The Clausius–Mossotti formula

Duerinckx, Mitia; Gloria, Antoine

doi:10.3233/asy-231840

Cited by 1 publication

(3 citation statements)

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“…As outlined in Section 2.2, our proof is variational and amounts to proving lower and upper bounds on x B that match with Id C x B 1 to the required accuracy. (For the case of the Clausius-Mossotti conductivity formula, we refer to [14], where we provide a PDE version of the present variational argument.) This new approach is very robust: it yields the first optimal error estimate and allows to cover the most general setting regarding particle separation assumptions.…”

Section: Assumptions (H Mommentioning

confidence: 99%

“…Remark 2.8. In [14], we refer to Lemma 2.7 above 3 in the following slightly different form: given a general stationary ergodic point process z (which can possibly be empty), we have for all stationary random fields with EOEj j < 1,…”

Section: Preliminary Lemmasmentioning

confidence: 99%

“…We start by slightly changing the point of view for correctors, focussing on particle positions rather than on particle indices in the notation: given a set Y Q L of "background" positions such that dist B.y/; B.y 0 / > 2 ; dist B.y/; @Q L > ; for all y; y 0 2 Y; y ¤ y 0 ; (4. 14) we denote by Y L 2 H 1 per .Q L / d the solution of the following periodic corrector problem, using the shorthand notation…”

Section: Estimates On Single-particle Contributionsmentioning

confidence: 99%

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On Einstein's Effective Viscosity Formula

Duerinckx,

Gloria

2023

Memoirs of the European Mathematical Society

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In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability.In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

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Section: Assumptions (H Mommentioning

confidence: 99%

Section: Preliminary Lemmasmentioning

confidence: 99%

Section: Estimates On Single-particle Contributionsmentioning

confidence: 99%

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