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Register a new userFouriers Law for a Granular Fluid
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James W. Dufty
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Newton viscosity law for the momentum flux and Fouriers law for the heat flux define Navier-Stokes hydrodynamics for a simple, one component fluid. There is ample evidence that a hydrodynamic description applies as well to a mesoscopic granular fluid with the same form for Newtons viscosity law. However, theory predicts a qualitative difference for Fouriers law with an additional contribution from density gradients even at uniform temperature. The reasons for the absence of such terms for normal fluids are indicated, and a related microscopic explanation for their existence in granular fluids is presented.
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The response of an isolated granular fluid to small perturbations of the hydrodynamic fields is considered. The corresponding linear response functions are identified in terms of a formal solution to the Liouville equation including the effects of the cooling reference state. These functions are evaluated exactly in the asymptotic long wavelength limit and shown to represent hydrodynamic modes. More generally, the linear granular Navier-Stokes equations for the response functions and related Langevin equations are obtained from an extension of Moris identity. The resulting Green-Kubo expressions for transport coefficients are compared and contrasted with those for a molecular fluid. Next the response functions are described in terms of an effective dynamics in the single particle phase space. A closed linear kinetic equation is obtained formally in terms of a linear two particle functional. This closure is evaluated for two examples: a short time Markovian approximation, and a low density expansion on length and time scales of the mean free time and mean free path. The former is a generalization of the revised Enskog kinetic theory to include velocity correlations. The latter is an extension of the Boltzmann equation to include the effects of recollisions (rings) among the particles.
Granular fluids consist of collections of activated mesoscopic or macroscopic particles (e.g., powders or grains) whose flows often appear similar to those of normal fluids. To explore the qualitative and quantitative description of these flows an idealized model for such fluids, a system of smooth inelastic hard spheres, is considered. The single feature distinguishing granular and normal fluids being explored in this way is the inelasticity of collisions. The dominant differences observed in real granular fluids are indeed captured by this feature. Following a brief introductory description of real granular fluids and motivation for the idealized model, the elements of nonequilibrium statistical mechanics are recalled (observables, states, and their dynamics). Peculiarities of the hard sphere interactions are developed in detail. The exact microscopic balance equations for the number, energy, and momentum densities are derived and their averages described as the origin for a possible macroscopic continuum mechanics description. This formally exact analysis leads to closed, macroscopic hydrodynamic equations through the notion of a normal state. This concept is introduced and the Navier-Stokes constitutive equations are derived, with associated Green-Kubo expressions for the transport coefficients. A parallel description of granular gases is described in the context of kinetic theory, and the Boltzmann limit is identified critically. The construction of the normal solution to the kinetic equation is outlined, and Navier-Stokes order hydrodynamic equations are re-derived for a low density granular gas.
A solution of the inelastic Enskog equation that goes beyond the weak dissipation limit and applies for moderate densities is used to determine the thermal diffusion factor of an intruder immersed in a dense granular gas under gravity. This factor provides a segregation criterion that shows the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the parameters of the system (masses, sizes, density and coefficients of restitution). The form of the phase-diagrams for the BNE/RBNE transition depends sensitively on the value of gravity relative to the thermal gradient, so that it is possible to switch between both states for given values of the parameters of the system. Two specific limits are considered with detail: (i) absence of gravity, and (ii) homogeneous temperature. In the latter case, after some approximations, our results are consistent with previous theoretical results derived from the Enskog equation. Our results also indicate that the influence of dissipation on thermal diffusion is more important in the absence of gravity than in the opposite limit. The present analysis extends previous theoretical results derived in the dilute limit case [V. Garzo, Europhys. Lett. {bf 75}, 521 (2006)] and is consistent with the findings of some recent experimental results.
A new segregation criterion based on the inelastic Enskog kinetic equation is derived to show the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the different parameters of the system. In contrast to previous theoretical attempts the approach is not limited to the near-elastic case, takes into account the influence of both thermal gradients and gravity and applies for moderate densities. The form of the phase-diagrams for the BNE/RBNE transition depends sensitively on the value of gravity relative to the thermal gradient, so that it is possible to switch between both states for given values of the mass and size ratios, the coefficients of restitution and the solid volume fraction. In particular, the influence of collisional dissipation on segregation becomes more important when the thermal gradient dominates over gravity than in the opposite limit. The present analysis extends previous results derived in the dilute limit case and is consistent with the findings of some recent experimental results.
We study the transport of heat along a chain of particles interacting through a harmonic potential and subject to heat reservoirs at its ends. Each particle has two degrees of freedom and is subject to a stochastic noise that produces infinitesimal changes in the velocity while keeping the kinetic energy unchanged. This is modelled by means of a Langevin equation with multiplicative noise. We show that the introduction of this energy conserving stochastic noise leads to Fouriers law. By means of an approximate solution that becomes exact in the thermodynamic limit, we also show that the heat conductivity $kappa$ behaves as $kappa = a L/(b+lambda L)$ for large values of the intensity $lambda$ of the energy conserving noise and large chain sizes $L$. Hence, we conclude that in the thermodynamic limit the heat conductivity is finite and given by $kappa=a/lambda$.
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