derivatives of the dissipative fluxes with respect to time and coordinates. Some consequence and physical effects, following from the obtained equations are analyzed.

Nonlinear resonant gas oscillation accompanied with evaporation and condensation
View Description Hide DescriptionResonant gas oscillation in a closed tube bounded by an oscillating plate and a vaporliquid interface is theoretically analyzed by applying the asymptotic theory to the ESBGK Boltzmann equation for the case of M_{ P } ∼ Kn ≪ 1 and a small evaporation coefficient , where M_{ P } and Kn are the Mach number of the plate and the Knudsen number, respectively. As a result, we derive a nonlinear integrodifferential equation for determining the wave profile with the evaporation and condensation including α. We numerically solve the integrodifferential equation with the method of Fourier series, and obtain a parameter plane for shock formation conditions and also the prediction of critical condition for shock formation.

On the secondorder slip and jump coefficients for the general theory of slip flow
View Description Hide DescriptionThe general theory of slip flow established in the late 1960s is revisited. For a long time, the complete set of data of the slip and jump coefficients up to the second order of the small Knudsen number have been available only for the BhatnagarGrossKrook model. The present paper provides the complete set of data of those coefficients for a hardsphere gas on the diffuse reflection boundary. The data are obtained by using the general identities that have been deduced from recently developed symmetry arguments. A few simple application examples are also presented.

Automated Boltzmann collision integrals for moment equations
View Description Hide DescriptionWe present a methodology to evaluate the moments of the Boltzmann collision term, in a general automated way, using the computer algebra software Mathematica. Based on Grad's distribution function with 26moments, we compute the nonlinear production terms for a simple gas and a granular gas, and the linear production terms for a binary mixture of gases. The results can be shown for general interaction potential, but, in this paper, they are given only for hardsphere interaction potential.

Model kinetic description for mixtures
View Description Hide DescriptionA consistent derivation of the model linearized collision operator for a multicomponent system is presented. In these results an ambiguity in the choice of coefficients is eliminated, in contrast to the BGK type models. A technique for reconstruction of the model collision integral form based on a known expression for the model linearized operator is proposed. It is shown that the model collision integral in the local (not complete) equilibrium approximation does not contain a complicated exponential, that is common for the BGK type integrals. Boltzmann’s Htheorem is proved for our model.

On the nonequilibrium thermodynamics of rarefied gases: Relationships between the ChapmanEnskog and moment methods
View Description Hide DescriptionThe assumptions are discussed under which the equations of the generalized nonequilibrium thermodynamics obtained by the moment and the ChapmanEnskog methods agree. The ChapmanEnskog method is known to be based on the perturbation theory with the Knudsen number as a small parameter. The parameter is introduced in the kinetic equation and the distribution function is expanded into a series in the small Knudsen number. We apply this procedure not to the distribution function, but to the expansion coefficients of the moment series. It can be shown that under special convolution of the resulting system of equations, one can pass from the moment equations to the ChapmanEnskog equations both in hydrodynamic and Burnett approximations. Here is exactly the same transition in the equations of generalized nonequilibrium thermodynamics. The structure of the heat and momentum fluxes, and nonphysical fluxes entered into the ChapmanEnskog method is reproduced from the moment equations.

Hot atom populations in the terrestrial atmosphere. A comparison of the nonlinear and linearized Boltzmann equations
View Description Hide DescriptionWe use a finite difference discretization method to solve the space homogeneous, isotropic nonlinear Boltzmann equation. We study the time evolution of the distribution function in relation to the solution of the linearized Boltzmann equation for three different initial conditions. The relaxation process is described in terms of the Laguerre moments and the spectral properties of the linearized collision operator. The motivation is the need to include selfcollisions in the study of suprathermal oxygen atoms in the terrestrial exosphere.

Transport processes and new types of boundary Knudsen layers in a gas flows through thin permeable membranes
View Description Hide DescriptionNew kinetic effects and new types of Knudsen boundary layers in a gas flows near permeable membranes with uniform temperature and with temperature difference between the surfaces were studied. Two flows were considered. The first one is a pressure driven gas flow through the membrane, and the second one is a pressure difference driven flow through the membrane, due to the temperature difference across it (thermomolecular pressure difference at zero flow through the membrane). The results of theoretical research of gas flow through the membranes with extremely thin perforated channels (pores) are considered. It is assumed that the gas flow regime inside the pores is a freemolecular one and is a transition one outside the membrane. Mathematical tool are DSMC and other MonteCarlo methods that describe the molecular motion outside the membrane and inside the membrane pores. The quantitative relationship between the flow parameters and the membrane properties is established. A new test for numerical methods to study the slow flows is proposed as follows: the states of gas around infinitely thin permeable membrane, the surfaces which of have different temperature.

