Stability of an Elastically Supported Cylinder in a Circular Viscous Fluid Flow

Abstract

A system consisting of two cylinders (the inner cylinder is elastically supported and the outer cylinder is rigidly fixed) and a circulation flow of viscous incompressible fluid between them is considered. In the equilibrium position the cylinders are coaxial and the inner one rotates at a constant angular velocity Ω. An expression is obtained of the force acting on the inner cylinder from the side of the fluid in the approximation of large Reynolds numbers at a small deviation of the inner cylinder from its equilibrium position. It is shown that the force has a hereditary character and depends on the entire trajectory of motion. The stability of the position of the inner elastically supported cylinder is studied in the linear approximation taking into account the hereditary force acting on the inner cylinder.