Kazuo Aoki

A unidirectional flow of a rarefied gas between two parallel plates driven by a uniform external force is investigated on the basis of kinetic theory with special interest in the behavior in the near continuum case. The BGK model of the Boltzmann equation and the diffuse reflection boundary condition are employed as the basic system. First, a systematic asymptotic analysis of the basic system for small Knudsen numbers and for a weak external force is carried out, and a system of fluid-dynamic type equations and the slip boundary conditions are derived up to the second order in the Knudsen number, together with the Knudsen-layer corrections near the plates. Then, an accurate numerical analysis of the original BGK system is performed by means of a finite-difference method. The behavior of the gas in the near continuum case is clarified on the basis of the fluid-dynamic type system as well as the numerical solution of the BGK system. In particular, it is shown that the bimodal shape of the temperature profile, pointed out in the previous papers [e.g., M. Malek Mansour, F. Baras, and A. L. Garcia, Physica A, 240, 255 (1997); M. Tij, M. Sabbane, and A. Santos, Phys. Fluids 10, 1021 (1998); D. Risso and P. Cordero, Phys. Rev. E, 58, 546 (1998)], is attributed to the higher-order correction to the Navier--Stokes solution. It is also shown that an infinitesimally weak external force can cause a flow with a finite Mach number in the continuum limit. The numerical analysis of the BGK system is also carried out for a wide range of the Knudsen number.

Florence Baras

Alain Barrat

We introduce the model of inelastic hard spheres with random restitution
coefficient (alpha), in order to account for the fact that, in a vertically
shaken granular system interacting elastically with the vibrating boundary,
the energy injected vertically is transferred
to the horizontal degrees of freedom through collisions only,
which leads to heating through collisions,
i.e. to inelastic horizontal collisions with an
effective restitution coefficient
that can be larger than one. This allows the system to
reach a non-equilibrium steady state, where we
focus in particular on the single particle velocity distribution *f*(*v*)
in the horizontal plane, and
on its deviation from a Maxwellian. Molecular Dynamics simulations
and Direct Simulation Monte Carlo (DSMC) show that,
depending on the distribution of (alpha), different shapes of *f*(*v*) can be
obtained, with very different high energy tails.
Moreover, the fourth cumulant of the
velocity distribution quantifying the deviations from Gaussian
statistics is obtained analytically from the Boltzmann equation and
successfully tested against the simulations.

Alfred Beylich

For many modern problems, flow situations arise where in (parts of ) the flow field the local Knudsen number (Kn, Knc =l/L; l=mean free path, L= macroscopic length, cell length) is no more small and, as a consequence, the classical hydrodynamic equations (Navier-Stokes, NSt) fail to describe the flow properly. Since, for large variations in Kn, particle tracing methods (PTM), being restricted to Knc > 1, also cannot be used efficiently, and since PTM-NSt patchwork is not satisfactory, a *unified approach* based on the kinetic equation is desirable.
When solving numerically flow problems with (large) variations of the (cell) Knudsen number, a system is needed which interlaces the *path-integral form* of the kinetic equation (lower level) with the moment equations resulting from the *equation of transfer* (upper level). We have developed a system for discrete grids in physical (r-space) and in velocity space (c-space). Any physical model of particle interaction can be used; starting with simple spherical potentials, such as hard sphere or Lennard-Jones (6-12), more complex systems with molecules having internal degrees of freedom (internal energy levels; spin vectors) and the consequences of increasing dimensionality will be discussed. Some examples of typical flows will be presented.

Graeme Bird

The DSMC method was applied by a number of investigators in the early '90s to the Taylor-Couette flow. In 1995, the author found that, for a case with a moderate supersonic wall speed, there was a transition at a Reynolds number of approximately 4,000 from steady vortices at large times to a permanently chaotic state that involved the continuous formation of new vortices at random time intervals. The axially symmetric assumption becomes unrealistic beyond this transition and three-dimensional calculations are required if the results are to be physically meaningful. However, the axially symmetric calculations required runs lasting hundreds of hours on the then fastest PC's and the three-dimensional calculations would have required equally long runs on the largest parallel machines that were then available.

