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

Techno-economic assessment of postcombustion carbon capture using high-performance nanoporous single-layer graphene membranes

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

A comprehensive review on micro- and nano-scale gas flow effects: Slip-jump phenomena, Knudsen paradox, thermally-driven flows, and Knudsen pumps

Ab initio simulation of rarefied flows of gaseous mixtures in the system of microbeams with different temperatures

In Knudsen pumps with geometric configuration of rectangle, gas flows are induced by temperature gradients along channel walls. In this paper, the direct simulation Monte Carlo (DSMC) method is used to investigate numerically the flow characteristics of H2–N2 mixtures in the Knudsen pump. The variable soft sphere (VSS) model is applied to depict molecular diffusion in the gas mixtures, and the results obtained are compared with those calculated from a variable hard sphere (VHS) model. It is demonstrated that pressure is crucial to affecting the variation of gas flow pattern, but the gas concentration in H2–N2 mixtures and the collision model do not change the flow pattern significantly. On the other hand, the velocity of H2 is larger than that of N2. The velocities of H2 and N2 increase if the concentration of H2 rises in the gas mixtures. The results of velocity and mass flow rate obtained from VSS and VHS models are different. Finally, a linear relation between the decrease of mass flow rate and the increase of H2 concentration is proposed to predict the mass flow rate in H2–N2 mixtures.

https://www.mdpi.com/2072-666X/11/9/784/pdf

Numerical Investigation into the Flow Characteristics of Gas Mixtures in Knudsen Pump with Variable Soft Sphere Model.

Numerical investigation of gas separation via thermally induced flows in ratchet-like patterned microchannels

Heat and drag reduction of single and combined opposing jets in hypersonic nonequilibrium flows

As a kind of fluid machinery, the roots vacuum pump is widely used in the metallurgical industry and semiconductor industry. The double-blade roots vacuum pump is widely used in the vacuum system due to its easy manufacture and low cost. The pumping process can be divided into four distinctive processes (i.e. sucking process, transferring process, backlashing process and exhausting process). The characteristics of each process are illustrated sufficiently in this paper. Based on the ideal gas laws, a thermodynamics model of pumping process is established and the influence of gas leakage on the pumping process is taken into consideration. Using this model, quantificational expressions of five gas thermodynamic parameters (volume, mass, pressure, temperature and internal energy) are given. The performance curves are plotted by MATLAB and the pumping characteristics of double-blade roots vacuum pump are discussed at last. This research can provide a theoretical reference on the design of double-blade roots vacuum pump.

https://iopscience.iop.org/article/10.1088/1742-6596/1519/1/012016/pdf

Feng Han

Papers

Numerical Investigation into the Flow Characteristics of Gas Mixtures in Knudsen Pump with Variable Soft Sphere Model.

Numerical investigation of gas separation via thermally induced flows in ratchet-like patterned microchannels

Heat and drag reduction of single and combined opposing jets in hypersonic nonequilibrium flows

Thermodynamics Analysis of Roots Vacuum Pump