Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge Texts in Applied Mathematics

Cercignani, Carlo; Michaelis, C

doi:10.1115/1.1399679

Cited by 183 publications

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“…Importantly, the physical conservation laws and the Boltzmann H-theorem also hold for the Boltzmann-BGK equation (Sone 2000). The insight afforded by this model equation is sufficient to describe the qualitative properties of the (true) flow in many cases of practical interest (Vincenti & Kruger 1965;Cercignani 2000;Sone 2007). Furthermore, the DSMC and LVDSMC methods discussed for the Boltzmann equation have been applied to study of the Boltzmann-BGK equation (Ramanathan & Koch 2009;Hadjiconstantinou, Radtke & Baker 2010).…”

Section: Introductionmentioning

confidence: 99%

“…This sole degree of freedom is insufficient to simultaneously set the correct gas viscosity and thermal conductivity. As such, the correct Prandtl number for a monatomic gas is not recovered (Cercignani 2000;Sone 2000).…”

Section: Introductionmentioning

confidence: 99%

“…Model equations have also been proposed to simplify the nature of the collision integral and preserve the qualitative behaviour of the flow (Bhatnagar, Gross & Krook 1954;Welander 1954;Lebowitz, Frisch & Helfand 1960;Holway 1963;Bouchut & Perthame 1993;Struchtrup 1997;Cercignani 2000;Sone 2000). The BGK collision model by Bhatnagar et al (1954) and Welander (1954) approximates the collision integral with a simple relaxation process.…”

Section: Introductionmentioning

confidence: 99%

“…The quadratic nonlinearity of the Boltzmann equation is recovered under certain conditions by the integro-differential model studied by Lebowitz et al (1960); see also Cercignani (2000). The correct Prandtl number is predicted by the ellipsoid statistical (ES) model proposed by Holway (1963), where the local equilibrium Maxwellian solution that appears in the BGK approximation is replaced by an anisotropic Gaussian defined in terms of the Prandtl number and the stress tensor of the gas (Holway 1966;Cercignani 2000;Brull & Schneider 2008;Gallis & Torczynski 2011). The BGK approximation is a special case of the ES model, that is recovered when the Prandtl number is set to unity.…”

Section: Introductionmentioning

confidence: 99%

“…First, the simplified collision integral depends on the local equilibrium Maxwellian distribution function, and is thus highly nonlinear (Cercignani 2000;Sone 2000). Second, the BGK approximation contains a single parameter, the inter-particle collision frequency.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

Nassios

Sader

2013

J. Fluid Mech.

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The Boltzmann equation provides a rigorous theoretical framework to study dilute gas flows at arbitrary degrees of rarefaction. Asymptotic methods have been applied to steady flows, enabling the development of analytical formulae. For unsteady (oscillatory) flows, two important limits have been studied: (i) at low oscillation frequency and small mean free path, slip models have been derived; and (ii) at high oscillation frequency and large mean free path, the leading-order dynamics are free-molecular. In this article, the complementary case of small mean free path and high oscillation frequency is examined in detail. All walls are solid and of arbitrary smooth shape. We perform a matched asymptotic expansion of the unsteady linearized Boltzmann-BGK equation in the small parameter ν/ω, where ν is the collision frequency of gas particles and ω is the characteristic oscillation frequency of the flow. Critically, an algebraic expression is derived for the perturbed mass distribution function throughout the bulk of the gas away from any walls, at all orders in the frequency ratio ν/ω. This is supplemented by a boundary layer correction defined by a set of first-order differential equations. This system is solved explicitly and in complete generality. We thus provide analytical expressions up to first order in the frequency ratio, for the density, temperature, mean velocity and stress tensor of the gas, in terms of the temperature and mean velocity of the wall, and the applied body force. In stark contrast to other asymptotic regimes, these explicit formulae eliminate the need to solve a differential equation for a body of arbitrary geometry. To illustrate the utility of these results, we study the oscillatory thermal creep problem for which we find a tangential boundary layer flow arises at first order in the frequency ratio.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

