BOLTZMANN-LIKE MODELLING OF A SUSPENSION

This paper deals with the presentation of a kinetic model for a suspension of identical hard spheres. Considering that the collisions between particles are instantaneous, binary, inelastic and taking the diameter of the spheres into account, a Boltzmann equation for the dispersed phase is proposed. It allows one to obtain the conservation of mass and momentum as well as, for slightly inelastic collisions, an H-theorem which conveys the irreversibility of the evolution. The problem of the boundary conditions for the Boltzmann equation is then introduced. From an anisotropic law of rebound characterizing the inelastic and non-punctual impact of a particle to the wall, a parietal behavior for the first moments of the kinetic equation is deduced.

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

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021177 ◽

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>

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Evaluation of Various Types of Wall Boundary Conditions for the Boltzmann Equation

42nd AIAA Thermophysics Conference ◽

10.2514/6.2011-3620 ◽

2011 ◽

Author(s):

Christopher Wilson ◽

Ramesh Agarwal ◽

Felix Tcheremissine

Keyword(s):

Boltzmann Equation ◽

Boundary Conditions ◽

Wall Boundary ◽

The Boltzmann Equation ◽

Wall Boundary Conditions

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The Boltzmann Equation and the H Theorem (1872–1875)

10.1093/oso/9780198816171.003.0004 ◽

2018 ◽

Author(s):

Olivier Darrigol

Keyword(s):

Heat Conduction ◽

Boltzmann Equation ◽

Transport Phenomena ◽

Dilute Gas ◽

The Boltzmann Equation ◽

H Theorem ◽

Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

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Approach to Equilibrium, the H-Theorem and Irreversibility

10.1093/oso/9780199592357.003.0003 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Boltzmann Equation ◽

Detailed Knowledge ◽

Approach To Equilibrium ◽

The Boltzmann Equation ◽

H Theorem ◽

Irreversible Evolution

Like most of the greatest equations in science, the Boltzmann equation is not only beautiful but also generous. Indeed, it delivers a great deal of information without imposing a detailed knowledge of its solutions. In fact, Boltzmann himself derived most if not all of his main results without ever showing that his equation did admit rigorous solutions. This Chapter illustrates one of the most profound contributions of Boltzmann, namely the famous H-theorem, providing the first quantitative bridge between the irreversible evolution of the macroscopic world and the reversible laws of the underlying microdynamics.

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