Motion of several slender rigid filaments in a Stokes flow

Höfer, Richard M.; Prange, Christophe; Sueur, Franck

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 327-380.

Nous étudions la dynamique de filaments rigides minces se déplaçant dans un fluide qui est décrit par un état de base régulier perturbé par la présence des solides selon les équations du système de Stokes stationnaire tridimensionnel. Plus précisément, nous considérons la limite dans laquelle l’épaisseur des corps solides tend vers zéro avec un taux commun $ϵ$, tandis que leur densité de masse volumétrique est maintenue constante, de sorte que les solides limites occupent des courbes que l’on suppose d’intersections deux à deux vides, et ont une masse nulle. Pour $ϵ>0$, la dynamique des solides est donnée par les équations de Newton et correspondent à des équations différentielles ordinaires du second ordre, couplées. Nous prouvons que les équations limites sont des équations différentielles ordinaires du premier ordre, découplées, dont les coefficients ne dépendent que des courbes limites et du flot de base. Nous déterminons également l’effet limite des courbes limites sur le fluide, dans l’esprit de la méthode des frontières immergées.

We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely, we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate $ϵ$, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive $ϵ$, the bodies’ dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method.

Classification : 74F10
Keywords: Slender rigid body, steady Stokes flow, fluid-solid interaction, singular perturbation
Mot clés : Filaments rigides minces, système de Stokes stationnaire, interaction entre fluide et solides, perturbation singulière

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