On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

Baranger, Céline; Bisi, Marzia; Brull, Stéphane; Desvillettes, Laurent

doi:10.1063/1.5119524

Cited by 12 publications

(21 citation statements)

References 21 publications

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“…results in a more familiar form of the Boltzmann collision operator for poly-atomic molecules modeled with a continuous energy variable [5,1,8]…”

Section: Collision Operatormentioning

confidence: 99%

“…More recently, compactness results were also obtained for monatomic multi-component mixtures [4], see also [2], and for polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable [2]. In this work, we consider polyatomic single species, where the polyatomicity is modeled by a continuous internal energy variable [5,1,8]. We restrict ourselves to the case when the number of internal degrees of freedom is greater or equal to two.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, the model considered is presented. The probabilistic formulation of the collision operator considered and its relation to a more classical formulation [5,1,8] is accounted for in Section 2.1 Some classical results for the collision operator in Section 2.2 and the linearized collision operator in Section 2.3 are reviewed. Section 3 is devoted to the main results of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Bernhoff¹

2022

Preprint

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The linearized collision operator of the Boltzmann equation can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator. Compactness of the integral operator for monatomic single species is a classical result, while corresponding result for mixtures is more recently obtained. In this work the compactness of the integral operator for polyatomic single species, for which the number of internal degrees of freedom is greater or equal to two and the polyatomicity is modeled by a continuous internal energy variable, is studied. Compactness of the integral operator is obtained by proving that its terms are, or, at least, can be approximated by, Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. Self-adjointness of the linearized collision operator follows. Moreover, bounds on -including coercivity of -the collision frequency, are obtained for some particular collision kernels -corresponding to hard sphere models, but also hard potential with cut-off models. Then it follows that the linearized collision operator is a Fredholm operator.

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“…results in a more familiar form of the Boltzmann collision operator for poly-atomic molecules modeled with a continuous energy variable [5,1,8]…”

Section: Collision Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Bernhoff¹

2022

Preprint

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“…In the last decade or so, the Boltzmann equation for mixtures, which was already mentioned in [10], attracted the attention of many works. The modelling issue, for both monatomic and polyatomic gases, was for instance discussed in [14,1,33] (see also the references therein). Many works focused on the analysis of the monatomic case, like [6,12,9,3], which were dedicated to compactness, hypocoercivity-related and stability results.…”

Section: Introductionmentioning

confidence: 99%

Energy method for the Boltzmann equation of monatomic gaseous mixtures

Boudin¹,

Grec²,

Pavić-Čolić³

et al. 2021

Preprint

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In this paper, we present an energy method for the system of Boltzmann equations in the multicomponent mixture case, based on a micro-macro decomposition. More precisely, the perturbation of a solution to the Bolzmann equation around a global equilibrium is decomposed into the sum of a macroscopic and a microscopic part, for which we obtain a priori estimates at both lower and higher orders. These estimates are obtained under a suitable smallness assumption.The assumption can be justified a posteriori in the higher-order case, leading to the closure of the corresponding estimate.

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“…Models based on a collision term for each type of interaction are more frequent in engineering literature, as [36]. Extensions to the polyatomic case are proposed in [48], or in [7].…”

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confidence: 99%

Kinetic models of BGK type and their numerical integration

Puppo¹

2019

Preprint

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This minicourse contains a description of recent results on the modelling of rarefied gases in weakly non equilibrium regimes, and the numerical methods used to approximate the resulting equations. Therefore this work focuses on BGK type approximations, rather than on full Boltzmann models. Within this framework, models for polyatomic gases and for mixtures will be considered. We will also address numerical issues characteristic of the difficulties one encounters when integrating kinetic equations. In particular, we will consider asymptotic preserving schemes, which are designed to approximate equilibrium solutions, without resolving the fast scales of the approach to equilibrium.Kinetic theory was initially developed to study the behaviour of rarefied gases, with applications, as a typical example, to flow in the higher levels of the atmosphere. Recently its scope has enlarged to include many non equilibrium phenomena, arising, for instance, in the study of microfluids, i.e. flows occurring in domains with microscales, where the equilibrium hypothesis underlying classical gas dynamics does not hold. But kinetic models have also been successfully applied to phenomena which do not stem from fluid dynamics. The attractive feature of kinetic theory beyond gas dynamics is its ability to start from the characteristics of interactions of particles at a microscopic scale, to develop equations for the collective behaviour. These new applications include social sciences, see the examples in [54] or natural sciences [8], or even traffic flow, [40]. For a short introduction to rarefied gas dynamics see [23][Chapt. 1]. A more in depth text is [22] This paper will review kinetic models of BGK type, and the numerical techniques which permit to obtain accurate and reliable approximate solutions. TheThe collisions conserve mass, momentum and energy, which, at the microscopic level, means

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On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases

Cited by 12 publications

References 21 publications

Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable

Energy method for the Boltzmann equation of monatomic gaseous mixtures

Kinetic models of BGK type and their numerical integration

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