Geodesic
Solvability of the unsteady problem of the motion of a~rigid body in a~flow of a~viscous incompressible fluid in a~pipe of arbitrary section
Sibirskij žurnal industrialʹnoj matematiki, Tome 20 (2017) no. 3, pp. 80-91.

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We prove the existence of a generalized weak solution to an unsteady problem of a motion of a rigid body in a flow of a viscous incompressible fluid. The flow of the fluid obeys the Navier–Stokes equations and tends to a Poiseuille flow at infinity. The body moves in accordance with the laws of classical mechanics under the action of the ambient fluid and the gravity force directed along the cylinder. Collisions of the body with the boundary of the flow domain are not allowed, and hence the problem is considered until the body approaches the boundary.
Keywords: Navier–Stokes equations, solid body, cylindrical pipe, noncompact boundary. 
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     title = {Solvability of the unsteady problem of the motion of a~rigid body in a~flow of a~viscous incompressible fluid in a~pipe of arbitrary section},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
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V. N. Starovoitov; B. N. Starovoitova. Solvability of the unsteady problem of the motion of a~rigid body in a~flow of a~viscous incompressible fluid in a~pipe of arbitrary section. Sibirskij žurnal industrialʹnoj matematiki, Tome 20 (2017) no. 3, pp. 80-91. http://geodesic.mathdoc.fr/item/SJIM_2017_20_3_a8/

[1] Hoffmann K.-H., Starovoitov V. N., “On a motion of a solid body in a viscous fluid. Twodimensional case”, Adv. Math. Sci. Appl., 9:2 (1999), 633–648 | MR | Zbl

[2] Desjardins B., Esteban M. J., “Existence of weak solutions for the motion of rigid bodies in a viscous fluid”, Arch. Rational Mech. Anal., 146:1 (1999), 59–71 | DOI | MR | Zbl

[3] Hoffmann K.-H., Starovoitov V. N., “Zur Bewegung einer Kugel in einer zähen Flüssigkeit”, Doc. Math., 5 (2000), 15–21 | MR | Zbl

[4] Starovoitov V. N., Neregulyarnye zadachi gidrodinamiki, Dis. $\dots$ dokt. fiz.-mat. nauk, Novosibirsk, 2000

[5] Conca C., San Martín J. A., Tucsnak M., “Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid”, Comm. Partial Differential Equations, 25:5–6 (2000), 1019–1042 | MR | Zbl

[6] Gunzburger M. D., Lee H.-C., Seregin G., “Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions”, J. Math. Fluid Mech., 2:3 (2000), 219–266 | DOI | MR | Zbl

[7] San Martín J. A., Starovoitov V. N., Tucsnak M., “Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid”, Arch. Rational Mech. Anal., 161:2 (2002), 113–147 | DOI | MR | Zbl

[8] Galdi G. P., “On the motion of a rigid body in a viscous liquid: a mathemati analysis with applications”, Handbook of Math. Fluid Dynamics, v. I, Amsterdam, 2002, 653–791 | DOI | MR | Zbl

[9] Takahashi T., Tucsnak M., “Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid”, J. Math. Fluid Mech., 6:1 (2004), 53–77 | DOI | MR | Zbl

[10] Plotnikov P. I., Sokolowski J., “Shape derivative of drag functional”, SIAM J. Control Optim., 48:7 (2010), 4680–4706 | DOI | MR | Zbl

[11] Plotnikov P. I., Sokolowski J., “Shape Optimization for Navier–Stokes Equations”, Control of coupled partial differential equations, Internat. Ser. Numer. Math., 155, Basel, 2007, 249–267 | MR | Zbl

[12] Starovoitov V. N., “Statsionarnoe reshenie zadachi o dvizhenii shara v stoksovom techenii Puazeilya”, Sib. zhurn. industr. matematiki, 18:3 (2015), 76–85 | DOI | MR | Zbl

[13] Hillairet M., Takahashi T., “Collisions in three-dimensional fluid structure interaction problems”, SIAM J. Math. Anal., 40:6 (2009), 2451–2477 | DOI | MR | Zbl

[14] Temam R., Matematicheskie zadachi teorii plastichnosti, Nauka, M., 1991 | MR

[15] Starovoitov V. N., “Behavior of a rigid body in an incompressible viscous fluid near a boundary”, Free Boundary Problems, Internat. Ser. Numer. Math., 147, Basel, 2004, 313–327 | MR