Erratum: Nonlinear theory of nonstationary low Mach number channel flows of freely cooling nearly elastic granular gases [Phys. Rev. E<b>77</b>, 021307 (2008)]

Meerson, Baruch; Fouxon, Itzhak; Vilenkin, Arkady

doi:10.1103/physreve.78.039902

Cited by 7 publications

(61 citation statements)

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“…The final outcome of our work is quite different both from the previous 1D results [36], and from the results of Ref. [37].…”

Section: Introductioncontrasting

confidence: 55%

“…Assuming a zero pressure gradient and using Lagrangian coordinates he showed that, in the framework of ideal hydrodynamics, thermal collapse persists in any dimension. In Lagrangian coordinates the flow singularity turns out to be identical to the one found in 1D [36]. Fouxon [37] also briefly considered the non-ideal, Navier-Stokes flow.…”

Section: Introductionmentioning

confidence: 99%

“…The authors of Refs. [23,36] took advantage of these salient features to greatly simplify the full 1D hydrodynamic problem, which takes a full account of heat diffusion and viscosity. It turns out that, at zero gravity, the heat diffusion arrests the thermal collapse in 1D [36].…”

Section: Introductionmentioning

confidence: 99%

“…[23,36] took advantage of these salient features to greatly simplify the full 1D hydrodynamic problem, which takes a full account of heat diffusion and viscosity. It turns out that, at zero gravity, the heat diffusion arrests the thermal collapse in 1D [36]. Instead of blowing up, the gas density approaches a time-independent Inhomogeneous Cooling Sate in which the heat diffusion balances the clustering instability brought about by the collisional energy loss.…”

Section: Introductionmentioning

confidence: 99%

“…Preceding our work is a series of papers [23,24,[28][29][30]36] which dealt with nonlinear clustering dynamics of a freely-cooling granular gas in a narrow channel. Fouxon et al [29,30] investigated the nonlinear clustering analytically and numerically, using ideal (Euler) granular hydrodynamics for a dilute gas which neglects heat diffusion and viscosity.…”

Section: Introductionmentioning

confidence: 99%

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Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

2010

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We employ Navier-Stokes granular hydrodynamics to investigate the long-time behavior of clustering instability in a freely cooling dilute granular gas in two dimensions. We find that, in circular containers, the homogeneous cooling state (HCS) of the gas loses its stability via a sub-critical pitchfork bifurcation. There are no time-independent solutions for the gas density in the supercritical region, and we present analytical and numerical evidence that the gas develops thermal collapse unarrested by heat diffusion. To get more insight, we switch to a simpler geometry of a narrowsector-shaped container. Here the HCS loses its stability via a transcritical bifurcation. For some initial conditions a time-independent inhomogeneous density profile sets in, qualitatively similar to that previously found in a narrow-channel geometry. For other initial conditions, however, the dilute gas develops thermal collapse unarrested by heat diffusion. We determine the dynamic scalings of the flow close to collapse analytically and verify them in hydrodynamic simulations. The results of this work imply that, in dimension higher than one, Navier-Stokes hydrodynamics of a dilute granular gas is prone to finite-time density blowups. This provides a natural explanation to the formation of densely packed clusters of particles in a variety of initially dilute granular flows.

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“…The final outcome of our work is quite different both from the previous 1D results [36], and from the results of Ref. [37].…”

Section: Introductioncontrasting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

2010

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Shear state of freely evolving granular gases

2008

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Hydrodynamic equations are used to identify the final state reached by a freely evolving granular gas above but close to its shear instability. The theory predicts the formation of a two bands shear state with a steady density profile. There is a modulation between temperature and density profiles as a consequence of the energy balance, the density fluctuations remaining small, without producing clustering. Moreover, the time dependence of the velocity field can be scaled out with the squared root of the average temperature of the system. The latter follows the Haff law, but with an effective cooling rate that is smaller than that of the free homogeneous state. The theoretical predictions are compared with numerical results for inelastic hard disks obtained by using the direct Monte Carlo simulation method, and a good agreement is obtained for low inelasticity.

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Finite-time collapse and localized states in the dynamics of dissipative gases

Fouxon

2009

Phys. Rev. E

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A study of the gas dynamics of a dilute collection of the inelastically colliding hard spheres is presented. When diffusive processes are neglected the gas density blows up in a finite time. The blowup is the mathematical expression for one of the possible mechanisms for cluster formation in dissipative gases. The way diffusive processes smooth the singularity has been studied. Exact localized states of the gas when heat diffusion balances nonlinear cooling are obtained. The presented results generalize previous findings for planar flows.

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Erratum: Nonlinear theory of nonstationary low Mach number channel flows of freely cooling nearly elastic granular gases [Phys. Rev. E77, 021307 (2008)]

Cited by 7 publications

References 0 publications

Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

Shear state of freely evolving granular gases

Finite-time collapse and localized states in the dynamics of dissipative gases

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