Transport coefficients of solid particles immersed in a viscous gas

Abstract

Transport properties of a suspension of solid particles in a viscous gas are studied. The dissipation in such systems arises from two sources: inelasticity in particle collisions and viscous dissipation due to the effect of the gas phase on the particles. Here we consider a simplified case in which the mean relative velocity between the gas and solid phases is taken to be zero, such that "thermal drag" is the only remaining gas-solid interaction. Unlike the previous, more general, treatment of the drag force [Garzó et al., J. Fluid Mech. 712, 129 (2012)]JFLSA70022-112010.1017/jfm.2012.404, here we take into account contributions to the (scaled) transport coefficients η^{*} (shear viscosity), κ^{*} (thermal conductivity), and μ^{*} (Dufour-like coefficient) coming from the temperature dependence of the (dimensionless) friction coefficient γ^{*} characterizing the amplitude of the drag force. At moderate densities, the thermal drag model (which is based on the Enskog kinetic equation) is solved by means of the Chapman-Enskog method and the Navier-Stokes transport coefficients are determined in terms of the coefficient of restitution, the solid volume fraction, and the friction coefficient. The results indicate that the effect of the gas phase on η^{*} and μ^{*} is non-negligible (especially in the case of relatively dilute systems) while the form of κ^{*} is the same as the one obtained in the dry granular limit. Finally, as an application of these results, a linear stability analysis of the hydrodynamic equations is carried out to analyze the conditions for stability of the homogeneous cooling state. A comparison with direct numerical simulations shows a good agreement for conditions of practical interest.

Transport coefficients for granular suspensions at moderate densities

“…These findings are likely the most significant contributions of the present work. In this context, this paper complements and extends previous papers on transport properties in granular suspensions [8,11,12].…”

“…However, for the sake simplicity, the temperature dependence of the scaled friction coefficient γ * = γ/ν(T ) (where ν ∝ T 1/2 is an effective collision frequency for hard spheres and T is the granular temperature) was implicitly neglected in the above calculations [8] to get analytic (explicit) expressions for the transport coefficients. The above temperature dependence of γ * was accounted for in a subsequent paper [11] but for a simplified model where only the drag force term was considered in the Enskog equation.A more careful study was carried out later in Ref. [12] where the transport coefficients were explicitly computed by considering both the temperature dependence of the reduced friction coefficient as well as the complete form of the suspension model.…”

Shear-rate-dependent transport coefficients in granular suspensions

“…In addition, as in previous works [40], since the friction coefficient γ does not induce any flux in the system, it is assumed then to be at least of zeroth order in the gradients. In this paper, only the first order approximation will be considered.…”

The Homogeneous Cooling State as a Verification Test for Kinetic Theory-Based Continuum Models of Gas–Solid Flows

“…Although the HCSt and the homogeneous cooling system (HCSys) are occasionally used interchangeably (possibly because they have the same HCS acronym), there is a significant distinction, which is of importance here (hence the separate acronyms). Due to the dissipative nature of particle collisions [2] and/or the viscous dissipation of an interstitial fluid [3], the homogeneous cooling state is linearly unstable, which causes a system-size dependence [4,5]. Similar to the transition to turbulence in pipe flow, if the system is sufficiently small the instability is suppressed, but if the system is larger than a critical length scale, instabilities develop commonly referred to as clustering, see Ref.…”

“…(2) The critical length scale necessary for onset of instabilities in the HCSys has been used more recently to quantitatively assess the ability of kinetic theory-based models to predict instabilities, e.g., see Refs. [5], [13], and [14]. Comparisons are typically made to numerical data generated from DEM; unlike DSMC, DEM involves balances for each particle, instead of relying on a kinetic equation.…”