Stability of Stationary Viscous Incompressible Flow Around a Rigid Body Performing a Translation

• Published in: Journal of mathematical fluid mechanics 2018-09, Vol.20 (3), p.937-967
• Main Author: Deuring, Paul
• Format: Article
• Language: English
• Subjects: Boundary value problems; Classical and Continuum Physics; Dirichlet problem; Eigenvalues; Flow stability; Fluid dynamics; Fluid flow; Fluid mechanics; Fluid- and Aerodynamics; Incompressible flow; Incompressible fluids; Linear operators; Mathematical Methods in Physics; Navier-Stokes equations; Physics; Physics and Astronomy; Rigid structures; Rigid-body dynamics; Theoretical mathematics; Velocity distribution
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-017-0350-5

Suppose a rigid body moves steadily and without rotation in a viscous incompressible fluid, and the flow around the body is steady, too. Such a flow is usually described by the stationary Navier–Stokes system with Oseen term, in an exterior domain. An Oseen term arises because the velocity field is scaled in such a way that it vanishes at infinity. In the work at hand, such a velocity field, denoted by U , is considered as given. We study a solution of the incompressible evolutionary Navier–Stokes system with the same right-hand side and the same Dirichlet boundary conditions as the stationary problem, and with U + u 0 as initial data, where u 0 is a H 1 -function. Under the assumption that the H 1 -norm of u 0 is small ( u 0 a “perturbation of U ”) and that the eigenvalues of a certain linear operator have negative real part, we show that ‖ ∇ ( v ( t ) - U ) ‖ 2 → 0 ( t → ∞ ) (“stability of v ”), where v denotes the velocity part of the solution to the initial-boundary value problem under consideration.