Computer speeds have increased by an order of magnitude since that time and, while parallel computation is still required, three-dimensional calculations now appear feasible and are in the planning stage. These calculations could again employ the Taylor-Couette geometry, but it has recently been found that similar instabilities can be produced in plane parallel flows that can be more easily related to boundary layers.

A flat surface is instantaneously inserted into a uniform flow at zero time and the flow is subjected to a vertical acceleration that is proportional to a power of the velocity deficit that develops over the plate. The flow boundaries normal to the stream direction are periodic boundaries and the side boundaries are either periodic or planes of symmetry. With an open outer boundary, the geometry is equivalent to Kramers problem and, with the product of the uniform flow velocity and time equated to distance from the leading edge, the flow may be directly related to the boundary layer over a flat plate. The imposed acceleration is analogous to the centrifugal acceleration produced by surface curvature and the instability mechanism in both this and Taylor-Couette flow is similar to the mechanism that produces Goertler vortices in flows over surfaces that are concave in the stream direction. Alternatively, if the outer boundary is a solid surface moving with the stream velocity, the flow is very similar to Taylor-Couette flow. However, there is freedom to vary many aspects of the flow that are inherently fixed in the axially symmetric flow. For example the acceleration may have an arbitrary dependence on the flow velocity and height above the surface. It may be varied with time or removed completely to determine whether a chaotic state is self-sustaining. Also, the flow in a thin slice normal to the undisturbed flow direction may be two-dimensional or, as in the axially symmetric case, have a thickness that varies with height in order to determine whether vortex stretching plays an important role.

A dedicated three-dimensional DSMC program has been written for this geometry and a version will be produced for parallel computations. However, there are many interesting flows that have no gradients in the stream direction. These are readily accessible to the existing program and a progress report will be presented.

Iain D. Boyd

A variety of plasma propulsion systems are used on spacecraft for attitude control and orbit maintenance. The most important devices presently under development are ion and Hall thrusters that use xenon as propellant. Numerical models of these systems are being developed to participate in the design of new thrusters and to help assess interactions between the plumes of the thrusters and the host spacecraft. The flow conditions of these thrusters indicate that a kinetic modelling approach is required that simulates ionization phenomena and charged particle motion in electromagnetic fields. In our present work we are using a number of particle techniques (Particle In Cell, Monte Carlo Collision, Direct Simulation Monte Carlo) to simulate the device and plume flows of these plasma propulsion systems. The present status will be described and discussion will be provided on remaining issues and future directions.

Javier Brey

The stationary state of a vibrated granular gas inside two connected compartments is studied both by using Molecular Dynamics simulations and a hydrodynamic description of the system. The existence of a phase transition for given values of the parameters is shown. The role played by an external field as well as the size of the hole connecting the compartments is discussed.

Patricio Cordero

Granular gases of inelastic hard spheres can be described from
the inelastic Boltzmann's equation by means of moment expansion
methods. Deriving hydrodynamics equations in dimension *d*
for the first *(1/2)(4+5d+d ^{2})* moments of the
distribution (14 moments in 3D and 9 moments in 2D) some known
results are reobtained and new ones emerge. The last moment
considered is the fourth cumulant

In our talk we will show the solutions that stem from such dynamics for a few illustrative simple cases: homogeneous cooling, steady states obtained heating the system from two parallel walls: (a) when there is no gravity, (b) when there is gravity, and finally we describe a quasi homogeneous stationary state. The latter has uniform density and temperature.

When *K>0* there is an overpopulation of slower particles, namely
the distribution is more peaked than the associated Maxwellian, while if
*K<0* there are relatively less slower particles and the distribution
is flatter. We reobtain that in the homogeneous cooling case *K* is
negative, while for steadily heated systems *K* is positive.

Andres Santos

Matthieu Ernst

We focus on the question to what extent it is possible to cast the dynamic equations for the unstable granular fluid in the form of the generic time-dependent Landau-Ginzburg theory, and to classify this fluid in one of the generic universality classes of critical and unstable systems, as given by Halpern and Hohenberg. It is shown how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau--Ginzburg (LG) models for critical and unstable fluids (e.g., spinodal decomposition).

