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.
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Notes
That is, \(\nabla \psi _E^\omega (x+y)=\nabla \psi _E^{\tau _{-y}\omega }(x)\) and \(\Sigma _E^\omega (x+y)\mathbb {1}_{{\mathbb {R}}^d\setminus {\mathcal {I}}^\omega }(x+y)=\Sigma _E^{\tau _{-y}\omega }(x)\mathbb {1}_{{\mathbb {R}}^d\setminus {\mathcal {I}}^{\tau _{-y}\omega }}(x)\) for all \(x,y,\omega \), cf. Remark 1.2.
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Acknowledgements
MD acknowledges financial support from the CNRS-Momentum program. AG has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 864066.
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Duerinckx, M., Gloria, A. Corrector Equations in Fluid Mechanics: Effective Viscosity of Colloidal Suspensions. Arch Rational Mech Anal 239, 1025–1060 (2021). https://doi.org/10.1007/s00205-020-01589-1
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DOI: https://doi.org/10.1007/s00205-020-01589-1