Dynamics of a small rigid body in a perfect incompressible fluid
Journées équations aux dérivées partielles (2014), article no. 3, 20 p.

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).

@article{JEDP_2014____A3_0,
author = {Glass, Olivier},
title = {Dynamics of a small rigid body in a perfect incompressible fluid},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2014},
doi = {10.5802/jedp.106},
language = {en},
url = {http://www.numdam.org/item/JEDP_2014____A3_0}
}

Glass, Olivier. Dynamics of a small rigid body in a perfect incompressible fluid. Journées équations aux dérivées partielles (2014), article  no. 3, 20 p. doi : 10.5802/jedp.106. http://www.numdam.org/item/JEDP_2014____A3_0/

[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. | Numdam | MR 202082 | Zbl 0148.45301

[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. | MR 108207 | Zbl 0128.21207

[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. | MR 2802302 | Zbl 1219.35181

[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. | MR 1748352 | Zbl 0970.35110

[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. | MR 2781594 | Zbl 1294.76082

[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. | MR 882068 | Zbl 0850.76730

[7] D. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92 (1970), 102–163. | MR 271984 | Zbl 0211.57401

[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

[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.

[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. | MR 3023405

[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. | MR 2888201 | Zbl 1261.76009

[13] O. Glass, F. Sueur, Uniqueness results for weak solutions of two-dimensional fluid-solid systems, Preprint 2012, arXiv:1203.2894. | MR 2888201

[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. | Numdam | MR 2961786

[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. | MR 1291115 | Zbl 0808.76015

[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. | MR 2719277 | Zbl 1200.35222

[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. | MR 1974460 | Zbl 1094.76007

[18] C. Lacave, Two-dimensional incompressible ideal flow around a small curve, Comm. Partial Diff. Eqns. 37:4 (2012), 690–731. | MR 2901062 | Zbl 1244.35107

[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. | MR 2529959 | Zbl 1189.35259

[20] H. Lamb, Hydrodynamics. Reprint of the 1932 sixth edition. Cambridge University Press, 1993. | MR 1317348 | Zbl 0828.01012

[21] C. Marchioro, M. Pulvirenti, Vortices and localization in Euler flows. Comm. Math. Phys. Volume 154 (1993), Issue 1, 49–61. | MR 1220946 | Zbl 0774.35058

[22] C. Marchioro, M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids. Applied Mathematical Sciences 96, Springer-Verlag, 1994. | MR 1245492 | Zbl 0789.76002

[23] A. Moussa, F. Sueur, On a Vlasov-Euler system for 2D sprays with gyroscopic effects. Asymptot. Anal. 81 (2013), no. 1, 53–91. | MR 3060032 | Zbl 1284.35440

[24] A. Munnier, Locomotion of Deformable Bodies in an Ideal Fluid: Newtonian versus Lagrangian Formalisms. J. Nonlinear Sci (2009), no. 19, 665–715. | MR 2559025 | Zbl 1178.74055

[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. | Numdam | MR 2286562 | Zbl 1168.35038

[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. | MR 2490232 | Zbl 1173.35105

[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. | MR 3158813 | Zbl 1293.35255

[28] B. Turkington, On the evolution of a concentrated vortex in an ideal fluid. Arch. Rational Mech. Anal. 97 (1987), no. 1, 75-87. | MR 856310 | Zbl 0623.76013

[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. | MR 2525759 | Zbl 1191.53055

[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. | MR 3084324 | Zbl 1273.35228

[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. | MR 158189 | Zbl 0147.44303