RAREFIED GAS DYNAMICS: 24th International Symposium on Rarefied Gas Dynamics

From Nice 1958 to Bari 2004: 46 years of RGD Symposia
View Description Hide DescriptionThe history and the most important scientific results obtained during 46 years of RGD symposia are critically reported.

Convergence to Equilibrium: Entropy Production and Hypocoercivity
View Description Hide DescriptionThese is the text for my Harold Grad lecture, delivered in the 24th Rarefied Gas Dynamics conference, Bari (July 2004). They describe recent developments about convergence to equilibrium in kinetic theory, related to Boltzmann’s H Theorem and entropy production. The style is intentionally informal, at the price of rigor and precision; full details can be found in the research papers quoted within the text. The text is partially based on my earlier contribution to the proceedings of the 14th International Congress of Mathematical Physics, Lisbon (July 2003).

Molecular Alignment in Gaseous Expansions and Anisotropy of Intermolecular Forces
View Description Hide DescriptionRecently we experimentally demonstrated how a planar molecule tends to travel as a “frisbee” when a gaseous mixture with lighter carriers expands into a vacuum, the orientation being due to collisions. The molecule is benzene, the prototype of aromatic chemistry .The demonstration is via two complementary experiments: interrogating benzene by IR‐laser light and controlling its orientation by selective scattering on a rare gas target. The results cast new light on the microscopic mechanism of collisional alignment (our earlier work regarded diatomic molecules, such as O_{2} and N_{2}) and suggest a useful way to produce intense beams of aligned molecules, permitting studies of steric effects in gas‐phase processes. This study has been extended to other simple hydrocarbons such as ethylene, acetylene and ethane, indicating that collisional alignment is effective also in these cases. Applications are in progress for molecule — surface scattering, opening expectations for nanocatalysis. Combining collisional studies of supersonic seeded beams of aligned molecules with results from scattering of rotationally hot molecular beams allows development of systematic characterization of intermolecular forces, particularly regarding molecular anisotropies and their roles is cluster formation: investigated systems include atmospheric gases, hydrocarbons, water. This extends our previous collisional studies on interactions of open‐shell atoms, aligned by a magnetic field. The review presented here concludes with a sketch of accompanying theoretical developments, and by perspectives for collisional orientations and even chiral discrimination in linear flows or vortices.

Historical Account And Branching To Rarefied Gas Dynamics Of Atomic and Molecular Beams : A Continuing And Fascinating Odyssey Commemorated By Nobel Prizes Awarded To 23 Laureates In Physics And Chemistry
View Description Hide DescriptionThis Historical Account derived in part from D. R. Herschbach was presented as an opening lecture of the Molecular Beam Session organized at the 24^{th} International Symposium on Rarefied Gas Dynamics held in Bari, Italy, in July 2004. The emphasis is on the impressive results due to the molecular beam techniques in the last century. The first section summarizes the historical beam experiments performed by 14 Nobel Prize laureates having used the thermally effusive sources to establish the basic principles of Modern Physics. The second section is on the branching of Molecular Beams to Rarefied Gas Dynamics having permitted to investigate the physics of supersonic free jets and transform the molecular beam techniques. Finally, the last section relates the spectacular molecular beam experiments in helium free jet ultracooling, molecular spectroscopy, chemical reaction dynamics, clustering and modification of low density matter, and biomolecule mass spectrometry, rewarded by nine Nobel Prizes in Chemistry from 1986 to 2002.

On the Quantum Boltzmann Equation
View Description Hide DescriptionIn this contribution I describe the problem of deriving a Boltzmann equation for a system of N interacting quantum particles under the weak‐coupling and the low density limits, showing that the limiting behavior of the one‐particle Wigner function is expected to satisfy a Boltzmann equation with a suitable cross‐section.
The problem is still open from a rigorous view point and only partial results are available till now. The talk is based on a sistematic collaboration with D. Benedetto, F. Castella and R. Esposito.

Kinetic Modelling of a Heterogeneous Dispersed Medium
View Description Hide DescriptionTwo‐phase dispersed media appear in a wide range of industrial processes and in many problems of pollution. Here, we are interested in the kinetic description for a suspension with two (or more) species of solid particles, the carrier fluid being a viscous gas. Attention is focused on the suspensions of solid spheres of different species. Boltzmann equations for species of hard spheres having binary, inelastic and non‐punctual collisions are written. For this purpose, the inelastic collision of two particles is described and the “pseudo‐inverse collision” is clearly introduced.
By considering the asymptotic case where the diameters of the particles are very small in front of the macroscopic length, simple expressions for the collision operators are given. Taylor expansions allow one to separate the collision term contributions in the global balance equations in a flux term and an in source term. In the momentum equation, the source term is zero and the contribution of the collisions may be written as a divergence term like in continuum mechanics.

