Poiseuille Flow in a Heated Granular Gas

Tij, Mohamed; Santos, Andrés

doi:10.1007/s10955-004-5710-x

Cited by 16 publications

(17 citation statements)

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“…It is clear that T increases monotonically with decreasing e n which implies that the degree of temperature bimodality becomes more pronounced with increasing dissipation/inelasticity. This finding is in apparent contradiction with Tij & Santos (2004) who predicted that T decreases with decreasing e n from the elastic limit up to a value of e n ∼ 0.4 and increases subsequently for e n < 0.4 (see § 3.2.4). …”

Section: The Phase Diagram: Rarefaction Versus Dissipationcontrasting

confidence: 71%

“…Let us make a quantitative comparison of our simulation with the analysis of Tij & Santos (2004) who solved the inelastic Boltzmann equation (with a BGK-type collision model) subjected to a constant gravitational force and a stochastic (white noise) thermostat. The role of the stochastic thermostat was to heat the granular gas such that it compensates the loss of energy due to collisional cooling, yielding a 'uniform' state about which a Chapman-Enskog-type expansion was carried out by treating the body force as a small parameter (which is proportional to the centreline Knudsen number Kn 0 ).…”

Section: Comparison Of Excess Temperature With Theorymentioning

confidence: 99%

“…Understanding the role of dissipation in rarefaction phenomena would help to formulate/test hydrodynamic-like theories and related boundary conditions for rarefied granular flows. Driven by this motivation, we focus on a simple flow configuration: the gravity-driven Poiseuille flow of a granular gas, for which no study exists on the Knudsen-minimum effect, but the temperature bimodality has been studied theoretically (Tij & Santos 2004). We shall show, via simulation, that (i) the effect of dissipation on the above two rarefied phenomena is non-trivial, (ii) the wall roughness has a crucial role in the Knudsen-minimum effect and (iii) the characteristic features of temperature bimodality are at variance with the theoretical analysis of Tij & Santos (2004).…”

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confidence: 99%

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On Knudsen-minimum effect and temperature bimodality in a dilute granular Poiseuille flow

2015

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The numerical simulation of gravity-driven flow of smooth inelastic hard disks through a channel, dubbed 'granular' Poiseuille flow, is conducted using event-driven techniques. We find that the variation of the mass-flow rate (Q) with Knudsen number (Kn) can be non-monotonic in the elastic limit (i.e. the restitution coefficient e n → 1) in channels with very smooth walls. The Knudsen-minimum effect (i.e. the minimum flow rate occurring at Kn ∼ O(1) for the Poiseuille flow of a molecular gas) is found to be absent in a granular gas with e n < 0.99, irrespective of the value of the wall roughness. Another rarefaction phenomenon, the bimodality of the temperature profile, with a local minimum (T min ) at the channel centerline and two symmetric maxima (T max ) away from the centerline, is also studied. We show that the inelastic dissipation is responsible for the onset of temperature bimodality (i.e. the 'excess' temperature, T = (T max /T min − 1) = 0) near the continuum limit (Kn ∼ 0), but the rarefaction being its origin (as in the molecular gas) holds beyond Kn ∼ O(0.1). The dependence of the excess temperature T on the restitution coefficient is compared with the predictions of a kinetic model, with reasonable agreement in the appropriate limit. The competition between dissipation and rarefaction seems to be responsible for the observed dependence of both the mass-flow rate and the temperature bimodality on Kn and e n in this flow. The validity of the Navier-Stokes-order hydrodynamics for granular Poiseuille flow is discussed with reference to the prediction of bimodal temperature profiles and related surrogates.

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Section: The Phase Diagram: Rarefaction Versus Dissipationcontrasting

confidence: 71%

Section: Comparison Of Excess Temperature With Theorymentioning

confidence: 99%

mentioning

confidence: 99%

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On Knudsen-minimum effect and temperature bimodality in a dilute granular Poiseuille flow

2015

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“…F ∼ 1) on the MFR. For granular gases driven by a gravitational force, a large value of F is possible because of the large disc size and mass, and the small thermal velocity (Tij & Santos 2004). Figure 9 shows that the normalized MFR reduces with increasing external acceleration.…”

