Motion of several slender rigid filaments in a Stokes flow
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 327-380.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.184
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

Richard M. Höfer 1; Christophe Prange 2; Franck Sueur 3

1 Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris 8 Place Aurélie Nemours, F75205 Paris Cedex 13, France
2 Laboratoire de Mathématiques AGM, UMR CNRS 8088, Cergy Paris Université 2 avenue Adolphe Chauvin, F95302 Cergy-Pontoise Cedex, France
3 Institut de Mathématiques de Bordeaux, UMR CNRS 5251, Université de Bordeaux 351 cours de la Libération, F33405 Talence Cedex, France & Institut Universitaire de France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2022__9__327_0,
     author = {Richard M. H\"ofer and Christophe Prange and Franck Sueur},
     title = {Motion of several slender rigid filaments in a {Stokes} flow},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {327--380},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.184},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.184/}
}
TY  - JOUR
AU  - Richard M. Höfer
AU  - Christophe Prange
AU  - Franck Sueur
TI  - Motion of several slender rigid filaments in a Stokes flow
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2022
SP  - 327
EP  - 380
VL  - 9
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.184/
DO  - 10.5802/jep.184
LA  - en
ID  - JEP_2022__9__327_0
ER  - 
%0 Journal Article
%A Richard M. Höfer
%A Christophe Prange
%A Franck Sueur
%T Motion of several slender rigid filaments in a Stokes flow
%J Journal de l’École polytechnique — Mathématiques
%D 2022
%P 327-380
%V 9
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.184/
%R 10.5802/jep.184
%G en
%F JEP_2022__9__327_0
Richard M. Höfer; Christophe Prange; Franck Sueur. Motion of several slender rigid filaments in a Stokes flow. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 327-380. doi : 10.5802/jep.184. https://jep.centre-mersenne.org/articles/10.5802/jep.184/

[1] G. Acosta, R. G. Durán & M. A. Muschietti - “Solutions of the divergence operator on John domains”, Adv. in Math. 206 (2006) no. 2, p. 373-401 | DOI | MR | Zbl

[2] V. Banica & L. Vega - “Selfsimilar solutions of the binormal flow and their stability”, in Singularities in mechanics: formation, propagation and microscopic description, Panoramas & Synthèses, vol. 38, Société Mathématique de France, Paris, 2012, p. 1-35 | MR | Zbl

[3] G. K. Batchelor - “Slender-body theory for particles of arbitrary cross-section in Stokes flow”, J. Fluid Mech. 44 (1970), p. 419-440 | DOI | MR | Zbl

[4] J. Berkowitz & C. S. Gardner - “On the asymptotic series expansion of the motion of a charged particle in slowly varying fields”, Comm. Pure Appl. Math. 12 (1959), p. 501-512 | DOI | MR | Zbl

[5] V. Bonnaillie-Noël, M. Dalla Riva, M. Dambrine & P. Musolino - “A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary”, J. Math. Pures Appl. (9) 116 (2018), p. 211-267 | DOI | MR | Zbl

[6] V. Bonnaillie-Noël, M. Dalla Riva, M. Dambrine & P. Musolino - “Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary”, Comm. Partial Differential Equations 46 (2021) no. 2, p. 282-309 | DOI | MR | Zbl

[7] R. G. Cox - “The motion of long slender bodies in a viscous fluid. I: General theory”, J. Fluid Mech. 44 (1970), p. 791-810 | DOI | Zbl

[8] M. Doi & S. F. Edwards - The theory of polymer dynamics, Intern. Series of Monographs in Physics, vol. 73, Clarendon Press, Oxford University Press, Oxford, 1988

[9] R. Elfasi, Y. Elimelech & A. D. Gat - “Propulsion and maneuvering of an artificial microswimmer by two closely spaced waving elastic filaments”, Phys. Rev. Fluids. 3 (2018), article ID 044203, 14 pages | DOI

[10] E. Feireisl, Y. Namlyeyeva & Š. Nečasová - “Homogenization of the evolutionary Navier-Stokes system”, Manuscripta Math. 149 (2016) no. 1-2, p. 251-274 | DOI | MR | Zbl

[11] G. P. Galdi - “On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications”, in Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, p. 653-791 | DOI | Zbl

[12] G. P. Galdi - An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, Springer Monographs in Math., Springer, New York, 2011 | DOI

[13] O. Glass & F. Sueur - “Dynamics of several rigid bodies in a two-dimensional ideal fluid and convergence to vortex systems”, 2019 | arXiv

[14] O. Gonzalez - “Theorems on the Stokesian hydrodynamics of a rigid filament in the limit of vanishing radius”, SIAM J. Appl. Math. 81 (2021) no. 2, p. 551-573 | DOI | MR | Zbl

[15] T. Götz - Interactions of fibers and flow: asymptotics, theory and numerics, Doctoral dissertation, University of Kaiserslautern, 2000

[16] G. J. Hancock - “The self-propulsion of microscopic organisms through liquids”, Proc. Roy. Soc. London Ser. A 217 (1953), p. 96-121 | DOI | MR | Zbl

[17] M. Higaki & C. Prange - “Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law”, Calc. Var. Partial Differential Equations 59 (2020) no. 4, article ID 131, 46 pages | DOI | MR | Zbl

