Maxwellian gas undergoing a stationary Poiseuille flow in a pipe

Sabbane, Mohamed; Tij, Mohamed; Santos, Andrés

doi:10.1016/s0378-4371(03)00513-2

Cited by 10 publications

(9 citation statements)

References 27 publications

(129 reference statements)

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“…The results show that the non-Newtonian features previously studied in the case of elastic particles 5,8,9,11,13,14,16 persist when inelasticity is present. In particular, the temperature profile T (y) exhibits a bimodal shape: it has a local minimum T 0 at the central layer and reaches two symmetric maxima T max at a distance |y max | of about three mean free paths.…”

Section: Discussionmentioning

confidence: 52%

Poiseuille Flow in a Heated Granular Gas

Tij

Santos

2004

J Stat Phys

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The planar Poiseuille flow induced by a constant external field (e.g., gravity) has been the subject of recent interest in the case of molecular gases. One of the predictions from kinetic theory (confirmed by computer simulations) has been that the temperature profile exhibits a bimodal shape with a local minimum in the middle of the slab surrounded by two symmetric maxima, in contrast to the unimodal shape expected from the Navier-Stokes (NS) equations. However, from a practical point of view, the interest of this non-Newtonian behavior in molecular gases is rather academic since it requires values of gravity extremely higher than the terrestrial one. On the other hand, gravity plays a relevant role in the case of granular gases due to the mesoscopic nature of the grains. In this paper we consider a dilute gas of inelastic hard spheres enclosed in a slab under the action of gravity along the longitudinal direction. In addition, the gas is subject to a white-noise stochastic force that mimics the effect of external vibrations customarily used in experiments to compensate for the collisional cooling. The system is described by means of a kinetic model of the inelastic Boltzmann equation and its steady-state solution is derived through second order in gravity. This solution differs from the NS description in that the hydrostatic pressure is not uniform, normal stress differences are present, a component of the heat flux normal to the thermal gradient exists, and the temperature profile includes a positive quadratic term. As in the elastic case, this new term is responsible for the bimodal shape of the temperature profile. The results show that, except for high inelasticities, the effect of inelasticity on the profiles is to slightly decrease the quantitative deviations from the NS results.

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Section: Discussionmentioning

confidence: 52%

Poiseuille Flow in a Heated Granular Gas

Tij

Santos

2004

J Stat Phys

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“…The inadequacy of the Navier-Stokes predictions in the case of plane Poiseuille flow was mentioned in 1992 [142][143][144][145]. We will focus our attention on the Burnett equations for dilute gases, the reader interested in other approaches will find further information in the recent work by Sabbane et al [143].…”

Section: Plane Poiseuille Flowmentioning

confidence: 99%

“…There have been several works on this subject [126][127][128][129][130][131][132][133][134][135] 14 and here we will be interested in the results obtained using the Burnett equations [141]. The inadequacy of the Navier-Stokes predictions in the case of plane Poiseuille flow was mentioned in 1992 [142][143][144][145]. We will focus our attention on the Burnett equations for dilute gases, the reader interested in other approaches will find further information in the recent work by Sabbane et al [143].…”

Section: Plane Poiseuille Flowmentioning

confidence: 99%

Beyond the Navier–Stokes equations: Burnett hydrodynamics

2008

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“…This force-driven Poiseuille flow [see Fig. 1(c)] has received much attention from computational [42,43,51,65,91,92,99] and theoretical [3,8,26,30,39,65,67,76,80,81,86,87,89,90,96,98] points of view. This interest has been mainly fueled by the fact that the force-driven Poiseuille flow provides a nice example illustrating the limitations of the NS description in the bulk domain (i.e., far away from the boundary layers).…”

Section: Force-driven Poiseuille Flowmentioning

confidence: 99%

Solutions of the moment hierarchy in the kinetic theory of Maxwell models

Santos

2009

Continuum Mech. Thermodyn.

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In the Maxwell interaction model the collision rate is independent of the relative velocity of the colliding pair and, as a consequence, the collisional moments are bilinear combinations of velocity moments of the same or lower order. In general, however, the drift term of the Boltzmann equation couples moments of a given order to moments of a higher order, thus preventing the solvability of the moment hierarchy, unless approximate closures are introduced. On the other hand, there exist a number of states where the moment hierarchy can be recursively solved, the solution generally exposing non-Newtonian properties. The aim of this paper is to present an overview of results pertaining to some of those states, namely the planar Fourier flow (without and with a constant gravity field), the planar Couette flow, the force-driven Poiseuille flow, and the uniform shear flow.

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Maxwellian gas undergoing a stationary Poiseuille flow in a pipe

Cited by 10 publications

References 27 publications

Poiseuille Flow in a Heated Granular Gas

Poiseuille Flow in a Heated Granular Gas

Beyond the Navier–Stokes equations: Burnett hydrodynamics

Solutions of the moment hierarchy in the kinetic theory of Maxwell models

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