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

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.