[18] M. Hillairet, A. Moussa & F. Sueur - “On the effect of polydispersity and rotation on the Brinkman force induced by a cloud of particles on a viscous incompressible flow”, Kinet. and Relat. Mod. 12 (2019) no. 4, p. 681-701 | DOI | MR | Zbl

[19] G. C. Hsiao & W. L. Wendland - Boundary integral equations, Applied Math. Sciences, vol. 164, Springer, Cham, 2021 | DOI

[20] G. B. Jeffery - “The motion of ellipsoidal particles immersed in a viscous fluid”, Proc. Roy. Soc. London Ser. A 102 (1923), p. 161-179 | DOI | Zbl

[21] R. L. Jerrard & C. Seis - “On the vortex filament conjecture for Euler flows”, Arch. Rational Mech. Anal. 224 (2017) no. 1, p. 135-172 | DOI | MR | Zbl

[22] R. E. Johnson - “An improved slender-body theory for Stokes flow”, J. Fluid Mech. 99 (1980) no. 2, p. 411-431 | DOI | MR | Zbl

[23] M. Junk & R. Illner - “A new derivation of Jeffery’s equation”, J. Math. Fluid Mech. 9 (2007) no. 4, p. 455-488 | DOI | MR | Zbl

[24] J. B. Keller & S. I. Rubinow - “Slender-body theory for slow viscous flow”, J. Fluid Mech. 75 (1976), p. 705-714 | DOI | Zbl

[25] S. Kim & S. J. Karrila - Microhydrodynamics. Principles and selected applications, Series in Chemical Engineering, Butterworth-Heinemann, 1991

[26] O. A. Ladyzhenskaya - The mathematical theory of viscous incompressible flow, Math. and its Applications, Vol. 2, Gordon and Breach Science Publ., New York-London-Paris, 1969

[27] E. Lauga & T. R. Powers - “The hydrodynamics of swimming microorganisms”, Reports on Progress in Physics 72 (2009) no. 9, article ID 096601 | DOI | MR

[28] Y. Man, L. Koens & E. Lauga - “Hydrodynamic interactions between nearby slender filaments”, EPL (Europhysics Letters) 116 (2016) no. 2, article ID 24002 | DOI

[29] Y. Man, W. Page, R. J. Poole & E. Lauga - “Bundling of elastic filaments induced by hydrodynamic interactions”, Phys. Rev. Fluids. 2 (2017), article ID 123101, 36 pages | DOI

[30] V. Maz’ya, S. Nazarov & B. Plamenevskij - Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. II, Operator Theory: Advances and Appl., vol. 112, Birkhäuser Verlag, Basel, 2000

[31] É. Miot - “Le flot binormal, l’équation de Schrödinger et les tourbillons filamentaires”, in Séminaire Bourbaki, Astérisque, vol. 390, Société Mathématique de France, Paris, 2017, p. 427-451 | MR | Zbl

[32] Y. Mori & L. Ohm - “An error bound for the slender body approximation of a thin, rigid fiber sedimenting in Stokes flow”, Res. Math. Sci. 7 (2020) no. 2, article ID 8, 27 pages | DOI | MR | Zbl

[33] Y. Mori & L. Ohm - “Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid”, 2020 | arXiv

[34] Y. Mori, L. Ohm & D. Spirn - “Theoretical justification and error analysis for slender body theory”, Comm. Pure Appl. Math. 73 (2020) no. 6, p. 1245-1314 | DOI | MR | Zbl

[35] Y. Mori, L. Ohm & D. Spirn - “Theoretical justification and error analysis for slender body theory with free ends”, Arch. Rational Mech. Anal. 235 (2020) no. 3, p. 1905-1978 | DOI | MR | Zbl

[36] L. Ohm, B. K. Tapley, H. I. Andersson, E. Celledoni & B. Owren - “A slender body model for thin rigid fibers: validation and comparisons”, 2019 | arXiv

[37] C. S. Peskin - “The immersed boundary method”, Acta Numer. 11 (2002), p. 479-517 | DOI | MR | Zbl

[38] C. Pozrikidis - Boundary integral and singularity methods for linearized viscous flow, Cambridge Texts in Applied Math., Cambridge University Press, Cambridge, 1992 | DOI

[39] D. Serre - “Chute libre d’un solide dans un fluide visqueux incompressible. Existence”, Japan J. Appl. Math. 4 (1987) no. 1, p. 99-110 | DOI | MR | Zbl

[40] D. Serre - Matrices. Theory and applications, Graduate Texts in Math., vol. 216, Springer, New York, 2010 | DOI

[41] J. Simon - “Domain variation for drag in Stokes flow”, in Control theory of distributed parameter systems and applications (Shanghai, 1990), Lect. Notes Control Inf. Sci., vol. 159, Springer, Berlin, 1991, p. 28-42 | DOI | MR | Zbl

[42] A.-K. Tornberg - “Accurate evaluation of integrals in slender-body formulations for fibers in viscous flow”, 2020 | arXiv

[43] A. Vibe & N. Marheineke - “Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles”, Kinet. and Relat. Mod. 11 (2018) no. 6, p. 1443-1474 | DOI | MR | Zbl

Cited by Sources: