On the validity of hydrodynamics in plane Poiseuille flows

Mansour, M. Malek; Baras, F.; Garcia, Alejandro L.

doi:10.1016/s0378-4371(97)00149-0

Cited by 92 publications

(83 citation statements)

References 29 publications

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“…Due to the presence of this force, this Poiseuille flow is more interesting than Couette or Fourier flows, as it presents a bi-modal temperature profile that the NSF equations fail to predict even qualitatively [20,[23][24][25][26]. To assess the modeling accuracy of the ES-BGK equation, g x ¼ 0:22 and 1 (which are also close to the values used in previous BGK simulations [20] and DSMC simulations [23]) are chosen in the following simulations. We focus on fully-developed flows, and the diffuse kinetic boundary condition is employed for the gas-wall interactions.…”

Section: Simulation Resultsmentioning

confidence: 99%

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Meng

Reese

et al. 2013

Journal of Computational Physics

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We investigate the accuracy of the ellipsoidal-statistical Bhatnagar-Gross-Krook (ES-BGK) kinetic model for planar force-driven Poiseuille flows. Our numerical simulations are conducted using the deterministic discrete velocity method, for Knudsen numbers (Kn) ranging from 0.05 to 10. While we provide numerically accurate data, our aim is to assess the accuracy of the ES-BGK model for these flows. By comparing with data from the direct simulation Monte Carlo (DSMC) method and the Boltzmann equation, the ES-BGK model is found to be able to predict accurate velocity and temperature profiles in the slip flow regime (0:01 < Kn 6 0:1), for both low-speed and high-speed flows. In the transition flow regime (0:1 < Kn 6 10), however, the model does not quantitatively capture the viscous heating effect.

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Section: Simulation Resultsmentioning

confidence: 99%

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Meng

Reese

et al. 2013

Journal of Computational Physics

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“…6 in spite that the heat flux vector takes the energy towards the walls. This minimum has been observed in simulations and an explanation was originally given using the BGK approximation to Boltzmann equation [11], see also [10]. Namely, near the center of the channel the system behaves as if the thermal conductivity was negative as in [5].…”

Section: Laminar Flowmentioning

confidence: 59%

Nonlinear effects in gases due to strong gradients

Cordero¹,

Risso²

2001

AIP Conference Proceedings

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Abstract. The behavior of a gas may be quite atypical if the velocity distribution function is significantly distorted by the presence of large gradients. Under such circumstances phenomena like shear thinning or negative effective thermal conductivity may be present. Hydrodynamic equations derived directly from Boltzmann's equation using a moment expansion method yield predictions which are sensitive to the value of Knudsen numbers associated to the macroscopic fields themselves and provide a rational for the peculiar behavior. In the case of sheared laminar flows the law of viscous flow predicts non-Newtonian effects including shear thinning and the law of energy transport (associated to what is usually called heat flux) is more general than Fourier's law: it is not linear and it implies a flux with a nontrivial component parallel to the isotherms. These nonlinear transport laws are well corroborated by molecular dynamic simulations based on straightforward Newtonian dynamics. More in general it is shown that the energy flux is naturally split in a component that removes heat from the system and a divergenceless component which represents a flux of energy which does not leave the system.

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“…[6] as corresponding to the Taylor series expansion of the normal (or Hilbert) solution around the center of the gap. The theoretical predictions of a bimodal temperature profile have been confirmed at a qualitative and semi-quantitative level by numerical Monte Carlo simulations of the Boltzmann equation [7,11] and by molecular dynamics simulations [9,13]. In the case of a dense gas, even though the nonmonotonic behavior of the temperature profile may disappear, the existence of a quadratic term coexisting with the classical quartic term is supported by molecular dynamics simulations [14].…”

Section: Introductionmentioning

confidence: 71%

“…The most interesting outcome of the solution, as first recognized by Malek Mansour et al [7], was the presence of a positive quadratic term in the temperature profile to second order in g, in addition to the negative quartic term predicted by the NS description. As a consequence of this new term, the temperature profile exhibits a bimodal shape, namely a local minimum at the middle of the channel surrounded by two symmetric maxima at a distance of a few mean free paths.…”

Section: Introductionmentioning

confidence: 72%

“…gravity) directed longitudinally. This latter mechanism for driving the Poiseuille flow does not produce longitudinal gradients and so is more convenient than the former in computer simulations as well as from the theoretical point of view, especially to assess the reliability of the continuum description [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. The first study of the Poiseuille flow driven by an external force we are aware of was carried out by Kadanoff et al [2], who simulated the flow with the FHP lattice gas automaton [17] to confirm the validity of a hydrodynamic description for lattice gas automata.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Maxwellian gas undergoing a stationary Poiseuille flow in a pipe

Sabbane

Tij

Santos

2003

Physica A: Statistical Mechanics and its Applications

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The hierarchy of moment equations derived from the nonlinear Boltzmann equation is solved for a gas of Maxwell molecules undergoing a stationary Poiseuille flow induced by an external force in a pipe. The solution is obtained as a perturbation expansion in powers of the force (through third order). A critical comparison is done between the Navier-Stokes theory and the predictions obtained from the Boltzmann equation for the profiles of the hydrodynamic quantities and their fluxes. The Navier-Stokes description fails to first order and, especially, to second order in the force. Thus, the hydrostatic pressure is not uniform, the temperature profile exhibits a non-monotonic behavior, a longitudinal component of the flux exists in the absence of longitudinal thermal gradient, and normal stress differences are present. On the other hand, comparison with the Bhatnagar-Gross-Krook model kinetic equation shows that the latter is able to capture the correct functional dependence of the fields, although the numerical values of the coefficients are in general between 0.38 and 1.38 times the Boltzmann values. A short comparison with the results corresponding to the planar Poiseuille flow is also carried out.

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On the validity of hydrodynamics in plane Poiseuille flows

Cited by 92 publications

References 29 publications

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

Nonlinear effects in gases due to strong gradients

Maxwellian gas undergoing a stationary Poiseuille flow in a pipe

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