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Identification of the Dilute Regime in Particle Sedimentation

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Abstract

We investigate the dynamics of rigid, spherical particles of radius R sinking in a viscous fluid. Both the inertia of the particles and the fluid are neglected. We are interested in a large number N of particles with average distance d≫R. We investigate in which regime (in terms of N and R/d) the particles do not significantly interact and approximately sink like single particles. We rigorously establish the lower bound for the critical number N crit of particles. This lower bound agrees with the heuristically expected N crit in terms of its scaling in R/d. The main difficulty lies in showing that the particles cannot get significantly closer over a relevant time scale. We use the method of reflection for the Stokes operator to bound the strength of the particle interaction.

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Authors and Affiliations

  1. Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Pierre-Emmanuel Jabin

  2. Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 10, 53115, Bonn, Germany

    Felix Otto

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  1. Pierre-Emmanuel Jabin
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  2. Felix Otto
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Communicated by H. Spohn

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Jabin, PE., Otto, F. Identification of the Dilute Regime in Particle Sedimentation. Commun. Math. Phys. 250, 415–432 (2004). https://doi.org/10.1007/s00220-004-1126-3

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  • DOI: https://doi.org/10.1007/s00220-004-1126-3

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