scholarly journals Variational derivation of thermal slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules and Cercignani–Lampis boundary conditions: Comparison with experimental results

2020 ◽  
Vol 32 (10) ◽  
pp. 102011 ◽  
Author(s):  
Nhu Ngoc Nguyen ◽  
Irina Graur ◽  
Pierre Perrier ◽  
Silvia Lorenzani
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Hard Sphere ◽  
Experimental Results ◽  
Thermal Slip ◽  
The Boltzmann Equation ◽  
Slip Coefficients

scholarly journals Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Corentin Le Bihan
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Hard Sphere ◽  
Hard Spheres ◽  
Compact Domain ◽  
The Boltzmann Equation ◽  
Grad Limit ◽  
Random Direction ◽  
Time Restriction ◽  
Short Time

<p style='text-indent:20px;'>In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with {isotropic} boundary conditions. We consider a system of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> hard spheres of diameter <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> in a box <inline-formula><tex-math id="M3">\begin{document}$ \Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2 $\end{document}</tex-math></inline-formula>. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling <inline-formula><tex-math id="M4">\begin{document}$ N\epsilon^2 = 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \epsilon\rightarrow 0 $\end{document}</tex-math></inline-formula> to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.</p>

scholarly journals Assessment and development of the gas kinetic boundary condition for the Boltzmann equation

2017 ◽  
Vol 823 ◽  
pp. 511-537 ◽  
Author(s):  
Lei Wu ◽  
Henning Struchtrup
Keyword(s):  
Experimental Data ◽  
Boundary Condition ◽  
Boltzmann Equation ◽  
Rarefied Gas ◽  
Thermal Slip ◽  
Thermal Transpiration ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation ◽  
Kinetic Boundary ◽  
Slip Coefficients

Gas–surface interactions play important roles in internal rarefied gas flows, especially in micro-electro-mechanical systems with large surface area to volume ratios. Although great progress has been made to solve the Boltzmann equation, the gas kinetic boundary condition (BC) has not been well studied. Here we assess the accuracy of the Maxwell, Epstein and Cercignani–Lampis BCs, by comparing numerical results of the Boltzmann equation for the Lennard–Jones potential to experimental data on Poiseuille and thermal transpiration flows. The four experiments considered are: Ewart et al. (J. Fluid Mech., vol. 584, 2007, pp. 337–356), Rojas-Cárdenas et al. (Phys. Fluids, vol. 25, 2013, 072002) and Yamaguchi et al. (J. Fluid Mech., vol. 744, 2014, pp. 169–182; vol. 795, 2016, pp. 690–707), where the mass flow rates in Poiseuille and thermal transpiration flows are measured. This requires that the BC has the ability to tune the effective viscous and thermal slip coefficients to match the experimental data. Among the three BCs, the Epstein BC has more flexibility to adjust the two slip coefficients, and hence for most of the time it gives good agreement with the experimental measurements. However, like the Maxwell BC, the viscous slip coefficient in the Epstein BC cannot be smaller than unity but the Cercignani–Lampis BC can. Therefore, we propose to combine the Epstein and Cercignani–Lampis BCs to describe gas–surface interaction. Although the new BC contains six free parameters, our approximate analytical expressions for the viscous and thermal slip coefficients provide useful guidance to choose these parameters.

scholarly journals Extraction of Tangential Momentum and Normal Energy Accommodation Coefficients by Comparing Variational Solutions of the Boltzmann Equation with Experiments on Thermal Creep Gas Flow in Microchannels

Fluids ◽  
2021 ◽  
Vol 6 (12) ◽  
pp. 445
Author(s):  
Tommaso Missoni ◽  
Hiroki Yamaguchi ◽  
Irina Graur ◽  
Silvia Lorenzani
Keyword(s):  
Boltzmann Equation ◽  
Gas Flow ◽  
Recent Experimental Data ◽  
Thermal Slip ◽  
The Boltzmann Equation ◽  
Variational Solutions ◽  
Experimental Apparatus ◽  
Accommodation Coefficients ◽  
Tangential Momentum ◽  
Slip Coefficients

In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients, σ1,T and σ2,T, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules. The Cercignani-Lampis scattering kernel of the gas-surface interaction has been considered in order to take into account the influence of the accommodation coefficients (αt, αn) on the slip parameters. Comparing our theoretical results with recent experimental data on the mass flow rate and the slip coefficient for five noble gases (helium, neon, argon, krypton, and xenon), we found out that there is a continuous set of values for the pair (αt, αn) which leads to the same thermal slip parameters. To uniquely determine the accommodation coefficients, we took into account a further series of measurements carried out with the same experimental apparatus, where the thermal molecular pressure exponent γ has been also evaluated. Therefore, the new method proposed in the present work for extracting the accommodation coefficients relies on two steps. First of all, since γ mainly depends on αt, we fix the tangential momentum accommodation coefficient in such a way as to obtain a fair agreement between theoretical and experimental results. Then, among the multiple pairs of variational solutions for (αt, αn), giving the same thermal slip coefficients (chosen to closely approximate the measurements), we select the unique pair with the previously determined value of αt. The analysis carried out in the present work confirms that both accommodation coefficients increase by increasing the molecular weight of the considered gases, as already highlighted in the literature.

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