Non-Newtonian transport properties of an inertial suspension of inelastic rough hard spheres under simple shear flow are determined from the Boltzmann kinetic equation. The influence of the interstitial gas on rough hard spheres is modeled via a Fokker-Planck generalized equation for rotating spheres accounting for the coupling of both the translational and rotational degrees of freedom of grains with the background viscous gas. The generalized Fokker-Planck term is the sum of two ordinary Fokker-Planck differential operators in linear $\mathbf{v}$ and angular $\boldsymbolω$ velocity space. As usual, each Fokker-Planck operator is constituted by a drag force term (proportional to $\mathbf{v}$ and/or $\boldsymbolω$) plus a stochastic Langevin term defined in terms of the background temperature $T_\text{ex}$. The Boltzmann equation is solved by two different but complementary approaches: (i) by means of Grad's moment method, and (ii) by using a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to inelastic rough hard spheres. As occurs in the case of \emph{smooth} inelastic hard spheres, our results show that both the temperature and the non-Newtonian viscosity increase drastically with increasing the shear rate (discontinuous shear thickening effect) while the fourth-degree velocity moments also exhibit an $S$-shape. In particular, while high levels of roughness may slightly attenuate the jump of the viscosity in comparison to the smooth case, the opposite happens for the rotational temperature. As an application of these results, a linear stability analysis of the steady simple shear flow solution is also carried out showing that there are regions of the parameter space where the steady solution becomes linearly unstable.
A systematic microscopic theory for the rheology of dense non-Brownian suspensions characterized by the volume fraction $φ$ is developed. The theory successfully derives the critical behavior in the vicinity of the jamming point (volume fraction $φ_{J}$), for both the pressure $P$ and the shear stress $σ_{xy}$, i.e. $P \sim σ_{xy} \sim \dotγη_0 δφ^{-2}$, where $\dotγ$ is the shear rate, $η_0$ is the shear viscosity of the solvent, and $δφ= φ_J - φ> 0$ is the distance from the jamming point. It also successfully describes the behavior of the stress ratio $μ= σ_{xy}/P$ with respect to the viscou…
We develop a constitutive model allowing for the description of the rheology of two-dimensional soft dense suspensions above jamming. Starting from a statistical description of the particle dynamics, we derive, using a set of approximations, a non-linear tensorial evolution equation linking the deviatoric part of the stress tensor to the strain-rate and vorticity tensors. The coefficients appearing in this equation can be expressed in terms of the packing fraction and of particle-level parameters. This constitutive equation rooted in the microscopic dynamic qualitatively reproduces a number of…
Using mode-coupling theory, we derive a constitutive equation for the nonlinear rheology of dense colloidal suspensions under arbitrary time-dependent homogeneous flow. Generalizing previous results for simple shear, this allows the full tensorial structure of the theory to be identified. Macroscopic deformation measures, such as the Cauchy-Green tensors, thereby emerge. So does a direct relation between the stress and the distorted microstructure, illuminating the interplay of slow structural relaxation and arbitrary imposed flow. We present flow curves for steady planar and uniaxial elongati…
We develop a statistical framework for the rheology of dense, non-Brownian suspensions, based on correlations in a space representing forces, which is dual to position space. Working with the ensemble of steady state configurations obtained from simulations of suspensions in two dimensions, we find that the anisotropy of the pair correlation function in force space changes with confining shear stress ($σ_{xy}$) and packing fraction ($ϕ$). Using these microscopic correlations, we build a statistical theory for the macroscopic friction coefficient: the anisotropy of the stress tensor, $μ= σ_{xy}…
Considering the rheology of two-dimensional soft suspensions above the jamming density, we derive a tensorial constitutive model from the microscopic particle dynamics. Starting from the equation governing the $N$-particle distribution, we derive an evolution equation for the stress tensor. This evolution equation is not closed, as it involves the pair and three-particle correlation functions. To close this equation, we first employ the standard Kirkwood closure relation to express the three-particle correlation function in terms of the pair correlation function. Then we use a simple and physi…
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)], 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 s…
There is now convincing evidence that inter-particle frictional contacts are essential for observing shear-thickening in concentrated suspensions of compact particles. While this has inspired many strategies to tailor the rheology in these systems, in the more general case, viz-a-viz suspensions of anisotropic particles, the mechanism of shear-thickening remains unclear. Here through simultaneous measurements of the bulk viscosity and the first Normal stress difference, we show that strong shear thickening in suspensions of colloidal rods is accompanied by large positive normal stresses, indic…
We formulate a microscopic, force-level, activated dynamics-based statistical-mechanical theory for the continuous startup nonlinear shear-rheology of ultra-dense glass-forming hard-sphere fluids and colloidal suspensions in the context of the ECNLE approach. Activated structural relaxation is described as a coupled local-nonlocal event involving caging and longer-range collective elasticity which controls the characteristic stress relaxation time. Theoretical predictions for the deformation-induced mobility enhancement, onset of relaxation acceleration at low values of stress, strain, or shea…
We study the rheology of cornstarch suspensions, a dense system of non-Brownian particles that exhibits shear thickening, i.e. a viscosity that increases with increasing shear rate. Using MRI velocimetry we show that the suspension has a yield stress. From classical rheology it follows that as a function of the applied stress the suspension is first solid (yield stress), then liquid and then solid again when it shear thickens. The onset shear rate for thickening is found to depend on the measurement geometry: the smaller the gap of the shear cell, the lower the shear rate at which thickening o…
The Granular Integration Through Transients (GITT) formalism gives a theoretical description of the rheology of moderately dense granular flows and suspensions. In this work, we extend the GITT equations beyond the case of simple shear flows studied before. Applying this to the particular example of extensional flows, we show that the predicted behavior is somewhat different from that of the more frequently studied simple shear case, as illustrated by the possibility of non monotonous evolution of the effective friction coefficient $μ$ with the inertial number $\mathcal{I}$. By the reduction o…
Tunable shear thickening in active frictional non-Brownian suspensions
School of Engineering, University of Edinburgh（爱丁堡大学工程学院）；Department of Physics, Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校物理系）
Colloidal suspensions have a rich rheology and can exhibit shear-thinning as well as shear-thickening. Numerical simulations recently suggested that shear-thickening may be attributed to the inertia of the colloids, besides the hydrodynamic interactions between them. Here, we consider the ideal limit of a dense bath of soft colloids following an underdamped Langevin dynamics. We use a mean-field equation for the colloidal density to get an analytical expression of the drag force felt by a probe pulled at constant velocity through the suspension. Our results show that inertia can indeed induce …
We consider the nonlinear rheology of dense colloidal suspensions under a time-dependent simple shear flow. Starting from the Smoluchowski equation for interacting Brownian particles advected by shearing (ignoring fluctuations in fluid velocity) we develop a formalism which enables the calculation of time-dependent, far-from-equilibrium averages. Taking shear-stress as an example we derive exactly a generalized Green-Kubo relation, and an equation of motion for the transient density correlator, involving a three-time memory function. Mode coupling approximations give a closed constitutive equa…
The dynamical mechanisms controlling the rheology of dense suspensions close to jamming are investigated numerically, using simplified models for the relevant dissipative forces. We show that the velocity fluctuations control the dissipation rate and therefore the effective viscosity of the suspension. These fluctuations are similar in quasi-static simulations and for finite strain rate calculations with various damping schemes. We conclude that the statistical properties of grain trajectories -- in particular the critical exponent of velocity fluctuations with respect to volume fraction ϕ-- o…