Nonlinear Acoustics to Second Order in Knudsen Number Without Unphysical Instabilities
View Description Hide DescriptionThe Burnett equations are consistently reformulated as a linearly stable first order system. The equations are then applied to study the nonlinear evolution of a sound wave. The initially sinusoidal wave is nonlinearly distorted and a shock wave develops. The shock is gradually dissolved by dissipation and a sinusoidal wave of smaller and decaying amplitude emerges. The amplitude of this old age solution is compared with the classical results from the Burgers equation of nonlinear acoustics and systematic deviations are found.

The Discrete Boltzmann Equation : The Regular Plane Model with Four Velocities
View Description Hide DescriptionFor a simple discrete model of Boltzmann equation, we study the derivatives of H‐Boltzmann function, and prove that all derivatives of odd order are negative, instead all derivatives of even order are postive. These result is a first and small generalisation of the classical H‐Boltzmann theorem.

Stability Analysis of a Multigroup Model for the Boltzmann Transport Equations of Carriers and Phonons
View Description Hide DescriptionWe present a direct solution method to the Bloch‐Boltzmann‐Peierls equations governing the transport of carriers and optical phonons in semiconductors. This approach is based on a multigroup formulation of the original equations, which still contains both the full quantum statistics of carriers and phonons and a very general description of the carrier band structure. It allows the investigation of the particle distributions of arbitrary anisotropies with respect to a main direction. Concerning the mathematical properties of the deduced transport model, we prove a Boltzmann H‐theorem for the obtained evolution equations. The equilibrium solution of the multigroup model is compared with that of the original Bloch‐Boltzmann‐Peierls equations. Numerical results are given for relaxation processes of hot electrons and hot phonons.

A reactive BGK‐type model: influence of elastic collisions and chemical interactions
View Description Hide DescriptionA BGK‐type model for a reactive multicomponent gas undergoing chemical bimolecular reactions is here presented. The mathematical and physical consistency of the model is stated in detail. The relaxation process towards local Maxwellians, depending on mass and numerical densities of each species, as well as on common mean velocity and temperature, is investigated with respect to chemical equilibrium. Such a trend is numerically tested within the hydrogen‐air reaction mechanism.

A semi‐continuous Boltzmann equation for particles with general dispersion relations
View Description Hide DescriptionThis paper presents a generalized semi‐continuous Boltzmann equation to describe the kinetics of particles with non‐classical dispersion relations. The kinetic equations are derived, and the conservation properties of the model are investigated. Numerical results for the relaxation of a hot electron gas in semiconductors are discussed.

A Method Of Joint Solution Of The Boltzmann And Navier‐Stokes Equations
View Description Hide DescriptionThe method is designed for gas flows in which the flow field can be divided on some relatively small and highly non equilibrium (kinetic) zones, and the main area where hydrodynamic approximation is valid. The above division is made from a priori analysis of a problem and then could be corrected after some trial computations. Inside the kinetic zones one solve the Boltzmann equation by regular finite‐difference method, and in the main area one apply the Navier‐Stokes equations. For matching the solutions at the interfaces of the regions the Chapman‐Enskog distribution function is used. As examples of application of the method two cases of interaction of rarefied gas flow with a periodic grid are computed.

Generalized Hydrodynamic Equations and the Problem of Boundary Conditions
View Description Hide DescriptionGeneralized Boltzmann equation (GBE) is applied for derivation of generalized hydrodynamic equations with taking into account the alternating gravitational field. The analog of Landau damping in gravitational field is considered on the basement of the generalized Boltzmann physical kinetics. The corresponding exact solution of dispersion equation is found. The problem of boundary conditions for generalized hydrodynamic equations is investigated.