The granular fluid, which is usually modeled as a fluid of inelastic hard
spheres (IHS), exhibits two instabilities: the spontaneous formation
of vortices and of high density clusters. As a first simplification the
clustering instability is suppressed by imposing constraints on the system
sizes. This is done to illustrate how LG-equations can be derived for the
order parameter, being the rate of deformation or shear rate tensor. This
tensor controls the formation of vortex patterns. From the shape of the energy
functional the stationary patterns in the flow field are obtained.
Quantitative predictions of this theory for the stationary states agree well
with molecular dynamics simulations on small systems of inelastic hard disks,
say *N* < 10000 particles.

Aldo Frezzotti

As is well known, the kinetic theory treatment of evaporation or condensation processes is limited to the description of the Knudsen layer formed at the boundary between the vapor and liquid phase. Since in most applications the vapor phase is dilute, the Boltzmann equation is used to describe gas motion above the liquid. Phenomenological models are used to deal with the molecular exchange processes at the gas-liquid interface whose structure is not resolved. In principle a unified study of both phases would be possible through MD simulations. However, this direct approach to the problem has been followed only recently by a very limited number of investigators and the interpretation of the simulation results is still not clear.

In this work an approximate kinetic equation is used
to provide a unified treatment of both phases during
evaporation or condensation. The advantage of using
a kinetic equation is twofold:

(a) Analytic or semi-analytic methods can be used
to find approximate solutions in some cases.

(b) Particle schemes can be used to find numerical
solutions with a numerical effort considerably
smaller than required by MD.

The approach is based on an idea by Karkhek and Stell who proposed a kinetic equation for a dense fluid whose molecules interact via a potential with a hard sphere part and a soft attractive tail. The hard sphere contribution gives a collision integral of Enskog type, whereas the tail is dealt with as a mean field term.

The equation can be solved by a DSMC particle scheme. The first results, aimed at understanding the equilibrium properties of the system, are encouraging.

Nicolas Hadjiconstantinou

We discuss recent progress towards understanding the extension of classical fluid mechanical phenomena in two-dimensional channels to the transition regime. The direct simulation Monte Carlo has been a key ingredient of our investigation. We have studied transition-regime wave propagation in narrow channels, and derived expressions for the wave speed and attenuation coefficient. We have also investigated the convective heat-transfer characteristics of two-dimensional channels in the slip-flow and transition regimes, under constant-wall-temperature and constant-wall-heat-flux conditions. Finally, we present an expression for the skin-friction coefficient in two-dimensional channels for arbitrary Knudsen numbers.

Siegfried Hess

Temperature profiles computed in simulations of a Poiseuille flow show significant devations from the thermo-hydrodynamical expression based on the standard Fourier law for heat conduction. This observation has been made both for rarefied gases [1] and dense fluids [2,3]. Appropriate generalizations of the Fourier law are discussed. For gases, the point of departure of a kinetic theory is the Boltzmann equation. The desired relations are derived by the moment method as used in Ref. [4]. For dense fluids, some heuristic arguments are given to explain the observed phenomena. Similarities and differences in the behavior of gases and dense fluids are pointed out. The generalized Fourier law is also of importance for the temperature profile in shock waves.

[1] M. Malek Mansour, F. Baras, A.L. Garcia, Physica A[2] A. Baranyai, D.J. Evans, and P.J. Daivis, Phys.Rev. E

[3] B.D. Todd and D.J. Evans, Phys.Rev. E

[4] S. Hess and M. Malek Mansour, Physica A

William G. Hoover

Strong shock waves with weak repulsive forces provide highly nonequilibrium profiles (longitudinal and transverse temperatures different by a factor of two with negative Fourier heat conductivity on the hot side of the shock). Stopping power simulations provide some interpretation for these results while Smooth Particle Applied Mechanics provides a formal underlying structure for analysis.

