Mild assumptions for the derivation of Einstein's effective viscosity formula

Abstract

We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of n rigid balls, n ≫ 1, set in a volume of size 1. So far, most justifications were carried under a strong assumption on the minimal distance between the balls:We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition.

The Influence of Einstein's Effective Viscosity on Sedimentation at Very Small Particle Volume Fraction

“…This result has been generalized to polydispersed particles of more general shape by Hillairet and Wu in [HW19], where they also removed any condition φ N → 0. Gérard-Varet [GV19] and Gérard-Varet and the first author [GVH20] were able to considerably relax the separation condition (H1) allowing to treat a large class of random particle configurations.…”

On Einstein's effective viscosity formula

“…• Einstein's formula: The low-density regime is understood as the smallness of the volume fraction of suspended particles, but we show that this diluteness assumption alone does not ensure the validity of Einstein's formula. We identify the relevant necessary assumption in terms of the relative size of the two-point intensity, viewed as some local independence condition, and we establish Einstein's formula in its most general form, thus recovering a recent result by Gérard-Varet and Höfer [22]. • Corrections and renormalization: We pursue the low-density expansion to higher orders in form of a cluster expansion in the spirit of e.g.…”

“…This type of low-density expansion was not new in physics at the time: it is indeed similar to the Clausius-Mossotti formula for the effective dielectric constant [44,45,9], the Maxwell formula for the effective conductivity in electrostatics [43], the Lorentz-Lorenz formula for the effective refractive index in optics [41,40], or the Bruggeman formula for the effective stiffness in linear elasticity [8]; we refer to [42] for the historical context. We focus here on the effective viscosity problem, which has attracted much attention in mathematics in the recent years [26,47,27,21,23,22]. The extension of our approach to other situations is straightforward and yields in particular drastic improvements over our previous results [13,10] on the effective conductivity problem; the adaptation is essentially transparent and is omitted for the sake of brevity.…”

Derivation of Batchelor-Green formula for random suspensions

“…Let us point out that, as in [16], part of the work carried in [15,17] is not related to homogenization. More precisely, one does not impose a priori some specific random stationary or periodic structure on the points x i : roughly, as shown in [17], a necessary and sufficient condition for the existence of a o(φ 2 ) effective approximate viscosity is the existence of the mean field limit at the r.h.s.…”

“…Many works, especially over the last two years, have been devoted to the justification of this claim, trying to identify milder and milder geometrical assumptions on the particles configuration under which particles interaction is indeed neglectible [29,26,20,28,21]. To our knowledge, justification of Einstein's formula under the current mildest requirements is found in [16]. Let us note that this work is of a deterministic nature: it does not use the existence of an effective viscosity, or some specific random structure of the set of particles.…”