Coupling Direct Boltzmann and Continuum Flow Solvers
View Description Hide DescriptionThis paper describes the development of a Unified Flow Solver (UFS) that automatically switches between and couples the continuum‐fluid‐dynamic and kinetic solvers needed for the continuum and kinetic regimes. A Direct Numerical Solver (DNS) of the Boltzmann Equation (BE) is used in regions of moderate and high local Knudsen number, while Kinetic schemes of gas dynamics are used elsewhere. The use of kinetic schemes instead of traditional continuum solvers facilitates automatic domain decomposition for coupling the BE and the continuum solvers. Solutions of several problems with small Knudsen number and in the continuum regime are presented. Criteria for switching between the kinetic and continuum solvers are formulated and tested. The efficiency and numerical stability of the UFS is attained by using similar computational techniques for the kinetic and continuum solvers, and by employing intelligent domain decomposition algorithms. Results obtained with the UFS for different problems are presented.

Macroscopic Equations for High‐Speed Rarefied Monatomic Gas Flows past Cold Bodies
View Description Hide DescriptionThe new form of breakdown parameter is proposed and the applicability of this parameter to shock wave flow and to shear flow has been established. This form of parameter may be applied to solutions of Boltzmann equation or to Navie‐Stokes equations. Burnett constitutive relations for stresses and heat flux were analyzed in shear flow, shock wave flow, flows past circular cylinder and past plane plate at zero angle of attack. The generalization of Newton‐Fourier (Navier‐Stokes) relations for plane nonequilibrium flow (based on macroscopic equations established for Couette flow and for cylindrical expansion into vacuum) is proposed.

Continuous inverse kinetic theory for incompressible fluids
View Description Hide DescriptionA fundamental aspect of CFD for incompressible fluids and magnetofluids is the algorithmic complexity which characterizes both conventional direct simulation methods and kinetic approaches based on microscopic or phenomenological models based on asymptotic kinetic theories. A possible solution to this problem, can be provided by so‐called inverse kinetic theories suitably constructed in such a way to avoid the computational complexity of previous numerical approaches, in particular due to the Poisson equation for the fluid pressure, which typically (for example, for so‐called fractional step due to Kim and Moin), involves an algorithmic complexity of order N ^{2}, being N the grid number characterizing the numerical solution. This may explain why, in massive parallel DNS simulations of turbulent flows, the maximum grid number remains substantially limited. Goal of the present Note is propose a solution to this problem, pointing out a new inverse kinetic theory which does not require the solution of Poisson equation and avoids the N ^{2}‐complexity.

Exact Forms of Representation of Boltzmann Collision Integral as a Divergence of the Flow in Velocity Space
View Description Hide DescriptionOn the basis of a recently discovered collision group, new exact renormalized forms of the Boltzmann equation are obtained. Boltzmann collision integral is rewritten exactly as a divergence of the flow in velocity space. This allows to consider the distribution function as a density of the points in the phase space moving along smooth trajectories under the influence of a nonlocal force. The points do not jump any more as it was in the case of the classical Boltzmann equation. It is shown that near the equilibrium the Boltzmann collision integral universally tends to the Landau‐Fokker‐Plank collision integral.

On One‐dimensional Discrete Velocity Models of The Boltzmann Equation For Mixtures
View Description Hide DescriptionIn this work the theorem that exists only two symmetric one‐dimensional discrete velocity models for two‐species mixture without spurious invariants is proved only for certain class of models. Additive invariants are classified for the Boltzmann equation and its discrete models. Approximation of the Boltzmann equation by discrete model in one‐dimensional case is considered and quadrature formula for the collision integral is proposed. Also we consider the size of frame, used for arrangement of values of particles’ momentum belonging to symmetric discrete velocity model of the Boltzmann equation, which has an exchange of energy between the components of mixture.

On the Application of the Moment equations to the Thermally Induced Flows
View Description Hide DescriptionMoment equations derived from the Boltzmann equation for the two‐dimensional flows were formulated and generalized slip boundary conditions for the moments were derived. Fifty one moments relevant to the eigenfunctions included in the Chapman‐Enskog solutions of the second order approximation were taking into account. These moment equations were applied to thermally induced flows in a two‐dimensional vessel. The moment equations were solved using the MacCormack’s difference scheme. Present results showed that the moment equations and the slip boundary conditions were applicable for the two‐dimensional flows in the transition regime. Thermally induced slip flow adjacent to the solid wall decreased linearly in accordance with the decrease of the Knudsen number relevant to the size of the vessel. The values of obtained slip coefficient, velocity/(gradient of temperature along the solid wall), increased as the number of moment equations increased. Present results suggested that moments should be selected so as to decrease the number of equivalent equations in the moment equation system.