Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions

Abstract

The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions.The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients Dy and Dz, respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of Dy on ϕ, from ϕ2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for Dz but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.

Rheology of sheared suspensions of rough frictional particles

“…Such surface asperities promote an early contact and impose the surface-to-surface separation between particles and the magnitude of lubrication stress consequently. Surface roughness can therefore significantly modify the rheological properties and the microstructure of suspensions, as confirmed by many numerical studies (Sierou & Brady 2002;Drazer et al 2002Drazer et al , 2004DaCunha & Hinch 1996). Theoretical studies in dilute regimes show that large roughness can decrease viscosity as well as increase |N 1 | and |N 2 | (Wilson 2005;Wilson & Davis 2002;Davis et al 2003;Zarraga & Leighton Jr 2001).…”

“…Among the most widely employed method for low Reynolds number suspensions is the Stokesian Dynamics (SD) (Bossis & Brady 1984;Brady & Bossis 1985, 1988. SD has provided among the most relevant results in the field of suspensions (Drazer et al 2002(Drazer et al , 2004Sierou & Brady 2002). This technique resembles a molecular dynamics method in which the driving forces are the hydrodynamic interactions, possibly complemented with Brownian or collisional forces.…”

Shear-induced self-diffusion in non-colloidal suspensions

“…viscosity and normal stress differences), especially for high-volume fractions, simulations using other parameters gave very similar results for the self-diffusion tensor. (At very low-volume fractions, however, where two-particle interactions dominate, the range and amplitude of the repulsive force can strongly influence the self-diffusivities (Drazer et al 2002)). …”

“…It has been suggested in the past (Marchioro & Acrivos 2001) that even in the absence of an interparticle force, diffusive-like characteristics could be present owing to the chaotic nature of the many-body hydrodynamic interactions. If that is indeed the case, the order φ correction to the diffusivity might not be dominant even for extremely small volume fractions (Drazer et al 2002).…”

Microstructure and velocity fluctuations in sheared suspensions

“…Thus, it is clear that the spurious contribution to the measured velocity fluctuations due to the angular rotation of the spheres eventually becomes important, and even dominant, at low enough concentrations. For example, if the scattering sites were distributed uniformly inside the spheres, which rotate with mean angular velocity ω 3 , the measured previous work, proved sufficiently long to allow us to study the system in the asymptotic, fully developed steady state (Drazer et al 2002).…”