Hostname: page-component-77f85d65b8-g98kq Total loading time: 0 Render date: 2026-04-22T18:21:57.147Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence for an Unsteady Fluid-StructureInteraction Problem

Published online by Cambridge University Press:  15 April 2002

Céline Grandmont  and
Yvon Maday
Céline Grandmont
Affiliation:
Ceremade, Université Paris Dauphine, 75775 Paris Cedex 16, France.
Yvon Maday
Affiliation:
Laboratoire ASCI, Bât. 506, Université Paris Sud, 91405 Orsay, Cedex, France et ONERA, 29 avenue Division Leclerc, 92322 Châtillon, France.
Get access

Abstract

We study the well-posedness of an unsteady fluid-structure interaction problem.We consider a viscous incompressible flow, which is modelled by theNavier-Stokes equations. The structure is a collection of rigid moving bodies. The fluiddomain depends on time and is defined by the position of the structure, itself resultingfrom a stress distribution coming from the fluid. The problem is thennonlinear and the equations we deal with are coupled. We prove its localsolvability in time through two fixed point procedures.

Keywords

Navier-Stokesfluid structure interaction.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 609 - 636
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

R.A. Adams, Sobolev spaces. Academic Press, New York NY, San Francisco CA, London (1975).
G. Allain, Un problème de Navier-Stokes avec surface libre. Thèse de troisième cycle de l'Université Paris VI, France (1983).
G. Allain, Small-time existence for the Navier-Stokes equations with a free surface. Appl. Math. Optim., 16 (1987) 37-50.
J.T. Beale, The Initial Value Problem for the Navier-Stokes Equation with a Free Surface. Comm. Pure Appl. Math., XXXIV (1981) 359-392.
H. Brezis, Analyse fonctionnelle: Théorie et applications. Masson, Paris (1983).
Conca, C., San Martin, J. and Tucsnak, M., Motion of a rigid body in viscous fluid. C. R. Acad. Sci. Paris Série I 32 (1999) 473-478. CrossRef
B. Desjardins and M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59-71.
B. Desjardins and M.J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Partial Differ. Eq. (to appear).
G. Duvaut, Mécanique des milieux continus. Masson, Paris, Milan, Barcelone, Mexico (1990).
V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin Heidelberg (1986).
R. Glowinski and B. Maury, Fluid-particule flow: a symmetric formulation. C. R. Acad. Sci. Paris Sér. I Math. t. 324, (1997) 1079-1084.
C. Grandmont and Y. Maday, Existence de solutions d'un problème de couplage fluide-structure bidimensionnel instationnaire. C. R. Acad. Sci. Paris Sér. I Math. t. 326, (1998) 525-530.
Heywood, J. and Rannacher, R., Finite-element approximation of the nonstationnary Navier-Stokes problem. Part III. Smoothing property and higher order error estimates for spatial discretisation. SIAM J. Numer. Anal. 25 (1988) 489-512. CrossRef
J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. T. I et II, Dunod, Paris (1968).
Serre, D., Chute libre d'un solide dans un fluide visqueux incompressible : Existence. Japan J. Appl. Math. 4 (1987) 33-73.
Solonnikov, V.A., Solvability of a Problem on the Motion of a Viscous Incompressible Fluid Bounded by a Free Surface. Math. USSR Izvestiya 4-1 (1977) 1388-1424.
Solonnikov, V.A., On the Transiant Motion of an Isolated Volume of Viscous Incompressible Fluid. Math. USSR Izvestiya 31 (1988) 381-405. CrossRef
Solonnikov, V.A., Unsteady motion of a finite mass of fluid, bounded by a free surface. J. Soviet Math. 40 (1988) 672-686. CrossRef
R. Temam, Navier-Stokes Equations. North-Holland Publishing Company (1977).