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 salient features of the rheology of jammed soft suspensions, including the presence of yield stresses for the shear component of the stress and for the normal stress difference. More complex protocols like the relaxation after a preshear are also considered, showing a smaller stress after relaxation for a stronger preshear.
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…
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 explore the rheology predicted by a recently proposed constitutive model for jammed suspensions of soft elastic particles derived from particle-level dynamics [Cuny et al., Phys. Rev. Lett. 127, 218003 (2021)]. Our model predicts that the orientation of the anisotropy of the microstructure, governed by an interplay between advection and contact elasticity, plays a key role at yielding and in flow. It generates normal stress differences contributing significantly to the yield criterion and Trouton ratio. It gives rise to non-trivial transients such as stress overshoots in step increases of s…
This is a comment on the recent letter by Wang and Brady on "Constant stress and pressure rheology of colloidal suspensions", Phys. Rev. Lett. 115, 158301 (2015).
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}…
Suspensions of cornstarch in water exhibit strong dynamic shear-thickening. We show that partly replacing water by ethanol strongly alters the suspension rheology. We perform steady and non-steady rheology measurements combined with atomic force microscopy to investigate the role of fluid chemistry on the macroscopic rheology of the suspensions and its link with the interactions between cornstarch grains. Upon increasing the ethanol content, the suspension goes through a yield-stress fluid state and ultimately becomes a shear-thinning fluid. On the cornstarch grain scale, atomic force microsco…
We probe the three dimensional flow structure and rheology of gravitational (non-density matched) suspensions for a range of driving rates in a split-bottom geometry. We establish that for sufficiently slow flows, the suspension flows as if it were a dry granular medium, and confirm recent theoretical modeling on the rheology of split-bottom flows. For faster driving, the flow behavior is shown to be consistent with the rheological behavior predicted by the recently developed "inertial number approaches for suspension flows.
Using a novel bi-axial confocal rheoscope, we investigate the structure and rheology of sheared colloidal suspensions under confinement. Consistent with previous work [X. Cheng \textit{et al., Proc. Natl. Acad. Sci. U. S. A.}, 2011, 109, 63], we observe a vorticity-aligned string phase in moderate concentrated colloidal suspensions under uniaxial shear. Using bi-axial shear protocols, we directly manipulate the orientation and morphology of the string structures. Simultaneously, we measure the suspension rheology along both the flow and vorticity directions with a bi-axial force measurement de…
The dynamics and rheology of suspensions of fluid vesicles or red blood cells is investigated by a combination of molecular dynamics and mesoscale hydrodynamics simulations in two dimensions. The vesicle suspension is confined between two no-slip walls, which are driven externally to generate a shear flow with shear rate $\dotγ$. The flow behavior is studied as a function of $\dotγ$, the volume fraction of vesicles, and the viscosity contrast between inside and outside fluids. Results are obtained for the encounter and interactions of two vesicles, the intrinsic viscosity of the suspension, an…
Many dense particulate suspensions show a stress induced transformation from a liquid-like state to a solid-like shear jammed (SJ) state. However, the underlying particle-scale dynamics leading to such striking, reversible transition of the bulk remains unknown. Here, we study transient stress relaxation behaviour of SJ states formed by a well-characterized dense suspension under a step strain perturbation. We observe a strongly non-exponential relaxation that develops a sharp discontinuous stress drop at short time for high enough peak-stress values. High resolution boundary imaging and norma…
We present an experimental study of the rheology and structure of a confined suspension of deformable particles flowing in a quasi-two-dimensional Poiseuille flow. Thanks to a precise microfluidic viscosimetry technique combined with measurements of concentration profiles, our study provides the first experimental confirmation {with three-dimensional particles} of a strong relationship between structuring effects and rheology, previously only reported in numerical simulations of purely two-dimensional systems. In conditions where strong structuring effects take place due to confinement, the ev…
The rheology of suspensions of Brownian, or colloidal, particles (diameter $d \lesssim 1$ $μ$m) differs markedly from that of larger grains ($d \gtrsim 50$ $μ$m). Each of these two regimes has been separately studied, but the flow of suspensions with intermediate particle sizes (1 $μ\textrm{m} \lesssim d \lesssim 50$ $μ$m), which occur ubiquitously in applications, remains poorly understood. By measuring the rheology of suspensions of hard spheres with a wide range of sizes, we show experimentally that shear thickening drives the transition from colloidal to granular flow across the intermedia…
We study the rheology of confined suspensions of neutrally buoyant rigid monodisperse spheres in plane-Couette flow using Direct Numerical Simulations. We find that if the width of the channel is a (small) integer multiple of the sphere's diameter, the spheres self-organize into two-dimensional layers that slide on each other and the suspension's effective viscosity is significantly reduced. Each two-dimensional layer is found to be structurally liquid-like but their dynamics is frozen in time.