Song Hi Lee and Raymond Kapral

The structural properties and dynamics of molecular clusters embedded in a mesoscopic solvent will be described. The solvent interactions are accounted for through a multi-particle collision operator that conserves mass, momentum and energy and the solvent dynamics is updated at discrete time intervals.[1] The cluster particles interact among themselves and with the solvent molecules through continuous intermolecular forces.[2] The general discussion of the mesoscopic dynamics will be given as well as a description of how such dynamics can be coupled to particle motion. As an example, the properties of large and small Lennard-Jones clusters interacting with the mesoscopic solvent through repulsive Lennard-Jones interactions will be investigated as a function of the potential parameters. Modifications of both the cluster structure and solvent structure as a result of solute-solvent interactions will be discussed. Since the solvent dynamics correctly reduces to that given by the hydrodynamic equations on long distance and time scales, the effects of hydrodynamic interactions on single-particle and multi-particle diffusion coefficients can be considered. Other generalizations of the mesoscopic solvent model will be described.

[1] A. Malevanets and R. Kapral, ``Mesoscopic Model for Solvent Dynamics", J. Chem. Phys., 110, 8605 (1999).

[2] A. Malevanets and R. Kapral, ``Solute Molecular Dynamics in a Mesoscale Solvent", J. Chem. Phys., 112, 7260 (2000).

Markus Kraft

We propose an improved stochastic algorithm for temperature-dependent homogeneous gas phase reactions. By combining forward and reverse reaction rates, a significant gain in computational efficiency is achieved. Two modifications of modelling the temperature dependence (with and without conservation of enthalpy) are introduced and studied quantitatively. The algorithm is tested for the combustion of n-heptane, which is a reference fuel component for internal combustion engines. The convergence of the algorithm is studied by a series of numerical experiments and the computational cost of the stochastic algorithm is compared with the DAE code DASSL. If less accuracy is needed the stochastic algorithm is faster on short simulation time intervals. The new stochastic algorithm is significantly faster than the original direct simulation algorithm in all cases considered.

Anthony JC Ladd

I will present an overview of the application of lattice-Boltzmann methods to particle-fluid systems. I will briefly summarize the key ideas and illustrate the range of possible applications. I will describe some recent innovations that improve the accuracy and computational efficiency of the method, and indicate some areas where technical challenges still exist.

Annie Lemarchand

In order to improve the mesoscopic description of thermochemical systems, we build a new master equation including the stochastic modelling of energy transfer. We give the explicit expression of the master equation in the case of Semenov model for explosive reaction and Newtonian heat exchange with the walls of the reactor. It has a complicated integro-differential form taking into account the continuous spectrum of possible temperature transitions due to Newtonian cooling. Contrary to the previous derivations of master equations for exothermal reactions [G. Nicolis and M. Malek Mansour, Phys. Rev. A 29, 2845 (1984)], our approach does not require any matching to the deterministic description in the macroscopic limit.

The main hypothesis on which our mesoscopic approach relies is the assumed Maxellian form of the particle velocity distribution. The good agreement between the results deduced from Monte Carlo simulations of the master equation and from simulations (DSMC) of the microscopic particle dynamics allows us to validate the mesoscopic description we propose for conditions preserving the Maxwellian character of velocity distribution.

In the parameter domain where Semenov model admits a single stationary state and in the vicinity of the bifurcation associated with the emergence of bistability, the master equation allows us to quantify the dispersion of induction times as a function of system size. Marked stochastic effects are observed in connection with the existence of transient bimodality [G. Nicolis and F. Baras, J. Stat. phys. 48, 1071 (1987)] for the probability distribution of temperature.

James Lutsko

The Enskog approximation of hard-sphere kinetic theory may be heuristically derived by introducing a factorization approximation to the two-body distribution function appearing in the first BBGKY equation. However , this does not require the factorization of the full two-body distribution but only of the pre-colliional part of the distribution. In fact, the post-collisional part of the distribution can be calculated within the Enskog approximation and used to estimate the nonequilibrium contributions to the pair distribution function at contact. This can then be used in combination with standard ideas from equilibrium liquid-state theory to define models for the full pair distribution function of nonequilibrium hard core fluids. In this talk, details of this model will be given and the results compared to molecular dynamics simulations of two strongly nonequilibrium systems: a granular fluid and a sheared fluid of elastic hard spheres.

Jaroslaw Piasecki

We consider a fluid composed of inelastic hard spheres moving in a thermostat modelled by a hard sphere gas. The losses of energy due to inelastic collisions are balanced by the energy transfer via elastic collisions from the thermostat particles.