Section: For Various Values Of L/σ and The Global Knudsen Numbermentioning

confidence: 99%

Non-equilibrium dynamics of dense gas under tight confinement

Liu

Reese

et al. 2016

J. Fluid Mech.

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The force-driven Poiseuille flow of dense gases between two parallel plates is investigated through the numerical solution of the generalized Enskog equation for two-dimensional hard discs. We focus on the competing effects of the mean free path λ, the channel width L and the disc diameter σ . For elastic collisions between hard discs, the normalized mass flow rate in the hydrodynamic limit increases with L/σ for a fixed Knudsen number (defined as Kn = λ/L), but is always smaller than that predicted by the Boltzmann equation. Also, for a fixed L/σ , the mass flow rate in the hydrodynamic flow regime is not a monotonically decreasing function of Kn but has a maximum when the solid fraction is approximately 0.3. Under ultra-tight confinement, the famous Knudsen minimum disappears, and the mass flow rate increases with Kn, and is larger than that predicted by the Boltzmann equation in the free-molecular flow regime; for a fixed Kn, the smaller L/σ is, the larger the mass flow rate. In the transitional flow regime, however, the variation of the mass flow rate with L/σ is not monotonic for a fixed Kn: the minimum mass flow rate occurs at L/σ ≈ 2-3. For inelastic collisions, the energy dissipation between the hard discs always enhances the mass flow rate. Anomalous slip velocity is also found, which decreases with increasing Knudsen number. The mechanism for these exotic behaviours is analysed.

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“…Next, we consider the steady base states that may be generated from energy input in our geometry. Independently of the nature of the boundary conditions, and if there is no pressure drop source or gravitational field in the horizontal directions (which may generate Poiseuille flows; see for example the recent works by Tij & Santos 2004;Santos & Tij 2006;Alam & Chikkadi 2010), the spatial dependence of these steady base states will occur only in the coordinate y, perpendicular to both walls (we call it vertical direction). Moreover, the flow velocity is expected to be parallel to the walls, i.e., u(y) = u x (y)e x .…”

Section: Boltzmann Kinetic Theory and General Balance Equationsmentioning

confidence: 99%

Steady base states for non-Newtonian granular hydrodynamics

2013

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We study in this work steady laminar flows in a low density granular gas modelled as a system of identical smooth hard spheres that collide inelastically. The system is excited by shear and temperature sources at the boundaries, which consist of two infinite parallel walls. Thus, the geometry of the system is the same that yields the planar Fourier and Couette flows in standard gases. We show that it is possible to describe the steady granular flows in this system, even at large inelasticities, by means of a (nonNewtonian) hydrodynamic approach. All five types of Couette-Fourier granular flows are systematically described, identifying the different types of hydrodynamic profiles. Excellent agreement is found between our classification of flows and simulation results. Also, we obtain the corresponding non-linear transport coefficients by following three independent and complementary methods: (1) an analytical solution obtained from Grad's 13-moment method applied to the inelastic Boltzmann equation, (2) a numerical solution of the inelastic Boltzmann equation obtained by means of the direct simulation Monte Carlo method and (3) event-driven molecular dynamics simulations. We find that, while Grad's theory does not describe quantitatively well all transport coefficients, the three procedures yield the same general classification of planar Couette-Fourier flows for the granular gas. IntroductionThere have been in the recent years a large number of studies on the dynamics of granular gases, where 'granular gas' is a term used to refer to a low density system of many mesoscopic particles that collide inelastically in pairs. Due to inelasticity in the collisions, the granular gas particles tend to collapse to a rest state, unless there is some kind of energy input. In particular, Goldhirsch & Zanetti (1993) showed that clustering instabilities spontaneously appear in a freely evolving granular gas. Nevertheless, most situations of practical interest involve an energy input to compensate for the energy loss and sustain, in some cases, the 'gas' condition of the granular system. This type of problem has been extensively studied, giving rise to a subfield of granular dynamics: 'rapid granular flows ' (Jenkins & Savage 1983;Wang, Jackson & Sundaresan 1996;Goldhirsch 2003;Aranson & Tsimring 2006). Furthermore, it has been shown that rapid granular flows can attain steady states, some of which, under appropriate circumstances and for simple geometries, can give rise to laminar flows, in the same way as a regular gas does (see, for instance, the work by Tij, Tahiri, Montanero, Garzó, Santos & Dufty † Email address for correspondence: fvega@unex.es

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Poiseuille Flow in a Heated Granular Gas

Cited by 16 publications

References 50 publications

On Knudsen-minimum effect and temperature bimodality in a dilute granular Poiseuille flow

On Knudsen-minimum effect and temperature bimodality in a dilute granular Poiseuille flow

Non-equilibrium dynamics of dense gas under tight confinement

Steady base states for non-Newtonian granular hydrodynamics

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