# American Institute of Mathematical Sciences

August  2019, 12(4): 681-701. doi: 10.3934/krm.2019026

## On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow

 1 Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, France 2 Laboratoire Jacques-Louis Lions, Sorbonne Université & Université Paris-Diderot SPC, CNRS, UMR 7598, France 3 Institut de Mathématiques de Bordeaux, Université de Bordeaux, CNRS, UMR 5251, France

Received  May 2017 Revised  July 2018 Published  May 2019

In this paper, we are interested in the collective friction of a cloud of particles on the viscous incompressible fluid in which they are moving. The particle velocities are assumed to be given and the fluid is assumed to be driven by the stationary Stokes equations. We consider the limit where the number $N$ of particles goes to infinity with their diameters of order $1/N$ and their mutual distances of order $1/ N^{1/3}$. The rigorous convergence of the fluid velocity to a limit which is solution to a stationary Stokes equation set in the full space but with an extra term, referred to as the Brinkman force, was proven in [5] when the particles are identical spheres in prescribed translations. Our result here is an extension to particles of arbitrary shapes in prescribed translations and rotations. The limit Stokes-Brinkman system involves the particle distribution in position, velocity and shape, through the so-called Stokes' resistance matrices.

Citation: Matthieu Hillairet, Ayman Moussa, Franck Sueur. On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow. Kinetic and Related Models, 2019, 12 (4) : 681-701. doi: 10.3934/krm.2019026
##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065. [2] H. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow, Turbulence and Combustion, 1 (1949), 27. doi: 10.1007/BF02120313. [3] K. Carrapatoso and M. Hillairet, On the derivation of a stokes-brinkman problem from stokes equations around a random array of moving spheres, arXiv: 1804.10498, April 2018. [4] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, In Nonlinear partial differential equations and their applications. Collége de France Seminar, Vol. II (Paris, 1979/1980), volume 60 of Res. Notes in Math., pages 98–138,389–390. Pitman, Boston, Mass.-London, 1982. [5] L. Desvillettes, F. 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. [6] L. Diening, E. Feireisl and Y. Lu, The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system, ESAIM Control Optim. Calc. Var., 23 (2017), 851-868.  doi: 10.1051/cocv/2016016. [7] E. Feireisl and Y. Lu, Homogenization of stationary Navier-Stokes equations in domains with tiny holes, J. Math. Fluid Mech., 17 (2015), 381-392.  doi: 10.1007/s00021-015-0200-2. [8] E. Feireisl, Y. Namlyeyeva and and v. S. Nečasová, Homogenization of the evolutionary Navier-Stokes system, Manuscripta Math., 149 (2016), 251-274.  doi: 10.1007/s00229-015-0778-y. [9] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics. Springer, New York, second edition, 2011. Steady-state problems. doi: 10.1007/978-0-387-09620-9. [10] M. Hillairet, On the homogenization of the stokes problem in a perforated domain, Archive for Rational Mechanics and Analysis, 230 (2018), 1179-1228.  doi: 10.1007/s00205-018-1268-7. [11] R. M. Höfer and J. J. L. Velázquez, The method of reflections, homogenization and screening for Poisson and Stokes equations in perforated domains, Arch. Ration. Mech. Anal., 227 (2018), 1165-1221.  doi: 10.1007/s00205-017-1182-4. [12] P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432.  doi: 10.1007/s00220-004-1126-3. [13] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 102 (1922), 161-179. [14] D. J. Jeffrey and Y. Onishi, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-reynolds-number flow, Journal of Fluid Mechanics, 139 (1984), 261-290.  doi: 10.1017/S0022112084000355. [15] S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Courier Corporation, 2013. [16] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 6, Pergamon Press, Oxford, second edition, 1987, Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. Reid. [17] Y. Lu and S. Schwarzacher, Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes, J. Differential Equations, 265 (2018), 1371-1406.  doi: 10.1016/j.jde.2018.04.007. [18] V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, volume 46 of Progress in Mathematical Physics, Birkhäuser Boston, Inc., Boston, MA, 2006. Translated from the 2005 Russian original by M. Goncharenko and D. Shepelsky. [19] A. Mecherbet and M. Hillairet, ${L}^{p}$ estimates for the homogenization of stokes problem in a perforated domain, J. Inst. Math. Jussieu, 2018, 1–8. [20] V. Ricci, Modelling of systems with a dispersed phase: "Measuring" small sets in the presence of elliptic operators, In From Particle Systems to Partial Differential Equations. III, volume 162 of Springer Proc. Math. Stat., pages 285–300. Springer, [Cham], 2016. doi: 10.1007/978-3-319-32144-8_14. [21] J. Rubinstein, On the macroscopic description of slow viscous flow past a random array of spheres, J. Statist. Phys., 44 (1986), 849-863.  doi: 10.1007/BF01011910.

