Dense non-Brownian Suspensions Research Articles

Heterogeneous solvent dissipation coupled with particle rearrangement in shear-thinning non-Brownian suspensions.

Dense non-Brownian suspensions exhibit significant shear thinning, although a comprehensive understanding of the full scope of this phenomenon remains elusive. This study numerically reveals intimate heterogenous coupled dynamics between many-body particle motions and solvent hydrodynamics in shear-thinning non-Brownian suspensions. In our simulation systems, we do not account for frictional contact forces, reflecting experimental conditions under low shear rates where shear thinning occurs, while hydrodynamic interactions are directly incorporated using the Smoothed Profile Method. We demonstrate the spatially correlated viscous dissipation and particle motions; they share the same characteristic length, which decreases with increasing shear rate. We further show that, at lower shear rates, significant particle density changes are induced against the incompressibility of the solvent, suggesting the cooperative creation and annihilation of gaps and flow channels. We discuss that hydrodynamic interactions may substantially restrict particle rearrangements even in highly dense suspensions, influencing the quantitative aspects of macroscopic rheology.

Rheology of Dense Suspensions under Shear Rotation.

Dense non-Brownian suspensions exhibit a spectacular and abrupt drop in viscosity under change of shear direction, as revealed by shear inversions (reversals) or orthogonal superposition. Here, we introduce an experimental setup to systematically explore their response to shear rotations, where one suddenly rotates the principal axes of shear by an angle θ, and measure the shear stresses with a biaxial force sensor. Our measurements confirm the genericness of the transient decrease of the resistance to shear under unsteady conditions. Moreover, the orthogonal shear stress, which vanishes in steady state, takes non-negligible values with a rich θ dependence, changing qualitatively with solid volume fraction ϕ and resulting in a force that tends to reduce or enhance the direction of flow for small or large ϕ. These experimental findings are confirmed and rationalized by particle-based numerical simulations and a recently proposed constitutive model. We show that the rotation angle dependence of the orthogonal stress results from a ϕ-dependent interplay between hydrodynamic and contact stresses.

Two-fluid modeling of heat transfer in flows of dense suspensions

We develop a two-fluid model (TFM) for heat transfer in dense non-Brownian suspensions. Specifically, we propose closure relations for the inter-phase heat transfer coefficient and the thermal diffusivity of the particle phase based on calibration against experimental data. The model is then employed to simulate non-isothermal flow in an annular Couette cell. We find that, when the shear rate is controlled by the rotation of the inner cylinder, both the shear and thermal gradients are responsible for particle migration. Within the TFM framework, we identify the origin and functional form of a “thermo-rheological” migration force that rationalizes our observations. Furthermore, we apply our model to flow in eccentric Couette cells. Our simulations reveal that the system’s heat transfer coefficient is affected by both the classic shear-induced migration of particles and the newly identified thermo-rheological migration effect.

Directional shear jamming of frictionless ellipses.

In this work we study shear reversals of dense non-Brownian suspensions composed of cohesionless elliptical particles. By numerical simulations, we show that a new fragility appears for frictionless ellipses in the flowing states, where particles can flow indefinitely in one direction at applied shear stresses but shear jam in the other direction upon shear stress reversal. This new fragility, absent in the isotropic particle case, is linked to the directional order of the elongated particles at steady shear and its reorientation at shear stress reversal, which forces the suspensions to pass through a more disordered state with an increased number of contacts in which it might get arrested.

Evolution of granular materials under isochoric cyclic simple shearing.

By means of 3D particle dynamics simulations, we analyze the microstructure of granular materials subjected to isochoric (constant volume) cyclic shearing, which drives the system towards a liquefaction state characterized by loops of jamming-unjamming transition with periodic loss of strength and irreversible accumulation of shear strain. We first show that the macroscopic response obtained by these simulations agrees well with the most salient features of the well-known cyclic behavior of granular materials both before and after liquefaction. Then we investigate the evolution of particle connectivity, force transmission, and anisotropies of contact and force networks. The onset of liquefaction is marked by partial collapse of the force-bearing network with rapid drop of the coordination number and nonrattler fraction of particles, and significant broadening of the contact force probability density function, which begins in the preliquefaction period. We find that the jamming transition in each cycle occurs for a critical value of the coordination number that can be interpreted as the percolation threshold of the contact network and appears to be independent of the initial mean stress, void ratio, and cyclic shear amplitude. We show that upon unjamming in each cycle an isotropic loss of contacts occurs and is followed by the development of high contact anisotropy and a large proportion of particles with only two or three contacts. The higher mobility of the particles also involves a lower degree of frustration of particle rotations and thus lower friction mobilization and tangential force anisotropy. These findings are relevant to both undrained cyclic deformations of saturated soils and rheology of dense non-Brownian suspensions where volume change is coupled with pore liquid drainage conditions.

