Transport properties for driven granular fluids in situations close to homogeneous steady states

Abstract

The transport coefficients of a granular fluid driven by a stochastic bath with friction are obtained by solving the inelastic Enskog kinetic equation from the Chapman-Enskog method. The heat and momentum fluxes as well as the cooling rate are determined to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the collisional cooling cannot be compensated locally for the heat produced by the external driving force, the reference distribution f (0) (zeroth-order approximation) depends on time through its dependence on temperature. This fact gives rise to conceptual and practical difficulties not present in the undriven case. On the other hand, to simplify the analysis and given that we are interested in computing transport in the first order of deviations from the reference state, the steady-state conditions are considered to get explicit forms for the transport coefficients and the cooling rate. A comparison with recent Langevin dynamics simulations for driven granular fluids shows an excellent agreement for the kinematic viscosity although some discrepancies are observed for the longitudinal viscosity and the thermal diffusivity at large densities. Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. As expected, no instabilities are found thanks to the presence of the external bath.

Enskog kinetic theory for multicomponent granular suspensions

“…On the other hand, as already pointed out in previous works [10,14,37], the evaluation of the transport coefficients under unsteady conditions requires one to know the complete time dependence of the first-order corrections to the mass, momentum, and heat fluxes. This is quite an intricate problem.…”

“…i , ζ (0) is given by Eq. (37). An accurate estimate of ζ (0) i is obtained by considering the Maxwellian approximation (47) to ϕ i .…”

“…III. However, as noted in previous works [10,14,37,52], since the densities n i (r, t) and the granular temperature T (r, t) are defined separately in the local reference state f Here, the derivative ∂ϕ i /∂θ is taken at constant c, ζ * 0 = ℓζ (0) /v 0 , γ * i = λ i θ −1/2 , and upon deriving Eq. (67) use has been of the property…”

Hydrodynamics of a Granular Gas in a Heterogeneous Environment

“…Notice that in (2) we have ξ = ξ(r) (a space-dependent noise intensity). The noisy term appears in the inelastic Boltzmann equation as a Fokker-Planck-like operator [43,44] …”

“…Multiplying by velocity momenta the kinetic Equation (6) and performing velocity integrals, we may easily obtain from (6) the mass, momentum and energy balance equations [43,47] ∂P yy ∂y…”