Segregation by thermal diffusion of an intruder immersed in a sheared granular gas is analyzed from the (inelastic) Boltzmann equation. Segregation is induced by the presence of a temperature gradient orthogonal to the shear flow plane and parallel to gravity. We show that, like in analogous systems without shear, the segregation criterion yields a transition between upwards segregation and downwards segregation. The form of the phase diagrams is illustrated in detail showing that they depend sensitively on the value of gravity relative to the thermal gradient. Two specific situations are considered: i) absence of gravity, and ii) homogeneous temperature. We find that both mechanisms (upwards and downwards segregation) are stronger and more clearly separated when compared with segregation criteria in systems without shear.
A low density binary mixture of granular gases is considered within the Boltzmann kinetic theory. One component, the intruders, is taken to be dilute with respect to the other, and thermal segregation of the two species is described for a special exact solution to the Boltzmann equation. This solution has a macroscopic hydrodynamic representation with a constant temperature gradient and is referred to as the Fourier state. The thermal diffusion factor characterizing conditions for segregation is calculated without the usual restriction to Navier-Stokes hydrodynamics. Molecular dynamics simulat…
Surface Flows of Granular Mixtures: III. Canonical Model
本文推广了Boutreux和de Gennes提出的颗粒混合物表面流动的极小模型[J. Phys. I France 6，1295（1996）]。极小模型适用于仅在表面性质上不同的晶粒。本模型还考虑了晶粒尺寸的差异。我们应用该模型来研究偏析在二维筒仓的颗粒混合物的大小和/或表面性质不同。当尺寸差异较小时，该模型预测在筒仓填充期间在静态相中出现连续偏析。当尺寸差异很大时，我们考虑到在轧制阶段晶粒的偏析，模型预测完全偏析和分层与实验观察一致。
Diffusion induced by a thermal gradient in a granular binary mixture is analyzed in the context of the (inelastic) Enskog equation. Although the Enskog equation neglects velocity correlations among particles which are about to collide, it retains spatial correlations arising from volume exclusion effects and thus it is expected to apply to moderate densities. In the steady state with gradients only along a given direction, a segregation criterion is obtained from the thermal diffusion factor $Λ$ measuring the amount of segregation parallel to the thermal gradient. As expected, the sign of the …
Segregation induced by a thermal gradient of an impurity in a driven low-density granular gas is studied. The system is enclosed between two parallel walls from which we input thermal energy to the gas. We study here steady states occurring when the inelastic cooling is exactly balanced by some external energy input (stochastic force or viscous heating), resulting in a uniform heat flux. A segregation criterion based on Navier-Stokes granular hydrodynamics is written in terms of the tracer diffusion transport coefficients, whose dependence on the parameters of the system (masses, sizes and coe…
We study segregation of an impurity in a driven granular fluid under two types of \emph{steady} states. In the first state, the granular gas is driven by a stochastic volume force field with a Fourier-type profile while in the second state, the granular gas is sheared in such a way that inelastic cooling is balanced by viscous heating. We compare theoretical results derived from a solution of the (inelastic) Boltzmann equation at Navier-Stokes (NS) order with those obtained from the Direct Monte Carlo simulation (DSMC) method and molecular dynamics (MD) simulations. Good agreement is found bet…
The self-diffusion coefficient of a granular gas in the homogeneous cooling state is analyzed near the shearing instability. Using mode-coupling theory, it is shown that the coefficient diverges logarithmically as the instability is approached, due to the coupling of the diffusion process with the shear modes. The divergent behavior, which is peculiar of granular gases and disappears in the elastic limit, does not depend on any other transport coefficient. The theoretical prediction is confirmed by molecular dynamics simulation results for two-dimensional systems.
The mass flux of a low-density granular binary mixture obtained previously by solving the Boltzmann equation by means of the Chapman-Enskog method is considered further. As in the elastic case, the associated transport coefficients $D$, $D_p$ and $D'$ are given in terms of the solutions of a set of coupled linear integral equations which are approximately solved by considering the first and second Sonine approximations. The diffusion coefficients are explicitly obtained as functions of the coefficients of restitution and the parameters of the mixture (masses, diameters and concentration) and t…
A recent segregation criterion [V. Garzó, Phys. Rev. E \textbf{78}, 020301(R) (2008)] based on the thermal diffusion factor $Λ$ of an intruder in a heated granular gas described by the inelastic Enskog equation is revisited. The sign of $Λ$ provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE). The present theory incorporates two extra ingredients not accounted for by the previous theoretical attempt. First, the theory is based upon the second Sonine approximation to the transport coefficients of the mass flux of intruder. Second, …
Applications of the kinetic theory for a model of a confined quasi-two dimensional granular mixture: Stability analysis and thermal diffusion segregation
Shearing with a finite shear rate a compressed granular system results in a region of grains flowing over a compact, static assembly. Perforce this region is dilated to a degree that depends on the shear rate, the loading pressure, gravity, various material parameters, and the preparation protocol. In spite of numerous studies of granular flows a predictive theory of the amount of dilation is still lacking. Here, we offer a scaling theory that is focused on such a prediction as a function of shear rate and the dissipative parameters of the granular assembly. The resulting scaling laws are univ…
We numerically investigate the rheological properties of jammed frictionless granular materials under an oscillatory shear in terms of our simulation of the distinct element method. It is demonstrated that the constitutive relation between the shear stress and the shear strain of grains strongly depends on the amplitude and the frequency of the oscillatory shear. For a small amplitude region, the granular material behaves as a visco-elastic material characterized by the Kelvin-Voigt model, while it behaves as a yield stress liquid for a large amplitude region. We find that the rheology of grai…
Rheology of a dilute cohesive granular gas is theoretically and numerically studied. The flow curve between the shear viscosity and the shear rate is derived from the inelastic Boltzmann equation for particles having square-well potentials in a simple shear flow. It is found that (i) the stable uniformly sheared state only exists above a critical shear rate and (ii) the viscosity in the uniformly sheared flow is almost identical to that for uniformly sheared flow of hard core granular particles. Below the critical shear rate, clusters grow with time, in which the viscosity can be approximated …
Shear modulus of frictionless granular materials near the jamming transition under oscillatory shear is numerically investigated. It is found that the shear modulus $G$ satisfies a scaling law to interpolate between $G \sim (ϕ- ϕ_J)^{1/2}$ and $G \sim γ_0^{-1/2}(ϕ-ϕ_J)$ for a linear spring model of the elastic interaction between contacting grains, where $ϕ$, $ϕ_J$, and $γ_0$ are, respectively, the volume fraction of grains, the fraction at the jamming point, and the amplitude of the oscillatory shear. The linear relation between the shear modulus and $ϕ- ϕ_J$ can be understood by slip avalanc…
Experimental measurements of the onset of granular flow are directly compared to predictions of the "shear transformation zone" (STZ) theory of amorphous plasticity. The STZ equations make it possible, on a coarse grained level, to incorporate the anisotropy of the particle contact network and its dynamic evolution including changes in the contact network direction under shear reversal. The STZ theory and experiment show qualitative agreement in the transient and steady state shear forces and velocity profiles during the start of shear and shear reversal.
Dry granular materials such as sand, gravel, pills, or agricultural grains, can become rigid when compressed or sheared. At low density, one can distort the shape of a container of granular material without encountering any resistance. Under isotropic compression, the material will reach a certain {\it jamming} density and then resist further compression. {\em Shear jamming} occurs when resistance to shear emerges in a system at a density lower than the jamming density, and the elastic properties of such states have important implications for industrial and geophysical processes. We report on …
