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

Rheology of granular particles immersed in a molecular gas under uniform shear flow

Non-Newtonian transport properties of a dilute gas of inelastic hard spheres immersed in a molecular gas are determined. We assume that the granular gas is sufficiently rarefied, and hence the state of the molecular gas is not disturbed by the presence of the solid particles. In this situation, one can treat the molecular gas as a bath (or thermostat) of elastic hard spheres at a given temperature. Moreover, in spite of the fact that the number density of grains is quite small, we take into account their inelastic collisions among themselves in its kinetic equation. The system (granular gas plus a bath of elastic hard spheres) is subjected to a simple (or uniform) shear flow. In the low-density regime, the rheological properties of the granular gas are determined by solving the Boltzmann kinetic equation by means of Grad's moment method. These properties turn out to be highly nonlinear functions of the shear rate and the remaining parameters of the system. Our results show that the kinetic granular temperature and the non-Newtonian viscosity present a discontinuous shear thickening effect for sufficiently high values of the mass ratio m/m_{g} (m and m_{g} being the mass of grains and gas particles, respectively). This effect becomes more pronounced as the mass ratio m/m_{g} increases. In particular, in the Brownian limit (m/m_{g}→∞) the expressions of the non-Newtonian transport properties derived here are consistent with those previously obtained by considering a coarse-grained approach where the effect of gas phase on grains is through an effective force. Theoretical results are compared against computer simulations, showing an excellent agreement.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equation by means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equation for the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Homogeneous cooling and heating states of dilute soft-core gases under nonlinear drag

The temperature evolution of dilute soft inertial gas-solid suspensions is theoretically analyzed when the gas particles are influenced by a nonlinear drag force from a background fluid. The kinetic theory is extended to this system, and the time evolutions of the temperature and the kurtosis of the velocity distribution are derived. Molecular dynamics simulations are also performed to check the validity of the theory, and they show good agreement with the theoretical predictions.

Non-monotonic Mpemba effect in binary molecular suspensions

The Mpemba effect is a phenomenon in which an initially hotter sample cools sooner. In this paper, we show the emergence of a non-monotonic Mpemba-like effect in a molecular binary mixture immersed in a viscous gas. Namely, a crossover in the temperature evolution when at least one of the samples presents non-monotonic relaxation. The influence of the bath on the dynamics of the particles is modeled via a viscous drag force plus a stochastic Langevin-like term. Each component of the mixture interchanges energy with the bath depending on the mechanical properties of its particles. This discrimination causes the coupling between the time evolution of temperature with that of the partial temperatures of each component. The non-monotonic Mpemba effect—and its inverse and mixed counterparts—stems from this coupling. In order to obtain analytical results, the velocity distribution functions of each component are approximated by considering multitemperature Maxwellian distributions. The theoretical results derived from the Enskog kinetic theory show an excellent agreement with direct simulation Monte Carlo (DMSC) data.

Mpemba effect in inertial suspensions

The Mpemba effect (a counterintuitive thermal relaxation process where an initially hotter system may cool down to the steady state sooner than an initially colder system) is studied in terms of a model of inertial suspensions under shear. The relaxation to a common steady state of a suspension initially prepared in a quasiequilibrium state is compared with that of a suspension initially prepared in a nonequilibrium sheared state. Two classes of Mpemba effect are identified, the normal and the anomalous one. The former is generic, in the sense that the kinetic temperature starting from a cold nonequilibrium sheared state is overtaken by the one starting from a hot quasiequilibrium state, due to the absence of initial viscous heating in the latter, resulting in a faster initial cooling. The anomalous Mpemba effect is opposite to the normal one since, despite the initial slower cooling of the nonequilibrium sheared state, it can eventually overtake an initially colder quasiequilibrium state. The theoretical results based on kinetic theory agree with those obtained from event-driven simulations for inelastic hard spheres. It is also confirmed the existence of the inverse Mpemba effect, which is a peculiar heating process, in these suspensions. More particularly, we find the existence of a mixed process in which both heating and cooling can be observed during relaxation.

