Comptes Rendus
no. 11-12
Partial differential equations
Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp

Presented by: the Editorial Board

Matthieu Hillairet1; Diaraf Seck2, 3; Lamine Sokhna1, 2, 3
1 IMAG, Université de Montpellier, CNRS, Montpellier, France
2 Laboratoire de mathématiques de la décision et d'analyse numérique (LMDAN), FASEG, UCAD, Senegal
3 École doctorale de mathématiques et informatique, UCAD, Senegal
Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1156-1166.

In this note, we consider the fall of an axisymmetric body in a perfect fluid over a ramp. It was shown in [12] that the possibility of a collision between the body and the ramp is related to the asymptotics of the so-called added mass when the distance between the ramp and the body goes to 0. We propose here a new method to compute this added mass, which provides simultaneously an approximation of an associated fluid velocity field in the gap between the ramp and the body.

Dans cette note, nous considérons la chute d'un solide axisymétrique dans un fluide parfait au-dessus d'un plan. Il est connu [12] que l'éventualité d'un contact entre le solide et le plan est reliée à l'asymptotique de l'effet de masse ajoutée quand la distance entre le plan et le solide tend vers 0. Nous proposons une nouvelle méthode pour calculer cet effet de masse ajoutée, qui fournit simultanément l'asymptotique d'un champ de vitesses associé entre le solide et le plan.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.10.001

Matthieu Hillairet 1; Diaraf Seck 2, 3; Lamine Sokhna 1, 2, 3

1 IMAG, Université de Montpellier, CNRS, Montpellier, France
2 Laboratoire de mathématiques de la décision et d'analyse numérique (LMDAN), FASEG, UCAD, Senegal
3 École doctorale de mathématiques et informatique, UCAD, Senegal 
@article{CRMATH_2018__356_11-12_1156_0,
     author = {Matthieu Hillairet and Diaraf Seck and Lamine Sokhna},
     title = {Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1156--1166},
     publisher = {Elsevier},
     volume = {356},
     number = {11-12},
     year = {2018},
     doi = {10.1016/j.crma.2018.10.001},
     language = {en},
}
TY  - JOUR
AU  - Matthieu Hillairet
AU  - Diaraf Seck
AU  - Lamine Sokhna
TI  - Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 1156
EP  - 1166
VL  - 356
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2018.10.001
LA  - en
ID  - CRMATH_2018__356_11-12_1156_0
ER  - 
%0 Journal Article
%A Matthieu Hillairet
%A Diaraf Seck
%A Lamine Sokhna
%T Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp
%J Comptes Rendus. Mathématique
%D 2018
%P 1156-1166
%V 356
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2018.10.001
%G en
%F CRMATH_2018__356_11-12_1156_0
Matthieu Hillairet; Diaraf Seck; Lamine Sokhna. Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1156-1166. doi : 10.1016/j.crma.2018.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.10.001/

[1] D. Coutand Finite time singularity formation for moving interface Euler equations (preprint) | arXiv

[2] R.G. Cox The motion of suspended particles almost in contact, Int. J. Multiph. Flow, Volume 1 (1974), pp. 343-371

[3] E. Feireisl On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., Volume 3 (2003) no. 3, pp. 419-441

[4] D. Gérard-Varet; M. Hillairet Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., Volume 195 (2010) no. 2, pp. 375-407

[5] D. Gérard-Varet; M. Hillairet Computation of the drag force on a sphere close to a wall: the roughness issue, ESAIM: Math. Model. Numer. Anal., Volume 46 (2012) no. 5, pp. 1201-1224

[6] T.I. Hesla Collisions of Smooth Bodies in Viscous Fluids: A Mathematical Investigation, University of Minnesota, Twin Cities, MN, USA, October 2005 (PhD thesis)

[7] M. Hillairet; T. Kelaï Justification of lubrication approximation: an application to fluid/solid interactions, Asymptot. Anal., Volume 95 (2015) no. 3–4, pp. 187-241

[8] M. Hillairet; T. Takahashi Blow up and grazing collision in viscous fluid solid interaction systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 27 (2010) no. 1, pp. 291-313

[9] J.-G. Houot; A. Munnier On the motion and collisions of rigid bodies in an ideal fluid, Asymptot. Anal., Volume 56 (2008) no. 3–4, pp. 125-158

[10] G. Joseph Collisional Dynamics of Macroscopic Particles in a Viscous Fluid, California Institute of Technology, Pasadena, CA, USA, May 2003 (PhD thesis)

[11] C. Marchioro; M. Pulvirenti Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994

[12] A. Munnier; K. Ramdani Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid, SIAM J. Math. Anal., Volume 47 (2015) no. 6, pp. 4360-4403

[13] J.A. San Martín; V. Starovoitov; M. Tucsnak Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid, Arch. Ration. Mech. Anal., Volume 161 (2002) no. 2, pp. 113-147

[14] V.N. Starovoĭtov Nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid, J. Math. Sci., Volume 130 (2005) no. 4, pp. 4893-4898

[15] R. Temam Navier–Stokes Equations, North-Holland Publ. Co., Amsterdam, The Netherlands, 1977

Cited by Sources:

Comments - Policy