Skip to main content
SpringerLink
Log in

A Small Solid Body with Large Density in a Planar Fluid is Negligible

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this article, we consider a small rigid body moving in a viscous fluid filling the whole \(\mathbb R^2\). We assume that the diameter of the rigid body goes to 0, that the initial velocity has bounded energy and that the density of the rigid body goes to infinity. We prove that the rigid body has no influence on the limit equation by showing convergence of the solutions towards a solution of the Navier–Stokes equations in the full plane \(\mathbb {R}^{2}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chipot, M., Droniou, J., Planas, G., Robinson, J.C., Xue, W.: Limits of the Stokes and Navier–Stokes equations in a punctured periodic domain. arXiv:1407.6942 [math] (2014)

  2. Desjardins, B., Esteban, M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146(1), 59–71 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ervedoza, S., Hillairet, M., Lacave, C.: Long-time behavior for the two-dimensional motion of a disk in a viscous fluid. Commun. Math. Phys. 329(1), 325–382 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Glass, O., Lacave, C., Sueur, F.: On the motion of a small body immersed in a two dimensional incompressible perfect fluid. Bull. Soc. Math. France 142(3), 489–536 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Glass, O., Lacave, C., Sueur, F.: On the motion of a small light body immersed in a two dimensional incompressible perfect fluid with vorticity. Commun. Math. Phys. 341(3), 1015–1065 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Glass, O., Munnier, A., Sueur, F.: Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid. Invent. Math. 214(1), 171–287 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hoffmann, K.-H., Starovoitov, V.N.: On a motion of a solid body in a viscous fluid: two-dimensional case. Adv. Math. Sci. Appl. 9(2), 633–648 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Iftimie, D., Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Two dimensional incompressible ideal flow around a small obstacle. Commun. Partial Differ. Equ. 28(1–2), 349–379 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Iftimie, D., Lopes Filho, M .C., Nussenzveig Lopes, H .J.: Two-dimensional incompressible viscous flow around a small obstacle. Math. Ann. 336(2), 449–489 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lacave, C.: Two-dimensional incompressible viscous flow around a thin obstacle tending to a curve. Proc. R. Soc. Edinb. Sect. A Math. 139(6), 1237–1254 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lacave, C., Takahashi, T.: Small moving rigid body into a viscous incompressible fluid. Arch. Ration. Mech. Anal. 223(3), 1307–1335 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  13. San Martín, J., Starovoitov, V., Tucsnak, M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161(2), 113–147 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Serre, D.: Chute libre d’un solide dans un fluide visqueux lncompressible: existence. Jpn. J. Ind. Appl. Math. 4(1), 99–110 (1987)

    Article  MATH  Google Scholar 

  15. Takahashi, T., Tucsnak, M.: Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6(1), 53–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yudakov, N.V.: The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid. Dinamika Sploshnoi Sredy 18, 249–253 (1974)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

J.H. and D.I. have been partially funded by the ANR Project Dyficolti ANR-13-BS01-0003-01. D.I. has been partially funded by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Author information

Authors and Affiliations

  1. CNRS UMR 5208, Institut Camille Jordan, Université de Lyon, Université Lyon 1, 43 bd. du 11 Novembre 1918, 69622, Villeurbanne Cedex, France

    Jiao He & Dragoş Iftimie

Authors
  1. Jiao He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dragoş Iftimie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jiao He or Dragoş Iftimie.

Additional information

Dedicated to the memory of George R. Sell.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, J., Iftimie, D. A Small Solid Body with Large Density in a Planar Fluid is Negligible. J Dyn Diff Equat 31, 1671–1688 (2019). https://doi.org/10.1007/s10884-018-9718-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-018-9718-3

Keywords

Search

Navigation