The phenomenon of shear-induced jamming is a factor in the complex rheological behavior of dense suspensions. Such shear-jammed states are fragile, i.e., they are not stable against applied stresses that are incompatible with the stress imposed to create them. This peculiar flow-history dependence of the stress response is due to flow-induced microstructures. To examine jammed states realized under constant shear stress, we perform dynamic simulations of non-Brownian particles with frictional contact forces and hydrodynamic lubrication forces. We find clear signatures that distinguish these fr…
Dense suspensions are widespread in nature, manufacturing and process engineering. Particle-based simulations have proven to be an invaluable complement to experimental rheological characterisation, serving as a virtual rheometer that enables rapid exploration of parameter space and detailed scrutiny of microscopic dynamics. To maximise the utility of such simulations, it can be advantageous to exploit pre-existing, well-optimised, well-documented codes. Here we provide a simple description of how to use LAMMPS to study the rheology of dense, granular suspensions.
We experimentally characterize the impact response of concentrated suspensions of cornstarch and water. We hypothesize that the dynamically jammed region that propagates ahead of the impactor is responsible for the strong stress response to impact when it spans between solid boundaries. Using surface imaging and particle tracking at the boundary opposite the impactor, we observed that a visible structure and particle flow at the boundary occur with a delay after impact. We show the delay time is about the same time as the the strong stress response, confirming that the strong stress response r…
We consider active suspensions in the isotropic phase subjected to a shear flow. Using a set of extended hydrodynamic equations we derive a variety of {\em analytical} expressions for rheological quantities such as shear viscosity and normal stress differences. In agreement to full-blown numerical calculations and experiments we find a shear thickening or -thinning behaviour depending on whether the particles are contractile or extensile. Moreover, our analytical approach predicts that the normal stress differences can change their sign in contrast to passive suspensions.
In everyday life we encounter many complex fluids, from shear-thinning paint and toothpaste to shear-thickening starch suspensions. The study of their properties offers an opportunity for students to relate sophisticated physical concepts to their everyday experience. Modern rheology uses expensive equipment impractical for the teaching laboratory. Here we describe a rudimentary rheometer suitable for student laboratories that can demonstrate and quantify discontinuous shear thickening, the most dramatic property of complex fluids, and use it to measure the properties of starch suspensions.
We study relaxation and rheology of dense athermal suspensions of frictionless particles close below the jamming density. Our key quantity, the relaxation time---determined from the exponential decay of the energy after the shearing has suddenly been switched off---is argued to be a determining factor behind the algebraic divergence of various quantities as the jamming density is approached from below. We also define and measure the ``dissipation time'', which is obtained directly in shearing simulations and find that it behaves similarly to the relaxation time. Comparing shear viscosity with …
Dense suspensions of hard particles in a Newtonian liquid can be jammed by shear when the applied stress exceeds a certain threshold. However, this jamming transition from a fluid into a solidified state cannot be probed with conventional steady-state rheology because the stress distribution inside the material cannot be controlled with sufficient precision. Here we introduce and validate a method that overcomes this obstacle. Rapidly propagating shear fronts are generated and used to establish well-controlled local stress conditions that sweep across the material. Exploiting such transient fl…
We provide a detailed derivation of a recently developed first-principles approach to calculating averages in systems of interacting, spherical Brownian particles under time-dependent flow. Although we restrict ourselves to flows which are both homogeneous and incompressible, the time-dependence and geometry (e.g. shear, extension) are arbitrary. The approximations formulated within mode-coupling theory are particularly suited to dense colloidal suspensions and capture the slow relaxation arising from particle interactions and the resulting glass transition to an amorphous solid. The delicate …
Shear flow of dense, non-Brownian suspensions is simulated using the discrete element method, taking particle contact and hydrodynamic lubrication into account. The resulting flow regimes are mapped in the parametric space of solid volume fraction, shear rate, fluid viscosity and particle stiffness. Below a critical volume fraction $ϕ_c$, the rheology is governed by the Stokes number, which distinguishes between viscous and inertial flow regimes. Above $ϕ_c$, a quasistatic regime exists for low and moderate shear rates. At very high shear rates, the $ϕ$ dependence is lost and soft particle rhe…
Recent studies have highlighted that oscillatory and time-dependent shear flows might help increase flowability of dense suspensions. While most focus has been on cross-flows we here study a simple two-dimensional suspensions where we apply simultaneously oscillatory and stationary shear along the same direction. We first show that the viscosities in this set-up significantly decrease with an increasing magnitude of the oscillations, contrary to previous claims. The decrease can be attributed to the large decrease in the number of contacts and an altered microstructure as the magnitude of the …
As a paradigmatic model of active fluids, bacterial suspensions show intriguing rheological responses drastically different from their counterpart colloidal suspensions. Although the flow of bulk bacterial suspensions has been extensively studied, the rheology of bacterial suspensions under confinement has not been experimentally explored. Here, using a microfluidic viscometer, we systematically measure the rheology of dilute E. coli suspensions under different degrees of confinement. Our study reveals a strong confinement effect: the viscosity of bacterial suspensions decreases substantially …
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…
