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Abstract
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We consider a microscopic model of
identical axis-symmetric rigid Brownian particles suspended in a Stokes flow. We
rigorously derive in the homogenization limit of many small particles a classical
formula for the viscoelastic stress that appears in so-called Doi models which couple a
Fokker–Planck equation to the Stokes equations. We consider both Deborah numbers
of order 1 and very small Deborah numbers. Our microscopic model contains several
simplifications; most importantly, we neglect the time evolution of the particle
centers as well as hydrodynamic interaction for the evolution of the particle
orientations.
The microscopic fluid velocity is modeled by the Stokes equations with given
torques at the particles in terms of Stratonovich noise. We give a meaning to this
PDE in terms of an infinite-dimensional Stratonovich integral. This requires the
analysis of the shape derivatives of the Stokes equations in perforated domains, which
we accomplish by the method of reflections.
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Keywords
homogenization, Stokes flows, viscoelasticity, Doi model,
Brownian particles
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Mathematical Subject Classification
Primary: 76M50, 76D07, 35R60, 35Q70
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Milestones
Received: 21 February 2022
Revised: 13 July 2022
Accepted: 26 August 2022
Published: 26 June 2023
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