An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. I. Flow problems

Scherer, C. S.; Filho, João Francisco Prolo; Barichello, Liliane Basso

doi:10.1007/s00033-008-7084-4

Cited by 24 publications

(24 citation statements)

References 45 publications

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“…(51) and (52), the eigenvalue problem we find here, in Eq. (58) is exactly the same one we obtained when the ADO solution was developed for flow problems [41]. The defining expressions given in Eqs.…”

Section: A Discrete Ordinates Solutionsupporting

confidence: 58%

“…(67), which are, for instance, different of the flow problems case [41]. In earlier works where the ADO method was used for solving (scalar) problems in this field of rarefied gas dynamics [49,37] the derivation of the needed exact solutions was based on the procedure already proposed by Case and Zweifel [50] for conservative problems in neutron transport.…”

Section: A Discrete Ordinates Solutionmentioning

confidence: 99%

“…In a very recent work [41], we were able to develop an unified approach, based on the ADO method, which solution includes a whole class of kinetic equations, more precisely, the BGK model [33], the S model [34], the Gross-Jackson (GJ) model [42] and the MRS model [43]. In that work, the ADO solution was able to represent in an unified approach the solution of a wide a class of flow problems for all the four kinetic equations.…”

Section: Introductionmentioning

confidence: 99%

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An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

Scherer

Filho

Barichello

2008

Z. Angew. Math. Phys.

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The ADO method, an analytical version of the discrete-ordinates method, is used here to solve a heat-transfer problem in a rarefied gas confined in a channel, as well as to solve a half-space problem in order to evaluate the temperature jump at the wall. This work is an extension of a previous work, devoted to flow problems, where the complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. In particular, numerical results for heat-flow profile, temperature and density perturbations are obtained for channels (walls), defined by different materials, on which different temperatures are imposed.

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Section: A Discrete Ordinates Solutionsupporting

confidence: 58%

Section: A Discrete Ordinates Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

Scherer

Filho

Barichello

2008

Z. Angew. Math. Phys.

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“…Investigations of its solution and associated kinetic models have found much attention in the last few years [7]- [13]. Numerical and analytical approaches have been developed, mainly in Cartesian coordinates [14] [15].…”

Section: Introductionmentioning

confidence: 99%

A Closed-Form Solution of a Kinetic Integral Equation for Rarefied Gas Flow in a Cylindrical Duct

Kamphorst¹,

Rodrigues²,

Barichello³

2014

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A spectral method based on Hermite cubic splines expansions combined with a collocation scheme is used to develop a solution for the vector form integral S-model kinetic equation describing rarefied gas flows in cylindrical geometry. Some manipulations are made to facilitate the computational treatment of the singularities inherent to the kernel. Numerical results for the simulation of flows generated by pressure and thermal gradients, Poiseuille and thermal-creep problems, are presented.

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“…Boltzmann method [88,89], the Information Preservation method [90,51], the Analytical Discrete Ordinate method [91,92], moment methods [93,94], variational methods [95], extended hydrodynamic equation systems [48] etc. Even though these methods can also be very effective, limitations related to applicability range, generality, accuracy or complexity usually lead us to the selection of DVM or DSMC.…”

Section: Methodsmentioning

confidence: 99%

Simulation of transport phenomena in conditions far from thermodynamic equilibrium via kinetic theory with applications in vacuum technology and MEMS

Πανταζής¹

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The approval of the current dissertation by the Department of Mechanical Engineering of the University of Thessaly does not imply acceptance of the author's opinions (Law 5343/32 number 202 paragraph 2). Also, the views and opinions expressed herein do not necessarily reflect those of the European Commission. There are several other people who I feel that have contributed in the writing of this dissertation.The most important contributions, along with several helpful conversations, were made by Prof. F.Sharipov, who provided the basic concept of the channel end effect study, and Dr. Chr. Day along with the KIT personnel, who briefly introduced me in the world of experimental rarefied gas dynamics. It was also a priviledge to discuss and work for a brief period of time with Prof. A. Grecos, who tirelessly discussed with me on various subjects of statistical mechanics. Financial support for this Ph.D. has been provided by the Euratom Association -Hellenic Republic, to which I am deeply indepted. Also, partial support for conference mobility expenses has been received from the GASMEMS Marie Curie programme. In this work, non-equilibrium transport phenomena have been examined via the kinetic theory of gases. The behaviour of the gas in low pressure conditions or in low-dimensionality systems can not be captured by the usual Navier-Stokes formulation, in conjunction with the constitutive laws of Newton-Fourier-Fick. The limited number of intermolecular collisions results in departure from equilibrium and the particulate nature of the gas must be taken into account, greatly increasing the computational effort.The problem is described by the integro-differential Boltzmann equation, which is used to determine the distribution of particles in physical and molecular velocity space, as well as in time.There are difficulties associated with the seven-dimensional nature of the distribution function, as well as with the complexity of the collision term, which is usually substituted by an appropriate model.The most widely used and successful numerical methodologies, the Discrete Velocity Method (DVM) and the Direct Simulation Monte Carlo (DSMC), are applied here in the whole range of the Knudsen number. It is important to employ approaches based on kinetic theory, since these are the only ones which are valid for any rarefaction level.In this work, several problems including non-equilibrium phenomena are considered. The interaction of gases with solid surfaces is considered for several problems according to the CercignaniLampis boundary conditions. The non-linear form of this scattering kernel, which had not been pre- viously used in the literature, is applied on the problem of heat transfer between parallel plates and coaxial cylinders. Its linearized form has also been employed for both flow and heat transfer problems and a comparison with relevant experiments has lead to surface characterization with respect to argon and helium flows.Non-linear heat conduction phenomena are also studied for the parallel plates and coaxial c...

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An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. I. Flow problems

Cited by 24 publications

References 45 publications

An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

A Closed-Form Solution of a Kinetic Integral Equation for Rarefied Gas Flow in a Cylindrical Duct

Simulation of transport phenomena in conditions far from thermodynamic equilibrium via kinetic theory with applications in vacuum technology and MEMS

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