In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Kinetic Theory and Fluid Dynamics

XIV.OnProfessor Osborne Reynolds'spaper “on certain dimensional properties of matter in the gaseous state

The kinetic theory of gases and the gas laws

By just applying a temperature difference to a micro-system filled with rarefied gas, it is possible to engender a displacement or a compression of the gas in the temperature gradient direction. This is the Thermal Transpiration phenomenon. In the present work, thermal transpiration has been studied both through an experimental approach, which exploits an original measuring system, and through a numerical approach, which is modeled on the basis of the Shakhov model kinetic equation. In both studies, a circular cross section glass micro-tube is submitted to a temperature gradient. The obtained results for Helium, such as the thermal molecular pressure difference, the thermal molecular pressure ratio and the thermal molecular pressure exponent at the final zero-flow stage, are analyzed in the case of a tube submitted to a temperature difference of 51 K. Finally, the obtained thermal molecular pressure ratio results are also compared to the semi-empirical formulas of Liang (1951) and Takaishi and Sensui (1963). These semi-empirical formulas are still in use nowadays to introduce correction factors for pressure measurements done when the pressure gauge functions at different temperatures in respect to the temperature of the operating gas. For the here working pressure conditions and the used tube dimensions the gas rarefaction conditions go from near free molecular to slip regime.

An Experimental and Numerical Study of the Final Zero-Flow Thermal Transpiration Stage

Molecular tagging velocimetry (MTV) is an optic experimental technique widely employed for measuring the velocity field in fluid flows. The measuring principle is based on the tracking of molecules able to emit light in response to a laser excitation. By seeding the flow with this tracer, local velocity measurements can be carried out by following the displacement of the emitting molecules. While this technique has already been successfully applied in liquid and gas flows, the application to rarefied and confined gas flows is still a challenge due to the high molecular diffusion and the low emitted light from the tracer at low pressures. The interest in applying MTV in rarefied conditions derives from the absence of local experimental data that can allow a better understanding on the mechanisms of interaction between the gas molecules and the wall surface. In this work, an experimental analysis of the intensity and lifetime of the photoluminescence of the molecular tracers employed, i.e., acetone and diacetyl, is presented. This analysis allowed to estimate the best working conditions in order to be able to apply MTV to rarefied gas flows. Thus, MTV has been applied to gas-tracer mixtures at low pressures in a millimetric rectangular channel producing the first preliminary results in the slip flow regime.

Molecular tagging velocimetry in rarefied and confined gas flows

Ce manuscrit presente l'etude et l'analyse d'ecoulements de gaz rarefies, induits par la transpiration thermique. Le terme de transpiration thermique designe le mouvement macroscopique d'un gaz rarefie engendre par l'effet du seul gradient de temperature. L'aspect principal de ce travail est centre autour de la mesure du debit stationnaire declenche en soumettant un micro tube a un gradient de temperature applique le long de son axe. On a developpe a cet effet un appareillage experimental original ainsi qu'une methodologie experimentale innovatrice basee sur la dependance du phenomene, analyse dans son ensemble, a l'egard du temps. Les resultats obtenus pour le debit stationnaire initial de transpiration thermique et pour les parametres thermo-moleculaires caracterisant l'equilibre final de debit nul, ont ete compares aux resultats obtenus numeriquement par la resolution de l'equation cinetique modele de Shakhov et par la methode de simulation directe de Monte-Carlo.

Temperature gradient induced rarefied gas flow

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Temperature gradient induced rarefied gas flow