Kinetic theory of shear thickening for a moderately dense gas-solid suspension: From discontinuous thickening to continuous thickening

Abstract

The Enskog kinetic theory for moderately dense gas-solid suspensions under simple shear flow is considered as a model to analyze the rheological properties of the system. The influence of the environmental fluid on solid particles is modeled via a viscous drag force plus a stochastic Langevin-like term. The Enskog equation is solved by means of two independent but complementary routes: (i) Grad's moment method and (ii) event-driven Langevin simulation of hard spheres. Both approaches clearly show that the flow curve (stress-strain rate relation) depends significantly on the volume fraction of the solid particles. In particular, as the density increases, there is a transition from the discontinuous shear thickening (observed in dilute gases) to the continuous shear thickening for denser systems. The comparison between theory and simulations indicates that while the theoretical predictions for the kinetic temperature agree well with simulations for densities φ≲0.5, the agreement for the other rheological quantities (the viscosity, the stress ratio, and the normal stress differences) is limited to more moderate densities (φ≲0.3) if the inelasticity during collisions between particles is not large.

Kinetic derivation of diffuse-interface fluid models

“…Phase changes have notably been investigated by employing linearized Boltzmann equations with condensation-evaporation boundary conditions [18,19,20]. Boltzmann-Vlasov and Enskog-Vlasov type equations have been used to investigate more deeply spatial aspects of phase transition [21,22,23,24,25,26,27,28,29,30,31,32], as well as particulate flows [33]. Detailed molecular dynamics of Lennard-Jones fluids have further been performed by Frezzotti and Barbante and compared to capillary fluid models with a general very good agreement [7].…”

Time-dependent homogeneous states of binary granular suspensions

“…Although the drag coefficients γ i should be in general tensorial quantities (as a result of the hydrodynamic interactions between solid particles), here we will assume that those coefficients are scalar quantities independent of configuration of grains. As said in previous works [39], this simple model is expected to be reliable for describing inertial suspensions where the mean diameter of suspended particles ranges approximately from 1 to 70 µm.…”

“…Substitution of Eqs. (39) into Eqs. (34) and retaining only linear terms in δθ and δθ 1 , one obtains the set of linear differential equations…”

Enskog kinetic theory for multicomponent granular suspensions

“…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.…”

“…In addition, the functions R i are defined by Eq. (11). Figure 3 shows the α-dependence of the reduced diffusion coefficients D * ij , D T * 1 , and D U * Two different values of the mass ratio are considered: m1/m2 = 0.5 (a) and m1/m2 = 10 (b).…”