Transport coefficients for granular suspensions at moderate densities

Abstract

The Enskog kinetic theory for moderately dense granular suspensions is considered as a model to determine the Navier-Stokes transport coefficients. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. The suspension model is solved by means of the Chapman-Enskog method conveniently adapted to dissipative dynamics. The momentum and heat fluxes as well as the cooling rate are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the cooling terms (arising from collisional dissipation and viscous friction) cannot be compensated for by the energy gained by grains due to collisions with the interstitial gas, the reference distribution (zeroth-order approximation of the Chapman-Enskog solution) depends on time through its dependence on temperature. On the other hand, to simplify the analysis and given that we are interested in computing transport properties in the first order of deviations from the reference state, the steady-state conditions are considered. This simplification allows us to get explicit expressions for the Navier-Stokes transport coefficients. The present work extends previous results [Garzó et al. 2013, Phys Rev. E 87, 032201] since it incorporates two extra ingredients (an additional density dependence of the zeroth-order solution and the density dependence of the reduced friction coefficient) not accounted for by the previous theoretical attempt. While these two new ingredients do not affect the shear viscosity coefficient, the transport coefficients associated with the heat flux as well as the first-order contribution to the cooling rate are different from those obtained in the previous study. In addition, as expected, the results show that the dependence of the transport coefficients on both inelasticity and density is clearly different from that found in its granular counterpart (no gas phase). Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. In contrast to the granular case (no gas-phase), no instabilities are found and hence, the homogeneous steady state is (linearly) stable. * Electronic address: ruben@unex.es † Electronic address: vicenteg@unex.es; URL: http://www.unex.es/eweb/fisteor/vicente/ Boltzmann equation for inelastic Maxwell models [10]) compare quite well with molecular dynamics simulations [9] for conditions of practical interest. This good agreement highlights again the good performance of kinetic theory tools in reproducing the transport properties of gas-solid flows.On the other hand, to the best of our knowledge, most of the efforts in kinetic theory of granular suspensions has been mainly focused on non-Newtonian transport properties (which are directly related with the pressure tensor). In particular, much less is known about the energy transport in gas-solid flows. The knowledge of the transport coefficients associated with the heat flux ...

Enskog kinetic theory for multicomponent granular suspensions

“…Note that λ 1 /R 1 = λ 2 /R 2 . As expected from previous works [10,14,37,38], the dependence of the scaled distribution ϕ i,s on the temperature is not only through the dimensionless velocity c but also through the dimensionless parameter θ s . This scaling differs from the one assumed in free cooling systems [31,39] where all the temperature dependence of ϕ i,s is encoded through c.…”

“…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.…”

“…As in previous works on granular suspensions [9,11,14,27], the influence of the surrounding gas on the dynamics of grains is accounted for via an instantaneous fluid force. For low Reynolds numbers, we assume that the external force is composed by two independent terms.…”

“…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…”

“…Hence, while the drag force tries to model the friction of grains with the interstitial gas phase, the stochastic Langevin-like term mimics the energy transfer from the surrounding gas particles to the granular particles. The above suspension model has been recently [14] employed to get the Navier-Stokes transport coefficients of monocomponent granular suspensions by solving the Enskog equation for inelastic hard spheres by means of the Chapman-Enskog method [15] adapted to dissipative dynamics.…”