Reaction rates and reaction rate constant conception. Onetemperature case
View Description Hide DescriptionThe new method of getting a normal solution for the generalized Boltzmann equation for reacting gas mixtures was proposed. It is based on the following items: (i) slow variables are introduced via approximate summational invariants, defined within the method, (ii) kinetic equations are presented in the form of a singularly perturbed system for gasdynamic (slow) variables and for the "fast" part of the distribution function, (iii) collisional integral is not expanded into the series over the Knudsen number (no assumption is made that the part of the collisional integral, responsible for the chemical reactions, can be treated as the perturbation of its "elastic" part). While deriving the gasdynamic equations it is shown that the role of nonequilibrium effects is much more essential in our approach, than is generally accepted. By nonequilibrium effects we mean all kind of effects caused by deviation of the distribution function from its quasiequilibrium value. From several examples it was shown that nonequilibrium corrections and the traditional equilibrium rate constants could be of the same order of magnitude. In this paper we derive expressions for corrections to equilibrium rate constants for arbitrary mixtures and corresponding integral equations for corrections to the quasiequilibrium distributions. In situations where corrections to the reaction rates are not small nonequilibrium effects dramatically impacts the chemical kinetics of the reacting gas mixture. This leads to the necessity of the revision of the concept of getting information on the reaction rates from the experiments.

Numerical analysis of nonlinear acoustic wave propagation in a rarefied gas
View Description Hide DescriptionUnsteady motion of a rarefied gas in a half space, caused by an infinitely wide plate when it starts a longitudinal and harmonic oscillation, is investigated numerically on the basis of the BhatnagarGrossKrook (BGK) model of the Boltzmann equation. A deterministic method capable of describing the singularities in the molecular velocity distribution function produced by the oscillating plate, which was developed recently by the authors, is used as a solution method, and the unsteady behavior of the gas is obtained accurately. The streaming motion and the attenuation of the wave, observed in the existing work using the direct simulation Monte Carlo (DSMC) method (T. Ohwada and M. Kunihisa, in Rarefied Gas Dynamics , AIP, Melville, 2003, pp. 202209), are also obtained. In addition, some pieces of numerical evidence that clarify the longtime behavior of the gas are provided. For example, oneperiod averages of the momentum and energy fluxes across the oscillating plate tend to approach their values for a periodic state (a constant for the momentum flux and zero for the energy flux) slowly, the rate of approach being likely to be inversely proportional to the square root of time.

The sign change effect of the energy flux and other effects in the transitional regime for the Couette problem
View Description Hide DescriptionResults of Couette flow investigation are given for the freemolecule flow regime as well as or the transitional one. The numerical solution of Boltzmann equation by the direct Monte Carlo simulation method is used in our paper for investigating the Couette flow in a wide range of values of plate temperatures ratio, Knudsen number and the upper plate velocity. The dependences of the energy flux transferred towards the lower plate, of the shear stress, and of the flux of normal momentum on the Knudsen number are considered. It is found that for any Mach number there exists the interval of plate temperatures ratios such that the energy flux towards the plate changes the sign if the Knudsen number increases, previously reaching a local extremum. It is demonstrated as well that the effect of nonmonotonicity at high Mach number is observed not only for the shear stress and the energy flux but for the normal momentum flux as well. We also found the cases where the energy flux varies monotonically or even has a deep negative minimum, but the shear stress has a maximum.

Generating a wave from a wall with changing temperature
View Description Hide DescriptionUnder the purpose to provide a foundation for the numerical approach, the problem of wave generation from a wall with changing temperature is considered to analytically solve the linear BoltzmannBGK equation with diffuse reflection at a wall by successive approximation. Convergence is verified, and approximate solutions used in its proof are utilized to demonstrate its wavelike features observed in the numerical solution.

Influence of reaction heat on time dependent processes in a chemically reacting binary mixture
View Description Hide DescriptionIn this paper we study time dependent problems, like the propagation of sound waves or the behavior of small local wave disturbances induced by spontaneous internal fluctuations, in a binary mixture undergoing a chemical reaction of type A + A ⇌ B + B. The study is developed at the hydrodynamic Euler level, in a chemical regime of fast reactive process in which the chemical reaction is close to its final equilibrium state. The hydrodynamic state of the mixture is described by the balance equations for the mass densities of both constituents A and B, together with the conservation laws for the momentum and total energy of the mixture. The progress of the chemical reaction is specified by an Arrheniustype reaction rate which defines the net balance between production and consumption of each constituent. Assuming that the considered time dependent problems induce weak macroscopic deviations, the hydrodynamic equations are linearized through a normal mode expansion of the state variables around the equilibrium state. From the dispersion relation of the normal modes, we determine the free and forced phase velocities as well as the attenuation coefficients of the waves. We show that the dispersion and absorption of these waves depend explicitly on the heat of the chemical reaction, the concentrations of the constituents and the activation energy through the exponential factor of Arrhenius law.

An ESBGK model for a gas mixture with bimolecular chemical reaction
View Description Hide DescriptionA relaxationtimeapproximation of a Boltzmann kinetic model for a slow bimolecular chemical reaction in a collision dominated gas mixture is presented. The dominant mechanical collision operator, approximated by an ellipsoidal BGK operator recently introduced in the literature, is coupled to a kinetic relaxation model enforcing chemical equilibrium. The hydrodynamic limit up to the fluiddynamic reactive NavierStokes equations are worked out by a ChapmanEnskog asymptotic procedure, emphasizing reactive effects in the resulting reactiondiffusion transport equations.
