Corrector Equations in Fluid Mechanics: Effective Viscosity of Colloidal Suspensions
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Corrector Equations in Fluid Mechanics: Effective Viscosity of Colloidal Suspensions Mitia Duerinckx & Antoine Gloria Communicated by N. Masmoudi
Abstract Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result for when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.
Contents 1. Introduction and Main Results . . . 1.1. General Overview . . . . . . . 1.2. Main Results . . . . . . . . . 2. Construction of Correctors . . . . . 3. Proof of the Homogenization Result References . . . . . . . . . . . . . . .
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1. Introduction and Main Results 1.1. General Overview This article is devoted to the large-scale behavior of the steady Stokes equation for a fluid with a dense colloidal suspension of small rigid particles that are randomly distributed. The fluid and the particles interact via the action-reaction principle, and satisfy a no-slip condition at the particle boundaries. Suspended particles then act as obstacles, hindering the fluid flow and therefore increasing the flow resistance,
M. Duerinckx, A. Gloria
that is, the viscosity. The system is naturally expected to behave, on large scales, approximately like a Stokes fluid with some effective viscosity. Our main result in this contribution makes this statement precise and rigorously defines the effective viscosity in terms of a stochastic homogenization result. Let us first describe previous contributions on the topic, and emphasize our main motivation. In his PhD thesis, Einstein [14] was the first to analyze this effective viscosity problem; focussing on a dilute regime (that is, assuming that particles are scarce), he argued that the fluid indeed behaves at leading order like a Stokes fluid with some effective viscosity and that the latter can be explicitly computed at first order in the particle concentration in form of the so-called Einstein’s formula, which played a key role in the physics community at that time as it served as a basis for Perrin’s celebrated experiment to estimate the Avogadro number. Various contributions followed, in particular going beyond the first order, e.g. [3,4,7,25]. From a rigorous perspective, several recent contributions stand out. In [20] (see also the refined version [24]), Haines and Mazzucato provide bounds on the difference betw
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