Enskog kinetic theory of rheology for a moderately dense inertial suspension

The Enskog kinetic theory for moderately dense inertial suspensions under simple shear flow is considered as a model to analyze the rheological properties of the system. The influence of the background fluid on suspended particles is modeled via a viscous drag force plus a Langevin-like term defined in terms of the background temperature. In a previous paper [Hayakawa et al., Phys. Rev. E 96, 042903 (2017)10.1103/PhysRevE.96.042903], Grad's moment method with the aid of a linear shear-rate expansion was employed to obtain a theory which gave good agreement with the results of event-driven Langevin simulations of hard spheres for low densities and/or small shear rates. Nevertheless, the previous approach had a limitation of not being applicable to the high-shear-rate and high-density regime. Thus, in the present paper, we extend the previous work and develop Grad's theory including higher-order terms in the shear rate. This improves significantly the theoretical predictions, a quantitative agreement between theory and simulation being found in the high-density region (volume fractions smaller than or equal to 0.4).

Kinetic derivation of diffuse-interface fluid models

We present a full derivation of capillary fluid equations from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, and Dunn and Serrin's heat flux as well as viscous and heat dissipative fluxes. Starting from macroscopic equations obtained from the kinetic theory of dense gases, we use a second-order expansion of the pair distribution function in order to derive the diffuse interface model. The capillary extra terms and the capillarity coefficient are then associated with intermolecular forces and the pair interaction potential.

Shear jamming, discontinuous shear thickening, and fragile states in dry granular materials under oscillatory shear

We numerically study the linear response of two-dimensional frictional granular materials under oscillatory shear. The storage modulus G^{'} and the loss modulus G^{''} in the zero strain rate limit depend on the initial strain amplitude of the oscillatory shear before measurement. The shear jammed state (satisfying G^{'}>0) can be observed at an amplitude greater than a critical initial strain amplitude. The fragile state is defined by the emergence of liquid-like and solid-like states depending on the form of the initial shear. In this state, the observed G^{'} after the reduction of the strain amplitude depends on the phase of the external shear strain. The loss modulus G^{''} exhibits a discontinuous jump corresponding to discontinuous shear thickening in the fragile state.

Enskog kinetic theory for multicomponent granular suspensions

The Navier-Stokes transport coefficients of multicomponent granular suspensions at moderate densities are obtained in the context of the (inelastic) Enskog kinetic theory. The suspension is modeled as an ensemble of solid particles where the influence of the interstitial gas on grains is via a viscous drag force plus a stochastic Langevin-like term defined in terms of a background temperature. In the absence of spatial gradients, it is shown first that the system reaches a homogeneous steady state where the energy lost by inelastic collisions and viscous friction is compensated for by the energy injected by the stochastic force. Once the homogeneous steady state is characterized, a normal solution to the set of Enskog equations is obtained by means of the Chapman-Enskog expansion around the local version of the homogeneous state. To first order in spatial gradients, the Chapman-Enskog solution allows us to identify the Navier-Stokes transport coefficients associated with the mass, momentum, and heat fluxes. In addition, the first-order contributions to the partial temperatures and the cooling rate are also calculated. Explicit forms for the diffusion coefficients, the shear and bulk viscosities, and the first-order contributions to the partial temperatures and the cooling rate are obtained in steady-state conditions by retaining the leading terms in a Sonine polynomial expansion. The results show that the dependence of the transport coefficients on inelasticity is clearly different from that found in its granular counterpart (no gas phase). The present work extends previous theoretical results for dilute multicomponent granular suspensions [Khalil and Garzó, Phys. Rev. E 88, 052201 (2013)10.1103/PhysRevE.88.052201] to higher densities.

Rheology of dilute cohesive granular gases

Rheology of a dilute cohesive granular gas is theoretically and numerically studied. The flow curve between the shear viscosity and the shear rate is derived from the inelastic Boltzmann equation for particles having square-well potentials in a simple shear flow. It is found that (i) the stable uniformly sheared state only exists above a critical shear rate and (ii) the viscosity in the uniformly sheared flow is almost identical to that for uniformly sheared flow of hard core granular particles. Below the critical shear rate, clusters grow with time, in which the viscosity can be approximated by that for the hard-core fluids if we replace the diameter of the particle by the mean diameter of clusters.