The resulting stationary state is analyzed within the Boltzmann kinetic theory. A numerical iterative method permits to study the nature of deviations from the gaussian state. Some analytic results are obtained for a one-dimensional system.

Maria J. Ruiz-Montero

The steady state of a fluidized granular medium in the presence of gravity is studied by using both Molecular Dynamics and an hydrodynamic description. In the case of an open system, the density profile exhibits a maximum, while the temperature profile presents a minimum beyond which it increases with height. This behavior is predicted by the hydrodynamic equations, and the information about the value and the position of the temperature minimum is enough to build up the hydrodynamic profiles in the bulk of the system.

J.M. Salazar

A system of *N* spheres with an elastic hard core and surrounded by
a dissipative layer is studied. Our results show that if the dissipative
part of the interaction
is proportional to the relative velocity between particles then the
Boltzmann equation for
this system admits an exact gaussian velocity distribution in the
homogeneous cooling state.
Moreover, the transport equations were obtained by using a Chapman-Enskog
expansion around this state and the stability of the gaussian solution is
analyzed and a
generalization of these results to a more general inelastic interactions is
discussed.

J. M. Montanero, V. Garzo, A. Santos

A.V. Vasenkov and B. D. Shizgal

The non-hydrodynamic behavior of electrons and/or ions near an electrode is studied. A system of electrons or ions in argon and helium is considered in the positive one-dimensional spatial half-space with an absorbing boundary at the origin which represents an electrode. A flux of electrons or ions is assumed to originate at infinite distance from the boundary. Elastic collisions for electrons and charge-exchange collisions for ions with the background moderator are taken into account. We solve the kinetic Boltzmann equation for electron and ion distribution functions in space and velocity with two direct numerical methods and a Monte Carlo simulation. The density and temperature profiles are determined and the departure from hydrodynamic behavior near the boundary is studied. The objective of the present work is to construct a self-consistent model of a discharge which couples the Poisson equation for the electric field with the electron/ion Boltzmann equation.

Acknowledgment is made to the Donors of the Petroleum Research Fund, administrated by the American Chemical Society for support of this research (PRF 34689-AC6)

Rodrigo Soto

Short range velocity correlations in granular fluids, modeled by the inelastic hard sphere model, are studied both by computer simulations and by kinetic theory. Starting from states with uncorrelated velocities, it is shown that the collision law creates post-collisional velocity correlations, modifying the structure of the pair correlation function at contact. The post-collisional correlations are propagated dynamically, giving rise to velocity correlations of colliding pairs, that are detected in molecular dynamics simulations. Studying correlated sequences of collisions (ring events) it is possible to compute the pre-collisional velocity correlations, obtaining a satisfactory accord with simulations. The presence of pre-collisional velocity correlations produces a net decrease of the collision frequency and of the virial pressure of the fluid. Also, as these correlations are created by the dynamics, the validity of the Boltzmann and Enskog kinetic must be reconsidered.

Emmanuel Trizac

We study the dynamic scaling behavior of ballistic annihilation with continuous initial velocities, whereby colliding particles disappear at contact. In the framework of the molecular chaos ansatz, the leading correction due to the non Gaussian behavior of the velocity distribution is computed, and the resulting decay exponents for the density and kinetic energy are obtained analytically. For arbitrary dimensionality, these exponents are in excellent agreement with numerical solutions of the non-linear Boltzmann equation given by Monte Carlo Direct Simulations (DSMC), but more surprisingly with Molecular Dynamics simulations of the exact dynamics. We finally discuss the notion of "universality classes" in this process.

Christian Van den Broeck

We review the problem of the so-called adiabatic piston and present theoretical and numerical results that clarify its operation as a Brownian motor.

Wolfgang Wagner

A theoretical foundation for the Consistent Boltzmann Algorithm is established by deriving the limiting kinetic equation (J.Statist.Phys. 101:1065 (2000)). Besides its relation to the algorithm, this new equation serves as an alternative to the Enskog equation in the kinetic theory of dense gases. For a simplified model, the limiting equation is solved numerically, and very good agreement with the predictions of the theory is found.

* Speaker; + Workshop participant; $ CECAM Director

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