A review of the second normal-stress difference; its importance in various flows, measurement techniques, results for various complex fluids and theoretical predictions

Shear flow is ubiquitous. Not only is it arguably the most widely-used deformation type to characterise complex fluids in rheological studies but also, in practice, the deformation most likely to occur in the great majority of flows, e.g. involving fluid transport through pipes or conduits. In steady simple shear flow the rheological properties of a complex fluid are completely characterised in just three material functions; the variation with shear rate of the shear viscosity and the so-called first and second normal-stress differences. Despite requiring only three material functions to be completely characterised, most shear-flow rheological characterisations are usually restricted simply to the shear viscosity and, at best, the variation of the first normal-stress difference N1 with shear rate. The second normal-stress difference N2 remains very much neglected. For dilute polymer solutions where this quantity may be negligibly small in comparison to the first normal-stress difference, such neglect is justified but for a whole range of complex fluids – indeed even polymer solutions outside of the dilute regime and especially melts – it is not clear that N2 may be safely disregarded. Indeed, in this review article we spotlight a number of flows where second normal-stress differences are of importance and potentially major consequence. Following this attention, we review the many experimental techniques which have been proposed for its measurement and survey the available literature for measurements of this quantity for various complex fluids including the aforementioned polymeric solutions, melts, liquid crystals, dense non-Brownian suspensions (both with Newtonian and complex fluid bases), semi-dilute wormlike micellar fluids and magnetorheological fluids. Theoretical predictions for N2 from various commonly-used continuum constitutive equations – primarily from the polymer literature – are also given and their asymptotic predictions at low and high shear rates compared. Finally, we end with a brief summary and outlook.

Shear thickening in dense non-Brownian suspensions: Viscous to inertial transition

We present an experimental study on the viscous to inertial mode of shear thickening in dense non-Brownian suspensions. We design a model suspension consisting of monosized spherical particles within a Newtonian suspending fluid. We develop a protocol for the rheological characterization of dense suspensions using the conventional rheometry technique. Our results provide constitutive laws for suspensions with solid volume fractions close to jamming when both viscous and inertial effects at the particle scale are present. We perform atomic force microscopy to measure forces between the particles immersed in the suspending fluid and show that our system of study corresponds to the frictionless regime of dense suspensions in which viscous and collisional forces dissipate the energy. Finally, we show that the proposed empirical constitutive laws, when approaching jamming, predict the dynamics of dense suspensions in a transient boundary driven flow.

Simulation of dense non-Brownian suspensions with the lattice Boltzmann method: shear jammed and fragile states.

Dense non-Brownian suspensions, including both hydrodynamic interactions and frictional contacts between particles, are numerically studied under simple and oscillatory shears in terms of the lattice Boltzmann method. We successfully reproduce the discontinuous shear thickening (DST) under a simple shear for bulk three-dimensional systems. For our simulation of an oscillatory shear in a quasi-two-dimensional system, we measure the mechanical response after the reduction of the strain amplitude from the initial oscillations. Here, we find the existence of a shear-jammed state under this protocol in which the storage modulus G' is only finite for high initial strain amplitude γI0. We also find the existence of a fragile state in which both fluid-like and solid-like responses can be detected for an identical area fraction and an initial strain amplitude γI0 depending on the initial phase Θ (or the asymmetricity of the applied strain) of the oscillatory shear. We also observe a DST-like behavior under the oscillatory shear in the fragile state. Moreover, we find that the stress anisotropy becomes large in the fragile state. Finally, we confirm that a stress formula based on the angular distribution of the contact force recovers the contact contributions to the stress tensors for both simple and oscillatory shears with large strains.

Theory for the rheology of dense non-Brownian suspensions: divergence of viscosities and– rheology

A systematic microscopic theory for the rheology of dense non-Brownian suspensions characterized by the volume fraction$\unicode[STIX]{x1D711}$is developed. The theory successfully derives the critical behaviour in the vicinity of the jamming point (volume fraction$\unicode[STIX]{x1D711}_{J}$), for both the pressure$P$and the shear stress$\unicode[STIX]{x1D70E}_{xy}$, i.e. $P\sim \unicode[STIX]{x1D70E}_{xy}\sim \dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D702}_{0}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}^{-2}$, where$\dot{\unicode[STIX]{x1D6FE}}$is the shear rate,$\unicode[STIX]{x1D702}_{0}$is the shear viscosity of the solvent and$\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{J}-\unicode[STIX]{x1D711}>0$is the distance from the jamming point. It also successfully describes the behaviour of the stress ratio$\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70E}_{xy}/P$with respect to the viscous number$J=\dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D702}_{0}/P$.

