Dynamics of a small rigid body in a perfect incompressible fluid

Olivier Glass[1]

  • [1] CEREMADE, UMR CNRS 7534 Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France

Journées Équations aux dérivées partielles (2014)

  • page 1-20
  • ISSN: 0752-0360

Abstract

top
We consider a solid in a perfect incompressible fluid in dimension two. The fluid is driven by the classical Euler equation, and the solid evolves under the influence of the pressure on its surface. We consider the limit of the system as the solid shrinks to a point. We obtain several different models in the limit, according to the asymptotics for the mass and the moment of inertia, and according to the geometrical situation that we consider. Among the models that we get in the limit, we find Marchioro and Pulvirenti’s vortex-wave system and a variant of this system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force. These results are obtained in collaboration with Christophe Lacave (Paris-Diderot), Alexandre Munnier (Nancy) and Franck Sueur (Bordeaux).

How to cite

top

Glass, Olivier. "Dynamics of a small rigid body in a perfect incompressible fluid." Journées Équations aux dérivées partielles (2014): 1-20. <http://eudml.org/doc/275626>.

@article{Glass2014,
abstract = {We consider a solid in a perfect incompressible fluid in dimension two. The fluid is driven by the classical Euler equation, and the solid evolves under the influence of the pressure on its surface. We consider the limit of the system as the solid shrinks to a point. We obtain several different models in the limit, according to the asymptotics for the mass and the moment of inertia, and according to the geometrical situation that we consider. Among the models that we get in the limit, we find Marchioro and Pulvirenti’s vortex-wave system and a variant of this system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force. These results are obtained in collaboration with Christophe Lacave (Paris-Diderot), Alexandre Munnier (Nancy) and Franck Sueur (Bordeaux).},
affiliation = {CEREMADE, UMR CNRS 7534 Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France},
author = {Glass, Olivier},
journal = {Journées Équations aux dérivées partielles},
language = {eng},
pages = {1-20},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Dynamics of a small rigid body in a perfect incompressible fluid},
url = {http://eudml.org/doc/275626},
year = {2014},
}

TY - JOUR
AU - Glass, Olivier
TI - Dynamics of a small rigid body in a perfect incompressible fluid
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 20
AB - We consider a solid in a perfect incompressible fluid in dimension two. The fluid is driven by the classical Euler equation, and the solid evolves under the influence of the pressure on its surface. We consider the limit of the system as the solid shrinks to a point. We obtain several different models in the limit, according to the asymptotics for the mass and the moment of inertia, and according to the geometrical situation that we consider. Among the models that we get in the limit, we find Marchioro and Pulvirenti’s vortex-wave system and a variant of this system where the vortex, placed in the point occupied by the shrunk body, is accelerated by a lift force similar to the Kutta-Joukowski force. These results are obtained in collaboration with Christophe Lacave (Paris-Diderot), Alexandre Munnier (Nancy) and Franck Sueur (Bordeaux).
LA - eng
UR - http://eudml.org/doc/275626
ER -

