The coupled problem of a solid oscillating in a viscous fluid under the action of an elastic force

Abstract

The torsional oscillations are studied of a solid of revolution under the action of elastic torque inside a container with a viscous incompressible fluid. We prove the asymptotic stability of the static equilibrium. We use the two approaches: the direct Lyapunov and linearization methods. The global asymptotic stability is established using a one-parameter family of Lyapunov functionals. Then small oscillations are studied of the fluid-solid system. The linearized operator of the problem of a solid oscillating in a fluid can be realized as an operator matrix obtained by appending two scalar rows and two columns to the Stokes operator. This operator is therefore a two-dimensional bordering of the Stokes operator and inherits many properties of the latter; in particular, the spectrum is discrete. The eigenvalue problem for the linearized operator is reduced to solving a dispersion equation. Inspection of the equation shows that all eigenvalues lie inside the right (stable) half-plane. Basing on this, we justify the linearization. Using an abstract theorem of Yudovich, we prove the asymptotic stability in a scale of function spaces, the infinite differentiability of solutions, and the decay of all their derivatives in time.