Geodesic
Mathematical Model of Dynamics of an Elastic Body in a Viscous Incompressible Fluid
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 9 (2009) no. 4, pp. 76-89 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, a mathematical model that describes the dynamics of an elastic body immersed in a viscous incompressible fluid is proposed. The displacement of the body is composed of a rigid motion (translation and rotation) combined with elastic deformations. Assuming the deformations to be small, we suppose that the form of the body does not change during the motion. At the same time, the velocity of the elastic deformations is not small and cannot be ignored. By this reason, at each point of the body, we put an oscillator that is described by the equations of linear elasticity. At the boundary of the body, the fluid velocity is matched with the sum of the velocities of the rigid motion and the elastic oscillations.
Keywords: viscosity, elastic body, small deformations, large displacements. 
@article{VNGU_2009_9_4_a8,
     author = {V. N. Starovoitov and B. N. Starovoitova},
     title = {Mathematical {Model} of {Dynamics} of an {Elastic} {Body} in a {Viscous} {Incompressible} {Fluid}},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {76--89},
     year = {2009},
     volume = {9},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2009_9_4_a8/}
}
TY  - JOUR
AU  - V. N. Starovoitov
AU  - B. N. Starovoitova
TI  - Mathematical Model of Dynamics of an Elastic Body in a Viscous Incompressible Fluid
JO  - Sibirskij žurnal čistoj i prikladnoj matematiki
PY  - 2009
SP  - 76
EP  - 89
VL  - 9
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VNGU_2009_9_4_a8/
LA  - ru
ID  - VNGU_2009_9_4_a8
ER  - 
%0 Journal Article
%A V. N. Starovoitov
%A B. N. Starovoitova
%T Mathematical Model of Dynamics of an Elastic Body in a Viscous Incompressible Fluid
%J Sibirskij žurnal čistoj i prikladnoj matematiki
%D 2009
%P 76-89
%V 9
%N 4
%U http://geodesic.mathdoc.fr/item/VNGU_2009_9_4_a8/
%G ru
%F VNGU_2009_9_4_a8
V. N. Starovoitov; B. N. Starovoitova. Mathematical Model of Dynamics of an Elastic Body in a Viscous Incompressible Fluid. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 9 (2009) no. 4, pp. 76-89. http://geodesic.mathdoc.fr/item/VNGU_2009_9_4_a8/

[1] Grandmont C., Maday Y., “Existence for Unsteady Fluid-Structure Interaction Problem”, Math. Model. Numer. Anal., 34 (2000), 609–636 | DOI | MR | Zbl

[2] Grandmont C., “Existence for a Three-Dimensional Steady State Fluid-Structure Interaction Problem”, J. Math. Fluid Mech., 4 (2002), 76–94 | DOI | MR | Zbl

[3] Chambolle A., Desjardins B., Esteban M. J., Grandmont C., “Existence of Weak Solutions for an Usteady Fluid–Plate Interaction Problem”, J. Math. Fluid Mech., 7 (2005), 368–404 | DOI | MR | Zbl

[4] H. Beirão da Veiga, “On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem”, J. Math. Fluid Mech., 6:1 (2004), 21–52 | DOI | MR | Zbl

[5] Desjardins B., Esteban M. J., Grandmont C., Le Tallec P., “Weak Solutions for a Fluid-Elastic Structure Interaction Model”, Rev. Mat. Complut., 14:2 (2001), 523–538 | MR | Zbl

[6] Boulakia M., “Existence of Weak Solutions for the Motion of an Elastic Structure in an Incompressible Viscous Fluid”, C. R. Acad. Sci. Paris, Ser. I, 336:12 (2003), 985–990 | DOI | MR | Zbl

[7] Boulakia M., “Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid”, J. Math. Fluid Mech., 9 (2007), 262–294 | DOI | MR | Zbl

[8] Coutand D., Shkoller S., “Motion of an Elastic Solid Inside an Incompressible Viscous Fluid”, Arch. Ration. Mech. Anal., 176:1 (2005), 25–102 | DOI | MR | Zbl

[9] Syarle F., Matematicheskaya teoriya uprugosti, per. s angl., Mir, M., 1992

[10] Trusdell K., Pervonachalnyi kurs ratsionalnoi mekhaniki sploshnykh sred, per. s angl., Mir, M., 1975

[11] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, per. s fr., Mir, M., 1972 | MR

[12] Temam R., Matematicheskie zadachi teorii plastichnosti, per. s fr., Nauka, M., 1991 | MR | Zbl