References

top
  1. V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 1, 319–361. Zbl0148.45301MR202082
  2. 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), 501-512. Zbl0128.21207MR108207
  3. C. Bjorland, The vortex-wave equation with a single vortex as the limit of the Euler equation. Comm. Math. Phys. Volume 305 (2011), Issue 1, 131–151. Zbl1219.35181MR2802302
  4. Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000), no. 3-4, 737–754. Zbl0970.35110MR1748352
  5. M. Dashti, J. C. Robinson, The motion of a fluid-rigid disc system at the zero limit of the rigid disc radius, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 285–312. Zbl1294.76082MR2781594
  6. R. J. DiPerna, A. J. Majda, Concentrations in regularizations for 2-D incompressible flow. Comm. Pure Appl. Math. 40 (1987), no. 3, 301–345. Zbl0850.76730MR882068
  7. D. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92 (1970), 102–163. Zbl0211.57401MR271984
  8. O. Glass, C. Lacave, F. Sueur, On the motion of a small disk immersed in a two dimensional incompressible perfect fluid, to appear in Bull. SMF. arXiv:1104.5404 Zbl1329.76048
  9. O. Glass, C. Lacave, F. Sueur, On the motion of a small light body immersed in a two dimensional incompressible perfect fluid with vorticity, Preprint 2014. Zbl1329.76048
  10. O. Glass, A. Munnier, F. Sueur, Dynamics of a point vortex as limits of a shrinking solid in an irrotational fluid. Preprint 2014. arXiv:1402.5387 
  11. O. Glass, F. Sueur, On the motion of a rigid body in a two-dimensional irregular ideal flow. SIAM Journal Math. Analysis. Volume 44 (2013), Issue 5, 3101–3126. Zbl1325.76026MR3023405
  12. O. Glass, F. Sueur, The movement of a solid in an incompressible perfect fluid as a geodesic flow. Proc. Amer. Math. Soc. 140 (2012), no. 6, 2155–2168. Zbl1261.76009MR2888201
  13. O. Glass, F. Sueur, Uniqueness results for weak solutions of two-dimensional fluid-solid systems, Preprint 2012, arXiv:1203.2894. Zbl06481060MR2888201
  14. O. Glass, F. Sueur, T. Takahashi, Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid, Ann. Sci. de l’École Normale Supérieure 45 (2012), fasc. 1, 1–51. Zbl1311.35217MR2961786
  15. C. Grotta Ragazzo, J. Koiller, W. M. Oliva, On the motion of two-dimensional vortices with mass. Nonlinear Sci. 4 (1994), no. 5, 375–418. Zbl0808.76015MR1291115
  16. J.-G. Houot, J. San Martin, M. Tucsnak, Existence and uniqueness of solutions for the equations modelling the motion of rigid bodies in a perfect fluid. J. Funct. Anal. 259 (2010), no. 11, 2856–2885. Zbl1200.35222MR2719277
  17. D. Iftimie, M. C. Lopes Filho, H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Comm. Partial Diff. Eqns. 28 (2003), no. 1 & 2, 349–379. Zbl1094.76007MR1974460
  18. C. Lacave, Two-dimensional incompressible ideal flow around a small curve, Comm. Partial Diff. Eqns. 37:4 (2012), 690–731. Zbl1244.35107MR2901062
  19. C. Lacave, E. Miot, Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex. SIAM J. Math. Anal. 41 (2009), no. 3, 1138–1163. Zbl1189.35259MR2529959
  20. H. Lamb, Hydrodynamics. Reprint of the 1932 sixth edition. Cambridge University Press, 1993. Zbl0828.01012MR1317348
  21. C. Marchioro, M. Pulvirenti, Vortices and localization in Euler flows. Comm. Math. Phys. Volume 154 (1993), Issue 1, 49–61. Zbl0774.35058MR1220946
  22. C. Marchioro, M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, 1994. Zbl0789.76002MR1245492
  23. A. Moussa, F. Sueur, On a Vlasov-Euler system for 2D sprays with gyroscopic effects. Asymptot. Anal. 81 (2013), no. 1, 53–91. Zbl1284.35440MR3060032
  24. A. Munnier, Locomotion of Deformable Bodies in an Ideal Fluid: Newtonian versus Lagrangian Formalisms. J. Nonlinear Sci (2009), no. 19, 665–715. Zbl1178.74055MR2559025
  25. J. Ortega, L. Rosier, T. Takahashi, On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), no. 1, 139–165. Zbl1168.35038MR2286562
  26. C. Rosier, L. Rosier, Smooth solutions for the motion of a ball in an incompressible perfect fluid. Journal of Functional Analysis, 256 (2009), no. 5, 1618–1641. Zbl1173.35105MR2490232
  27. A. L. Silvestre, T. Takahashi, The motion of a fluid-rigid ball system at the zero limit of the rigid ball radius. Arch. Ration. Mech. Anal. 211 (2014), no. 3, 991–1012. Zbl1293.35255MR3158813
  28. B. Turkington, On the evolution of a concentrated vortex in an ideal fluid. Arch. Rational Mech. Anal. 97 (1987), no. 1, 75-87. Zbl0623.76013MR856310
  29. J. Vankerschaver, E. Kanso, J. E. Marsden, The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices. J. Geom. Mech. 1 (2009), no. 2, 223-266. Zbl1191.53055MR2525759
  30. Y. Wang, Z. Xin, Existence of weak solutions for a two-dimensional fluid-rigid body system. J. Math. Fluid Mech. 15 (2013), no. 3, 553–566. Zbl1273.35228MR3084324
  31. V. I. Yudovich, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 1032–1066 (in Russian). English translation in USSR Comput. Math. & Math. Physics 3 (1963), 1407–1456. Zbl0147.44303MR158189

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.