Benjamin Graille

KINETIC THEORY OF PLASMAS: TRANSLATIONAL ENERGY

In the present contribution, we derive from kinetic theory a unified fluid model for multicomponent plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and the reactive collisions. Given the strong disparity of mass between the electrons and heavy particles, such as molecules, atoms, and ions, we conduct a dimensional analysis of the Boltzmann equation. We then generalize the Chapman-Enskog method, emphasizing the role of a multiscale perturbation parameter on the collisional operator, the streaming operator, and the collisional invariants of the Boltzmann equation. The system is examined at successive orders of approximation, each of which corresponding to a physical time scale. The multicomponent Navier-Stokes regime is reached for the heavy particles, which follow a hyperbolic scaling, and is coupled to first order drift-diffusion equations for the electrons, which follow a parabolic scaling. The transport coefficients exhibit an anisotropic behavior when the magnetic field is strong enough. We also give a complete description of the Kolesnikov effect, i.e., the crossed contributions to the mass and energy transport fluxes coupling the electrons and heavy particles. Finally, the first and second principles of thermodynamics are proved to be satisfied by deriving a total energy equation and an entropy equation. Moreover, the system of equations is shown to be conservative and the purely convective system hyperbolic, thus leading to a well-defined structure.

Kinetic theory of partially ionized reactive gas mixtures

We investigate partially ionized reactive gas mixtures in the presence of electric and magnetic fields. Our starting point is a generalized Boltzmann equation with a chemical source term valid for arbitrary reaction mechanism. We study the Enskog expansion and obtain macroscopic equations in the zeroth- and first-order regimes, together with transport fluxes and transport coefficients. New bracket expressions are obtained for perpendicular/transverse diffusion, thermal diffusion and thermal conductivity coefficients as well as shear viscosity coefficients. A new definition of thermal diffusion ratios—consistent with the zero magnetic fields limit—is introduced. Positivity properties of multicomponent diffusion matrices are investigated and macroscopic entropy production is shown to be positive. The mathematical structure of the transport linear systems that are to be solved in order to evaluate transport coefficients is discussed. In particular, all transport coefficients are expressed as convergent series. These series yield by truncation accurate approximated coefficients relevant to computational models.

ASYMPTOTIC STABILITY OF EQUILIBRIUM STATES FOR AMBIPOLAR PLASMAS

We investigate a system of partial differential equations modeling ambipolar plasmas. The ambipolar — or zero current — model is obtained from general plasmas equations in the limit of vanishing Debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients. We establish that the governing equations can be written as a symmetric form by using entropic variables. The corresponding dissipation matrices satisfy the null space invariant property and the system of partial differential equations can be written as a normal form, i.e. in the form of a symmetric hyperbolic–parabolic composite system. By properly modifying the chemistry source terms and/or the diffusion matrices, asymptotic stability of equilibrium states is established and decay estimates are obtained. We also establish the continuous dependence of global solutions with respect to vanishing electron mass.

The kinetic theory of partially ionized reactive gas mixtures II

We investigate the kinetic theory of partially ionized reactive gas mixtures in strong magnetic fields following Giovangigli et al (2003 Physica A 327 313–48). A new tensor basis is introduced for expanding the perturbed distribution functions associated with the viscous tensor. New symmetry properties of transport coefficients are established as well as simplified bracket expressions. A variational framework is introduced for a direct evaluation of the thermal conductivity and the thermal diffusion ratios. The transport linear systems corresponding to the usual Sonine/Wang–Chang Uhlenbeck polynomial expansions are evaluated. The behavior of transport coefficients and transport fluxes for vanishing magnetic fields is investigated using series expansions. Practical implementation of iterative algorithms for solving the resulting complex symmetric constrained singular linear systems is discussed as well as various approximations of the transport coefficients.

Multicomponent transport in weakly ionized mixtures

We discuss transport coefficients in weakly ionized mixtures. We investigate the situations of weak and strong magnetic fields as well as electron temperature nonequilibrium. We present in each regime the Boltzmann equations, examples of transport fluxes, the structure of transport linear systems and discuss their solution by efficient iterative techniques. Numerical simulations are presented for partially ionized high-temperature air.

Approximation of mono-dimensional hyperbolic systems: A lattice Boltzmann scheme as a relaxation method

We focus on mono-dimensional hyperbolic systems approximated by a particular lattice Boltzmann scheme. The scheme is described in the framework of the multiple relaxation times method and stability conditions are given. An analysis is done to link the scheme with an explicit finite differences approximation of the relaxation method proposed by Jin and Xin. Several numerical illustrations are given for the transport equation, Burgers equation, the p-system, and full compressible Eulers system.

Kinetic theory derivation of transport equations for gases with internal energy

ows. In this paper, we propose a general description of the internal energy excitation of a molecular gas in thermal nonequilibrium by distinguishing between slow and fast collisions. A multiscale Chapman-Enskog method is used to study thermalization and derive Euler equations of conservation of mass, momentum, translational energy and internal energy. As opposed to conventional perturbation methods, the fast collision operator is expanded in the small parameter used to dene the threshold for the net energy for fast collisions. We show that the role of the fast collisions is to thermalize the translational and internal energy modes, whereas the role of the slow collisions is to contribute to the thermal relaxation of the translational and internal energy modes.

Thermo‐chemical dynamics and chemical quasi‐equilibrium of plasmas in thermal non‐equilibrium

We examine both processes of ionization by electron and heavy‐particle impact in spatially uniform plasmas at rest in the absence of external forces. A singular perturbation analysis is used to study the following physical scenario, in which thermal relaxation becomes much slower than chemical reactions. First, electron‐impact ionization is investigated. The dynamics of the system rapidly becomes close to a slow dynamics manifold that allows for defining a unique chemical quasi‐equilibrium for two‐temperature plasmas and proving that the second law of thermodynamics is satisfied. Then, all ionization reactions are taken into account simultaneously, leading to a surprising conclusion: the inner layer for short time scale (or time boundary layer) directly leads to thermal equilibrium. Global thermo‐chemical equilibrium is reached within a short time scale, involving only chemical reactions, even if thermal relaxation through elastic collisions is assumed to be slow.

Recovering the full Navier Stokes equations with lattice Boltzmann schemes

We consider multi relaxation times lattice Boltzmann scheme with two particle distributions for the thermal Navier Stokes equations formulated with conservation of mass and momentum and dissipation of volumic entropy.Linear stability is taken into consideration to determine a coupling between two coefficients of dissipation.We present interesting numerical results for one-dimensional strong nonlinear acoustic waves with shocks.

Hydrodynamic model for molecular gases in thermal nonequilibrium

We derive a hydrodynamic model for the internal energy excitation of molecular gases in thermal nonequilibrium based on kinetic theory. The co-existence of fast and slow collisions in the system results in thermal nonequilibrium between the translational and internal energy modes. A proper scaling for the Boltzmann equation that accounts for the different relaxation times is obtained from a dimensional analysis. The collisions are divided into fast and slow processes based on the magnitude of the net internal energy. As opposed to conventional perturbations methods, the fast collision operator is expanded in a small parameter used to define the threshold for the net energy for fast collisions. A lemma allows us to split the internal energy of all the levels into perturbed elastic and inelastic contributions for the fast collisions. The introduction of perturbed energy levels is crucial to separate the energy collision invariant into two types of fast collisional invariants. The gas particle population is ...