Sedimentation of random suspensions and the effect of hyperuniformity

Abstract

This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension d = 3 should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was put forward later on to explain additional screening of this divergence in steadystate observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature. The main difficulty is to account for the nonlinear multibody hydrodynamic interactions in a context when stochastic cancellations are crucial, and to show how the improvement provided by hyperuniformity on the linearized problem is not destroyed by the nonlinearity (as it would be in stochastic homogenization of linear elliptic equations, for instance). On the one hand, in order to analyze stochastic cancellations in this nonlinear setting, we introduce a suitable functional-analytic version of hyperuniformity, which is of independent interest. On the other hand, we appeal to a new annealed regularity theory for the steady Stokes equation describing the viscous fluid in presence of a random suspension. While all previous works build on deterministic approaches like the reflection method or other perturbative ideas to analyze interactions in dilute regimes, we rather appeal to a non-perturbative approach that takes advantage of randomness and yields optimal regularity properties upon averaging over the ensemble of particle positions. This is inspired by recent achievements in the quantitative theory of stochastic homogenization of divergence-form linear elliptic equations. As a corollary, we establish a homogenization result for a steady Stokes fluid in a finite tank with a dense suspension of small sedimenting particles, and we define the associated effective viscosity.

Derivation of Batchelor-Green formula for random suspensions

“…First, our setting is not restricted to the random deletion case, as g k is allowed to depend on φ (and is even fully arbitrary for k ≥ 6). It is neither covered by [11,Theorem 7]: the mixing conditions that we use, expressed through the asymptotic behaviour of the first correlation functions of the process, seem a bit weaker than the α-mixing used there. Furthermore, the general bounds given in [11,Theorem 7] would only imply in our context µ 2 = O(| log φ|), to be compared to the optimal bound µ 2 = O(1) in Proposition 2.2.…”

“…While such result has been known for long in the context of the Laplace equation [25, chapter 3], to our knowledge, the Stokes case was only analyzed in the recent paper [9]. See [5,2,14,7,18,11] for other stochastic homogenization results in fluid mechanics. In [9], the authors consider a stationary and ergodic point process (x i ) i∈N over R 3 , and B i,ε := εB i , where B i := B(x i , 1) is the closed unit ball centered at x i .…”

Quantitative homogenization theory for random suspensions in steady Stokes flow

“…Since energy is then pumped into the system, naïve energy estimates blow up, and the analysis crucially relies on stochastic cancellations. Annealed L p regularity in form of Theorem 5 below constitutes the main technical input in [15] for our analysis of this sedimentation problem. More precisely, in a general non-dilute regime, this allows us to obtain the first rigorous proof of the celebrated predictions by Batchelor [9] and by Caflisch and Luke [11] on the effective sedimentation speed and on individual velocity fluctuations, thus significantly extending the perturbative results of [24] (see also [32]).…”

“…We also mention the useful reformulation in form of annealed regularity in [19]. Based on these ideas, we develop corresponding quenched large-scale and annealed regularity theories for the steady Stokes problem (2.5), which constitute the key technical ingredient in our work [15] on sedimentation.…”

“…As in [15], we establish the following annealed version of the above quenched largescale L p regularity statement. The main merit of this estimate is that a stochastic L q (Ω) norm appears inside the spatial L p (R d ) norm and allows to remove local quadratic averages on the random minimal scale r * (up to a tiny loss of stochastic integrability), which is particularly convenient for applications.…”