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##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065. [2] H. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Flow, Turbulence and Combustion, 1 (1949), 27. doi: 10.1007/BF02120313. [3] K. Carrapatoso and M. Hillairet, On the derivation of a stokes-brinkman problem from stokes equations around a random array of moving spheres, arXiv: 1804.10498, April 2018. [4] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, In Nonlinear partial differential equations and their applications. Collége de France Seminar, Vol. II (Paris, 1979/1980), volume 60 of Res. Notes in Math., pages 98–138,389–390. Pitman, Boston, Mass.-London, 1982. [5] L. Desvillettes, F. 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. [6] L. Diening, E. Feireisl and Y. Lu, The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system, ESAIM Control Optim. Calc. Var., 23 (2017), 851-868.  doi: 10.1051/cocv/2016016. [7] E. Feireisl and Y. Lu, Homogenization of stationary Navier-Stokes equations in domains with tiny holes, J. Math. Fluid Mech., 17 (2015), 381-392.  doi: 10.1007/s00021-015-0200-2. [8] E. Feireisl, Y. Namlyeyeva and and v. S. Nečasová, Homogenization of the evolutionary Navier-Stokes system, Manuscripta Math., 149 (2016), 251-274.  doi: 10.1007/s00229-015-0778-y. [9] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer Monographs in Mathematics. Springer, New York, second edition, 2011. Steady-state problems. doi: 10.1007/978-0-387-09620-9. [10] M. Hillairet, On the homogenization of the stokes problem in a perforated domain, Archive for Rational Mechanics and Analysis, 230 (2018), 1179-1228.  doi: 10.1007/s00205-018-1268-7. [11] R. M. Höfer and J. J. L. Velázquez, The method of reflections, homogenization and screening for Poisson and Stokes equations in perforated domains, Arch. Ration. Mech. Anal., 227 (2018), 1165-1221.  doi: 10.1007/s00205-017-1182-4. [12] P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432.  doi: 10.1007/s00220-004-1126-3. [13] G. B. Jeffery, The motion of ellipsoidal particles immersed in a viscous fluid, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 102 (1922), 161-179. [14] D. J. Jeffrey and Y. Onishi, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-reynolds-number flow, Journal of Fluid Mechanics, 139 (1984), 261-290.  doi: 10.1017/S0022112084000355. [15] S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Courier Corporation, 2013. [16] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics. Vol. 6, Pergamon Press, Oxford, second edition, 1987, Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. Reid. [17] Y. Lu and S. Schwarzacher, Homogenization of the compressible Navier-Stokes equations in domains with very tiny holes, J. Differential Equations, 265 (2018), 1371-1406.  doi: 10.1016/j.jde.2018.04.007. [18] V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, volume 46 of Progress in Mathematical Physics, Birkhäuser Boston, Inc., Boston, MA, 2006. Translated from the 2005 Russian original by M. Goncharenko and D. Shepelsky. [19] A. Mecherbet and M. Hillairet, ${L}^{p}$ estimates for the homogenization of stokes problem in a perforated domain, J. Inst. Math. Jussieu, 2018, 1–8. [20] V. Ricci, Modelling of systems with a dispersed phase: "Measuring" small sets in the presence of elliptic operators, In From Particle Systems to Partial Differential Equations. III, volume 162 of Springer Proc. Math. Stat., pages 285–300. Springer, [Cham], 2016. doi: 10.1007/978-3-319-32144-8_14. [21] J. Rubinstein, On the macroscopic description of slow viscous flow past a random array of spheres, J. Statist. Phys., 44 (1986), 849-863.  doi: 10.1007/BF01011910.
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