On the homogenization of the Stokes equations in perforated domains with application to fluid/solid interaction problems
Matthieu Hillairet1
1 Institut Montpelliérain Alexander Grothendieck, Université de Montpellier
Séminaire Laurent Schwartz — EDP et applications (2016-2017), Talk no. 15, 15 p.

In these notes, we consider the Stokes equations on a perforated domain. The inclusions represent particles moving in a viscous fluid so that the partial differential equations in the bulk are completed with non-zero boundary conditions simulating the motion of the particles. We review recent results on the associated homogenization problem in the regime where the number of particles increases while their diameters converge to 0.

Published online:
DOI: 10.5802/slsedp.102
Matthieu Hillairet 1

1 Institut Montpelliérain Alexander Grothendieck, Université de Montpellier 
@article{SLSEDP_2016-2017____A15_0,
     author = {Matthieu Hillairet},
     title = {On the homogenization of the {Stokes} equations in~perforated domains with application to~fluid/solid~interaction problems},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:15},
     pages = {1--15},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2016-2017},
     doi = {10.5802/slsedp.102},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.102/}
}
TY  - JOUR
AU  - Matthieu Hillairet
TI  - On the homogenization of the Stokes equations in perforated domains with application to fluid/solid interaction problems
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:15
PY  - 2016-2017
SP  - 1
EP  - 15
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.102/
DO  - 10.5802/slsedp.102
LA  - en
ID  - SLSEDP_2016-2017____A15_0
ER  - 
%0 Journal Article
%A Matthieu Hillairet
%T On the homogenization of the Stokes equations in perforated domains with application to fluid/solid interaction problems
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:15
%D 2016-2017
%P 1-15
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.102/
%R 10.5802/slsedp.102
%G en
%F SLSEDP_2016-2017____A15_0
Matthieu Hillairet. On the homogenization of the Stokes equations in perforated domains with application to fluid/solid interaction problems. Séminaire Laurent Schwartz — EDP et applications (2016-2017), Talk no. 15, 15 p. doi : 10.5802/slsedp.102. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.102/

[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(3):209–259, 1990. | DOI | MR | Zbl

[2] H. C. Brinkman. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow, Turbulence and Combustion, 1(1):27, 1949. | DOI | Zbl

[3] L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa. Global existence of solutions for the coupled Vlasov and Navier-Stokes equations. Differential Integral Equations, 22(11-12):1247–1271, 2009. | Zbl

[4] L. Desvillettes. Some aspects of the modeling at different scales of multiphase flows. Comput. Methods Appl. Mech. Engrg., 199(21-22):1265–1267, 2010. | DOI | MR | Zbl

[5] L. Desvillettes, F. Golse and V. Ricci. The mean field limit for solid particles in a Navier-Stokes flow. J. Stat. Phys. 131: 941-967, 2008. | DOI | MR | Zbl

[6] G. P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations. Springer Monographs in Mathematics. Springer, New York, second edition, 2011. | DOI | MR | Zbl

[7] R. Höfer and J. L. Velázquez. The method of reflections, homogenization and screening for Poisson and Stokes equations in perforated domains. March 2016. | arXiv | DOI | MR | Zbl

[8] E. Guazzelli and J. F. Morris. A Physical Introduction To Suspension Dynamics. Cambridge Texts In Applied Mathematics. 2012 | DOI | Zbl

[9] I. Gallagher, L. Saint-Raymond and B. Texier. From Newton to Boltzmann: hard spheres and short-range potentials. Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2013. | DOI | Zbl

[10] M. Hillairet. On the homogenization of the Stokes problem in a perforated domain. August 2016. | arXiv | DOI | MR | Zbl

[11] M. Hillairet, A. Moussa and F. Sueur. On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow. May 2017. | arXiv | DOI | MR | Zbl

[12] L. D. Landau and E. M. Lifshitz. Fluid mechanics. Course of Theoretical Physics, Vol. 6. Pergamon Press, London, 1959.

[13] A. Mecherbet and M. Hillairet L p estimates for the homogenization of stokes problem in a perforated domain. November 2016. | arXiv | DOI | MR

Cited by Sources: