Particles suspended in a Newtonian fluid raise the viscosity and also generally give rise to a shear-rate dependent rheology. In particular, pronounced shear thickening may be observed at large solid volume fractions. In a recent article (R. Seto, R. Mari, J. F. Morris, and M. M. Denn., Phys. Rev. Lett., 111:218301, 2013) we have considered the minimum set of components to reproduce the experimentally observed shear thickening behavior, including Discontinuous Shear Thickening (DST). We have found frictional contact forces to be essential, and were able to reproduce the experimental behavior by a simulation including this physical ingredient along with viscous lubrication. In the present article, we thoroughly investigate the effect of friction and express it in the framework of the jamming transition. The viscosity divergence at the jamming transition has been a well known phenomenon in suspension rheology, as reflected in many empirical laws for the viscosity. Friction can affect this divergence, and in particular the jamming packing fraction is reduced if particles are frictional. Within the physical description proposed here, shear thickening is a direct consequence of this effect: as the shear rate increases, friction is increasingly incorporated as more contacts form, leading to a transition from a mostly frictionless to a mostly frictional rheology. This result is significant because it shifts the emphasis from lubrication hydrodynamics and detailed microscopic interactions to geometry and steric constraints close to the jamming transition.

Discontinuous shear thickening (DST) observed in many dense athermal suspensions has proven difficult to understand and to reproduce by numerical simulation. By introducing a numerical scheme including both relevant hydrodynamic interactions and granularlike contacts, we show that contact friction is essential for having DST. Above a critical volume fraction, we observe the existence of two states: a low viscosity, contactless (hence, frictionless) state, and a high viscosity frictional shear jammed state. These two states are separated by a critical shear stress, associated with a critical shear rate where DST occurs. The shear jammed state is reminiscent of the jamming phase of granular matter. Continuous shear thickening is seen as a lower volume fraction vestige of the jamming transition.Suspensions of particles at high volume fraction of solid, often termed dense suspensions, have a rich nonNewtonian rheology. This is particularly striking for the simple system of nearly rigid particles in a Newtonian fluid, which exhibits shear thinning, shear thickening, and normal stresses, the last associated with strong microstructural distortion, despite the dominant influence played in such mixtures by viscous (Stokes-flow) fluid mechanics [1]. The phenomenon of discontinuous shear thickening (DST) (see [2][3][4][5] and references therein) is especially fascinating. Suspensions exhibiting DST flow relatively easily with slow stirring, but become highly viscous or even seemingly solid if one tries to stir them rapidly. In a rheometer, the transition is seen at a critical shear rate for a given volume fraction. It is often found that DST is completely reversible [6]. DST typically occurs for a volume fraction that exceeds a threshold value φ c , which depends on the nature of the suspended particles: increased nonsphericity or surface roughness seem to lower φ c . Continuous shear thickening (CST) is observed below φ c , and becomes weaker with decreasing volume fraction. Although counterintuitive, the abrupt or discontinuous increase of viscosity with increase of shear rate is a generic feature of dense suspensions [3,7], occurring in both Brownian (colloidal) and non-Brownian suspensions. This ubiquity suggests the possibility of a single mechanistic basis applicable to the various types of suspension. DST has yet to be reproduced by a simulation method which can unambiguously point to the essential physical features necessary for its observation. This Letter presents a novel method able to identify these features.Several possible mechanisms have been proposed as the origin of DST. An order-disorder mechanism [8-10] describes a low shear rate ordered flow with few interactions between particles that becomes unstable at high shear rates and evolves to a disordered, highly interacting viscous flow. A hydroclustering [6,[11][12][13][14][15] or (hydro)contact network [16,17] mechanism attributes the thickening to clusters of particles "glued" together by the lubrication singularity. The competition between a force (Brown...

The role of normal stresses in causing particle migration and macroscopic spatial variation of the particle volume fraction in a mixture of rigid neutrally buoyant spherical particles suspended in Newtonian fluid is examined for curvilinear shear flows. The problem is studied for monodisperse noncolloidal Stokes-flow suspensions, i.e., for conditions of low-Reynolds-number flow and infinite Péclet number, Pe ϭ O(␥ a 3 /kT), where is the suspending fluid viscosity, ␥ is the shear rate, a is the particle radius, and kT is the thermal energy. Wide-gap Couette, parallel-plate torsional, and cone-and-plate torsional flows are studied. The entire dependence of the compressive shear-induced normal stresses is captured by a ''normal stress viscosity'' n (), which vanishes ͑as 2) at ϭ 0 and diverges at maximum packing in the same fashion as does the shear viscosity s (). Anisotropy of the normal stresses arising from the presence of the particles is modeled as independent of , so that ratios of any two particle contributions to the bulk normal stress components are constants, ⌺ 22 p /⌺ 11 p ϭ 2 and ⌺ 33 p /⌺ 11 p ϭ 3 ; the standard convention of ͑1,2,3͒ denoting the ͑flow, gradient, vorticity͒ directions is used so that, for example, ⌺ 11 p is the normal component of the particle stress ⌺ p in the flow direction. Predictions for the steady and unsteady flows are presented to demonstrate the influence of variation of the normal stress anisotropy parameters 2 and 3 , the rheological functions s and n , and the sedimentation hindrance function used to represent the resistance to relative motions of the phases during migration. Comparison with available experimental data shows that a single set of parameters for the rheological model is able to describe all qualitative features of the observed migrations in the flows considered.

The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe → ∞ under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe =γa 2 /2D is the Péclet number withγ the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe −1 ) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe 0.78 ) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a 3 Pe −1 ) layer volume, vanish as Pe −0.22 as Pe → ∞. If, however, particles are maintained at a minimum separation of 2b, with b > a, by a hard-sphere force there is also a boundary layer of thickness of O(aPe −1 ), but the asymmetry of the pair-distribution function for this situation is O(Pe), with an excess of particles along the compressional axes. The product of the asymmetric pair-distribution function and the thin boundary layer volume is now O(1) (with dependence on b/a) as Pe → ∞, thus yielding non-Newtonian rheology with normal stresses scaling as ηγ, where η is the fluid viscosity. For a dilute suspension without hydrodynamic interactions in a general linear flow, the bulk stress resulting from pair interactions is proportional to ηγφ 3 n is the thermodynamic volume fraction. Including hydrodynamic interactions, the hydrodynamic normal stress differences are O(ηγφ 2 ). The O(φ 2 ) hydrodynamic contribution to the viscosity due to the boundary layer is shear-thickening. The broken symmetry and boundarylayer structure also yield a shear-induced self-diffusivity of O(γa 2 φ) as Pe → ∞. At higher concentrations the boundary-layer structure is the same, with the pairdistribution function outside the boundary layer changed from its dilute value to a concentration-dependent function g ∞ (r; φ), which must be determined selfconsistently; the function g ∞ (r; φ) is not determined here. The appropriate Péclet number at high concentration is based on the concentration-dependent short-time selfdiffusivityPe =γa 2 /2D s 0 (φ). The stress contributions from the boundary layer scale asis the pair-distribution function at contact, and are argued to be dominant at high concentrations. The long-time self-diffusivity arising from the boundary-layer structure is predicted to scale asγa 2 φg ∞ (2; φ). † Present address: School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA. 104 J. F. Brady and J. F. Morris IntroductionThis work addresses the microstructure of low-Reynolds-number suspensions in strong shearing flow where the influence of Brownian motion is weak. We have analy...