Nassios

Sader

2013

J. Fluid Mech.

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Thermophoresis of an aerosol sphere perpendicular to two plane walls

Keh

Chang

2006

AIChE Journal

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in Wiley InterScience (www.interscience.wiley.com). The problem of the thermophoretic motion of a spherical particle in a gaseous medium situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The imposed temperature gradient is uniform and perpendicular to the plane walls. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface.The presence of the confining walls causes two basic effects on the particle velocity: first, the local temperature gradient on the particle surface is altered by the walls, thereby speeding up or slowing down the particle; second, the walls enhance the viscous retardation of the moving particle. A boundary collocation method is used to solve the thermal and hydrodynamic governing equations of the system. Numerical results for the thermophoretic velocity of the particle relative to that under identical conditions in an unbounded gaseous medium are presented for various cases. The collocation results agree well with the available approximate analytical solutions obtained by using a method of reflections. The presence of the walls can reduce or enhance the particle velocity, depending upon the relative thermal conductivity and surface properties of the particle as well as the relative particle-wall separation distances. The boundary effect on thermophoresis of a particle normal to two plane walls is found to be quite significant and generally stronger than that parallel to the confining walls. © 2006 American Institute of Chemical Engineers AIChE J, 52: 1690AIChE J, 52: -1704AIChE J, 52: , 2006 Keywords: thermophoresis, aerosol sphere, continuum regime, fluid mechanics, boundary effect, plane walls IntroductionA particle, when suspended in a gas possessing a temperature gradient, acquires a velocity relative to the gas in the direction of decreasing temperature. This phenomenon, known as thermophoresis, was first observed by Tyndall in 1870 when it was discovered that a dust-free space surrounded a hot body. 1 The thermophoretic effect can be explained in part by appealing to the kinetic theory of gases. 2 The higher energy molecules in the hot regions of the gas impinge on the particle with greater momenta than molecules coming from the cold regions, thus resulting in the migration of the particle in the direction of decreasing temperature. Being a mechanism for the capture of aerosol particles on cool surfaces, thermophoresis is of considerable importance in many practical applications, such as sampling of aerosol particles, 3 cleaning of air, 4 scale formation on surfaces of heat exchangers, 5 modified chemical vapor deposition, 6 manufacturing of microelectronics, 7 nuclear reactor safety, 8 and removal of soot aerosol particles for combustion exhaust gas systems. 9,10 On the basis of the assumptions of small Knudsen number (l/a, where a is the...

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Thermophoresis of axisymmetric aerosol particles along their axes of revolution

Chang

Keh

2008

AIChE Journal

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in Wiley InterScience (www.interscience.wiley.com).The axisymmetric thermophoretic motion of an aerosol particle of revolution in a uniformly prescribed temperature gradient is studied theoretically. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model. A method of distribution of a set of spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solutions for the temperature distribution and fluid velocity field. The jump/slip conditions on the particle surface are satisfied by applying a boundarycollocation technique to these general solutions. Numerical results for the thermophoretic velocity of the particle are obtained with good convergence behavior for various cases. For the axisymmetric thermophoresis of an aerosol spheroid with no temperature jump and frictional slip at its surface, the agreement between our results and the available analytical solutions is very good. The thermophoretic velocity of a spheroid along its axis of revolution in general increases with an increase in its axial-to-radial aspect ratio, but there are exceptions. For most practical cases of a spheroid with a specified aspect ratio, its thermophoretic mobility is not a monotonic function of its relative jump/slip coefficients and thermal conductivity.

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Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations. Cambridge Texts in Applied Mathematics

Cited by 183 publications

References 0 publications

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

Thermophoresis of an aerosol sphere perpendicular to two plane walls

Thermophoresis of axisymmetric aerosol particles along their axes of revolution

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