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Sedimentation of particles in Stokes flow

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  • In this paper, we consider $ N $ identical spherical particles sedimenting in a uniform gravitational field. Particle rotation is included in the model while fluid and particle inertia are neglected. Using the method of reflections, we extend the investigation of [11] by discussing the threshold beyond which the minimal particle distance is conserved for a short time interval independent of $ N $. We also prove that the particles interact with a singular interaction force given by the Oseen tensor and justify the mean field approximation in the spirit of [8] and [9].

    Mathematics Subject Classification: 35Q70, 76T20, 76D07, 35Q83.

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  • [1] G. K. Batchelor, Sedimentation in a dilute dispersion of spheres, J. Fluid Mech., 52 (1972), 245-268. 
    [2] L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solutions for the coupled Vlasov and Navier-Stokes equations, Differential Integral Equations, 22 (2009), 1247-1271. 
    [3] T. ChampionL. D. Pascale and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAMJ. Math. Anal, 40 (2008), 1-20.  doi: 10.1137/07069938X.
    [4] L. DesvillettesF. Golse and V. Ricci, The mean field limit for solid particles in a Navier-Stokes flow, J. Stat. Phys., 131 (2008), 941-967.  doi: 10.1007/s10955-008-9521-3.
    [5] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Second edition edition, Springer Monographs in Mathematics.Springer, New York, 2011, Steady-state problems. doi: 10.1007/978-0-387-09620-9.
    [6] E. Guazzelli and J. F. Morris, A Physical Introduction to Suspension Dynamics, Cambridge Texts In Applied Mathematics, 2012.
    [7] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396.
    [8] M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384.  doi: 10.1142/S0218202509003814.
    [9] M. Hauray and P. E. Jabin, Particle approximation of Vlasov equations with singular forces: Propagation of chaos, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 891-940.  doi: 10.24033/asens.2261.
    [10] M. Hillairet, On the homogenization of the stokes problem in a perforated domain, Arch Rational Mech Anal, 230 (2018), 1179-1228.  doi: 10.1007/s00205-018-1268-7.
    [11] R. M. Höfer, Sedimentation of inertialess particles in Stokes flows, Commun. Math. Phys., 360 (2018), 55-101.  doi: 10.1007/s00220-018-3131-y.
    [12] R. M. Höfer and J. J. L. Velàzquez, The method of reflections, homogenization and screening for Poisson and Stokes equations in perforated domains, Arch Rational Mech Anal, 227 (2018), 1165-1221.  doi: 10.1007/s00205-017-1182-4.
    [13] P. E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Communications in Mathematical Physics, 250 (2004), 415-432.  doi: 10.1007/s00220-004-1126-3.
    [14] S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Courier Corporation, 2005.
    [15] G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.
    [16] J. H. C. Luke, Convergence of a multiple reflection method for calculating Stokes flow in a suspension, Society for Industrial and Applied Mathematics, 49 (1989), 1635-1651.  doi: 10.1137/0149099.
    [17] M. Smoluchowski, Über die Wechelwirkung von Kugeln, die sich in einer zähen Flüssigkeit bewegen, Bull. Int. Acad. Sci. Cracovie, Cl. Sci. Math. Nat., Sér. A Sci. Math, (1911), 28–39.
    [18] C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.
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