Microstructural description of shear-thickening suspensions

Dynamic particle-scale numerical simulations are used to study the variation of microstructure with shear stress during shear thickening in dense non-Brownian suspensions. The microscale information is used to characterize the differences between the shear thickened (frictional) and non-thickened (lubricated, frictionless) states. Here, we focus on the force and contact networks and study the evolution of associated anisotropies with increase in shear stress. The force and contact networks are both more isotropic in the shear-thickened state than in non-thickened state. We also find that both force and structural anisotropies are rate independent for both low and high stress, while they are rate (or stress) dependent for the intermediate stress range where the shear thickening occurs. This behavior is similar to the evolution of viscosity with increasing stress, showing a clear correlation between the microstructure and the macroscopic rheology.

Shear thickening regimes of dense non-Brownian suspensions.

We propose a unifying rheological framework for dense suspensions of non-Brownian spheres, predicting the onsets of particle friction and particle inertia as distinct shear thickening mechanisms, while capturing quasistatic and soft particle rheology at high volume fractions and shear rates respectively. Discrete element method simulations that take suitable account of hydrodynamic and particle-contact interactions corroborate the model predictions, demonstrating both mechanisms of shear thickening, and showing that they can occur concurrently with carefully selected particle surface properties under certain flow conditions. Microstructural transitions associated with frictional shear thickening are presented. We find very distinctive divergences of both microstructural and dynamic variables with respect to volume fraction in the thickened and non-thickened states.

Flow regime transitions in dense non-Brownian suspensions: rheology, microstructural characterization, and constitutive modeling.

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 the 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 rheology is explored. The transitions between rheological regimes are associated with the evolving contribution of lubrication to the suspension stress. Transitions in microscopic phenomena, such as interparticle force distribution, fabric, and correlation length are found to correspond to those in the macroscopic flow. Motivated by the bulk rheology, a constitutive model is proposed combining a viscous pressure term with a dry granular model presented by Chialvo et al. [Phys. Rev. E 85, 021305 (2012)]. The model is shown to successfully capture the flow regime transitions.

Discontinuous Shear Thickening in Dense non-Brownian Suspensions

Discontinuous Shear Thickening in Dense non-Brownian Suspensions

Discontinuous Shear Thickening without Inertia in Dense Non-Brownian Suspensions

A consensus is emerging that discontinuous shear thickening (DST) in dense suspensions marks a transition from a flow state where particles remain well separated by lubrication layers, to one dominated by frictional contacts. We show here that reasonable assumptions about contact proliferation predict two distinct types of DST in the absence of inertia. The first occurs at densities above the jamming point of frictional particles; here, the thickened state is completely jammed and (unless particles deform) cannot flow without inhomogeneity or fracture. The second regime shows strain-rate hysteresis and arises at somewhat lower densities, where the thickened phase flows smoothly. DST is predicted to arise when finite-range repulsions defer contact formation until a characteristic stress level is exceeded.

Dynamics of non-Brownian fiber suspensions under periodic shear.

We report experiments studying the dynamics of dense non-Brownian fiber suspensions subjected to periodic oscillatory shear. We find that periodic shear initially causes fibers to collide and to undergo irreversible diffusion. As time progresses, the fibers tend to orient in the vorticity direction while the number of collisions decreases. Ultimately, the system goes to one of two steady states: an absorbing steady state, where collisions cease and the fibers undergo reversible trajectories; an active state, where fibers continue to collide causing them to diffuse and undergo irreversible trajectories. Collisions between fibers can be characterized by an effective volume fraction Φ with a critical volume fraction Φc that separates absorbing from active (diffusing) steady states. The effective volume fraction Φ depends on the mean fiber orientation and thus decreases in time as fibers progressively orient under periodic shear. In the limit that the temporal evolution of Φ is slow compared to the activity relaxation time τ, all the data for all strain amplitudes and all concentrations can be scaled onto a single master curve with a functional dependence well-described by t(-β/ν)R(e(-t)R), where tR is the rescaled time. As Φ → Φc, τ diverges. Therefore, for experiments in which Φ(t) starts above Φc but goes to a steady state below Φc, departures from scaling are observed for Φ very near Φc. The critical exponents are measured to be β = 0.84 ± 0.04 and ν = 1.1 ± 0.1, which is consistent with the Manna universality class for directed percolation.