Oscillatory shear has been widely used to study the rheological properties of suspensions under unsteady shear. Furthermore, recent works have shown that oscillatory flows can improve the flowability of dense suspensions. While most studies have been done under constant volume we here study oscillatory shear flows of a two-dimensional suspensions using a normal pressure-controlled set-up. To characterise the rheology, we introduce both a complex macroscopic friction coefficient $μ^{*}$, following the convention of the complex viscosity $η^{*}$, and a shear-rate averaged viscous number $J'$. Th…
We experimentally characterize the impact response of concentrated suspensions consisting of cornstarch and water. We observe that the suspensions support a large normal stress -- on the order of MPa -- with a delay after the impactor hits the suspension surface. We show that neither the delay nor the magnitude of the stress can yet be explained by either standard rheological models of shear thickening in terms of steady-state viscosities, or impact models based on added mass or other inertial effects. The stress increase occurs when a dynamically jammed region of the suspension in front of th…
Dense suspensions of soft colloidal particles display a broad range of physical and rheological properties which are still far from being fully understood. To elucidate the role of deformability on colloidal flow, we employ computer simulations to measure the apparent viscosity of a system of droplets of variable surface tension subjected to a pressure-driven flow. We confirm that our suspension generically undergoes discontinuous shear thinning, and determine the dependence of the onset of the discontinuity on surface tension. We find that the effective viscosity of the suspension is mainly d…
Using a simple model of self-propelled particle, the effective shear viscosity of a dilute, spatially homogeneous suspension of active particles is studied. We use formulation of non-linear Fokker-Planck equation to drive a kinetic description, including the effect of external flow field, rotary diffusion, and particle-particle hydrodynamic interactions in two dimensions. Analytical expressions are obtained for the stress tensor at small Peclet numbers, in a simple shear flow, up to second order of volume fraction of the swimmers, which shows the explicit dependence of shear viscosity on the d…
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…
The mixing of a powder of 10-50μm primary particles into a liquid to form a dispersion with the highest possible solid content is a common industrial operation. Building on recent advances in the rheology of such 'granular dispersions', we study a paradigmatic example of such powder incorporation: the conching of chocolate, in which a homogeneous, flowing suspension is prepared from an inhomogeneous mixture of particulates, triglyceride oil and dispersants. Studying the rheology of a simplified formulation, we find that the input of mechanical energy and staged addition of surfactants combine …
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 granular suspensions under a variety of extensional deformations and simple shear using numerical simulations. The viscosity and Trouton's ratio (the ratio of extensional to shear viscosity) are computed as functions of solids volume fraction $ϕ$ close to the limit of zero inertia. Suspensions of frictionless particles follow a Newtonian Trouton's ratio for $ϕ$ all the way up to $ϕ_0$, a universal jamming point that is independent of deformation type. In contrast, frictional particles lead to a deformation-type-dependent jamming fraction $ϕ_m$, which is largest for shear flows. Trouto…
Aqueous suspensions of Laponite, a system composed of disk-like nanoparticles, are found to develop optical birefringence over several days, well after the suspensions solidified due to jamming. The optical anisotropy is particularly enhanced near the air Laponite suspension interface over length-scales of several millimetres, which is beyond five orders of magnitude larger than the particle length scale, suggestive of large scale ordering influenced by the interface. The orientational order increases with time and is always greater for higher concentration of salt, higher concentration of Lap…
At the same solid volume fraction, bidisperse and polydisperse suspensions display lower viscosities, and weaker normal stress response, compared to monodisperse suspensions. The reduction of viscosity associated with size distribution can be explained by an increase of the maximum flowable, or jamming, solid fraction $ϕ_m$. In this work, dense suspensions are simulated under strong shearing, where thermal motion and repulsive forces are negligible, but we allow for particle contact with a mild frictional interaction with interparticle friction coefficient of $μ = 0.2$. Aspects of bidisperse s…
We simulate a densely jammed, athermal assembly of repulsive soft particles immersed in a solvent. Starting from an initial condition corresponding to a quench from a high temperature, we find non-trivial slow dynamics driven by a gradual release of stored elastic energy, with the root mean squared particle speed decaying as a power law in time with a fractional exponent. This decay is accompanied by the presence within the assembly of spatially localised and temporally intermittent `hot-spots' of non-affine deformation, connected by long-ranged swirls in the velocity field, reminiscent of the…
We explain the origins of yield stress, shear-thinning, and normal stress differences in rigid fiber suspensions. We investigate the interplay between the hydrodynamic, colloidal attractive and repulsive, and inter-fiber contact interactions. The shear-thinning viscosity and finite yield stress obtained from the computational model are in quantitative agreement with experiential results from the literature. In this study, we show that attractive interactions result in yield stress and shear-thinning rheology in the suspensions of rigid fibers. This is an important finding, given the ongoing di…
We numerically investigate the rheological properties of jammed frictionless granular materials under an oscillatory shear in terms of our simulation of the distinct element method. It is demonstrated that the constitutive relation between the shear stress and the shear strain of grains strongly depends on the amplitude and the frequency of the oscillatory shear. For a small amplitude region, the granular material behaves as a visco-elastic material characterized by the Kelvin-Voigt model, while it behaves as a yield stress liquid for a large amplitude region. We find that the rheology of grai…
