Comptes Rendus
no. 11-12
Mathematical Problems in Mechanics
The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions
[Équations de Navier–Stokes avec conditions aux limites de Navier pour décrire un fluide autour de corps en mouvement pouvant entrer en collision]

Présenté par : Gérard Iooss

Jiří Neustupa1 ; Patrick Penel2
1 Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
2 Université du Sud Toulon-Var, département de mathématique et laboratoire systèmes navals complexes, BP 20132, 83957 La Garde, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 685-690.

Dans cette Note, nous considérons les équations de Navier–Stokes avec des conditions aux limites de Navier dans un domaine borné temporellement variable contenant un nombre fini de corps compacts en mouvement. Le mouvement de ces corps est supposé connu, ainsi que la simulation de leurs contacts ou collisions éventuels (en nombre fini, entre eux ou avec la frontière du domaine). Nous établissons un résultat d'existence, globale en temps, des solutions faibles. Le choix des conditions aux limites est intéressant à commenter par comparaison avec les conditions standard de Dirichlet.

In this Note, we treat the Navier–Stokes equation with slip Navier's boundary condition in a time variable domain around a finite system of compact bodies moving in a container. The motion of the bodies is assumed to be a priori known. The bodies may collide at a finite number of time instants. We present the theorem on the global in time existence of a weak solution. It is remarkable that Navier's boundary condition enables us to consider a larger class of possible collisions of bodies with C2 front surfaces in comparison with the no-slip Dirichlet condition.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.03.021
Jiří Neustupa 1 ; Patrick Penel 2

1 Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
2 Université du Sud Toulon-Var, département de mathématique et laboratoire systèmes navals complexes, BP 20132, 83957 La Garde, France 
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Jiří Neustupa; Patrick Penel. The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 685-690. doi : 10.1016/j.crma.2009.03.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.021/

[1] H. Fujita; N. Sauer On existence of weak solutions of the Navier–Stokes equations in regions with moving boundaries, J. Fac. Sci. Univ. Tokyo, Sec. 1A, Volume 17 (1970), pp. 403-420

[2] G.P. Galdi On the motion of a rigid body in a viscous fluid: a mathematical analysis with applications (S. Friedlander; D. Serre, eds.), Handbook of Mathematical Fluid Dynamics I, Elsevier, 2002, pp. 653-791

[3] K.H. Hoffmann; V.N. Starovoitov On a motion of a solid body in a viscous fluid. Two-dimensional case, Adv. Math. Sci. Appl., Volume 9 (1999), pp. 633-648

[4] O.A. Ladyzhenskaya Initial-boundary value problem for the Navier–Stokes equations in domains with time-varying boundaries, Zapiski Nauchnykh Seminarov LOMI, Volume 11 (1968), pp. 97-128 (Russian)

[5] J. Neustupa, Existence of a weak solution to the Navier–Stokes equation in a general time-varying domain by the Rothe method, Math. Meth. Appl. Sci., in press

[6] J. Neustupa, P. Penel, Existence of a weak solution to the Navier–Stokes equation with Navier's boundary condition around striking bodies, preprint 2008

[7] V.N. Starovoitov Behavior of a rigid body in an incompressible viscous fluid near a boundary, Int. Ser. Numer. Math., Volume 147 (2003), pp. 313-327

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