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October  2019, 12(5): 995-1044. doi: 10.3934/krm.2019038

Sedimentation of particles in Stokes flow

IMAG, Montpellier University, Place Eugène Bataillon, Montpellier, 34090, France

Received  June 2018 Revised  April 2019 Published  July 2019

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].

Citation: Amina Mecherbet. Sedimentation of particles in Stokes flow. Kinetic & Related Models, 2019, 12 (5) : 995-1044. doi: 10.3934/krm.2019038
References:
[1]

G. K. Batchelor, Sedimentation in a dilute dispersion of spheres, J. Fluid Mech., 52 (1972), 245-268. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

E. Guazzelli and J. F. Morris, A Physical Introduction to Suspension Dynamics, Cambridge Texts In Applied Mathematics, 2012. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Courier Corporation, 2005.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[18]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

show all references

References:
[1]

G. K. Batchelor, Sedimentation in a dilute dispersion of spheres, J. Fluid Mech., 52 (1972), 245-268. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

E. Guazzelli and J. F. Morris, A Physical Introduction to Suspension Dynamics, Cambridge Texts In Applied Mathematics, 2012. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Courier Corporation, 2005.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[18]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. Google Scholar

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