This paper presents three-dimensional numerical simulations of non-Brownian concentrated suspensions in a Couette flow at zero Reynolds number using a fictitious domain method. Contacts between particles are modelled using a discrete element method (DEM)-like approach, which allows for a more physical description, including roughness and friction. This work emphasizes the effect of friction between particles and its role on rheological properties, especially on normal stress differences. Friction is shown to notably increase viscosity and second normal stress difference $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}|N_2|$ and decrease $|N_1|$, in better agreement with experiments. The hydrodynamic and contact contributions to the overall particle stress are particularly investigated. This shows that the effect of friction is mostly due to the additional contact stress since the hydrodynamic stress remains unaffected by friction. Simulation results are also compared with experiments, such as normal stresses or effective friction coefficient $\mu (I_v)$, and the agreement is improved when friction is accounted for. This suggests that friction is operative in actual suspensions.

We propose to explain shear thinning behaviour observed in most concentrated non-Brownian suspensions by variable friction between particles. Considering the low magnitude of the forces experienced by the particles of suspensions under shear flow, it is first argued that rough particles come into solid contact through one or a few asperities. In such a few-asperity elastic-plastic contact, the friction coefficient is expected not to be constant but to decrease with increasing normal load. Simulations based on the Force Coupling Method and including such a load-dependent friction coefficient are performed for various particle volume fractions. The results of the numerical simulations are compared to viscosity measurements carried out on suspensions of polystyrene particles (40 µm in diameter) dispersed in a Newtonian silicon oil. The agreement is shown to be satisfactory. Furthermore, the comparison between the simulations conducted either with a constant or a load-dependent friction coefficient provides a model for the shearthinning viscosity. In this model the effective friction coefficient µ ef f is specified by the effective normal contact force which is simply proportional to the bulk shear stress. As the shear stress increases, µ ef f decreases and the jamming volume fraction increases, leading to the reduction of the viscosity. At last, using this model, we show that it is possible to evaluate the microscopic friction coefficient for each applied shear stress from the rheometric measurements.

No abstract

International audienceWe perform particle scale simulations of suspensions submitted to shear reversal. The simulations are based on the Force Coupling Method, adapted to account for short range lubrication interactions together with direct contact forces between particles, including surface roughness, contact elasticity and solid friction. After shear reversal, three consecutive steps are identified in the viscosity transient: an instantaneous variation, followed by a rapid contact force relaxation, and finally a long time evolution. The separated contributions of hydrodynamics and contact forces to the viscosity are investigated during the transient, allowing a qualitative understanding of each step. In addition, the influence of the contact law parameters (surface roughness height and friction coefficient) on the transient are evaluated. Concerning the long time transient, the difference between the steady viscosity and minimum viscosity is shown to be proportional to the contact contribution to the steady viscosity, allowing in principle easy determination of the latter in experiments. The short time evolution is studied as well. After the shear reversal, the contact forces vanish over a strain that is very short compared to the typical strain of the long time transient, allowing to define an apparent step between the viscosity before shear reversal and after contact force relaxation. This step is shown to be an increasing function of the friction coefficient between particles. Two regimes are identified as a function of the volume fraction. At low volume fraction, the step is small compared to the steady contact viscosity, in agreement with a particle pair model. As the volume fraction increases, the value of the viscosity step increases faster than the steady contact viscosity, and, depending on the friction coefficient, may approach it

Low Reynolds number concentrated suspensions do exhibit an intricate physics which can be partly unraveled by the use of numerical simulation. To this end, a Lagrange multiplier-free fictitious domain approach is described in this work. Unlike some methods recently proposed, the present approach is fully Eulerian and therefore does not need any transfer between the Eulerian background grid and some Lagrangian nodes attached to particles. Lubrication forces between particles play an important role in the suspension rheology and have been properly accounted for in the model. A robust and effective lubrication scheme is outlined which consists in transposing the classical approach used in Stokesian Dynamics to our present direct numerical simulation. This lubrication model has also been adapted to account for solid boundaries such as walls. Contact forces between particles are modeled using a classical Discrete Element Method (DEM), a widely used method in granular matter physics. Comprehensive validations are presented on various one-particle, two-particle or three-particle configurations in a linear shear flow as well as some O(10 3 ) and O(10 4 ) particle simulations.