The Large Deformation of Nonlinearly Elastic Tubes in Two-Dimensional Flows

Abstract

This paper treats the large deformation of closed nonlinearly elastic cylindrical tubes (rings) under an external pressure field generated by the steady, irrotational, two-dimensional flow of an incompressible, inviscid fluid. The flow is assumed to have a prescribed velocity U and pressure P at infinity. The deformation of the tubes is described by a geometrically exact theory of rods. The parameters U and P and the deformed shape of the ring uniquely determine the velocity field of the steady flow exterior to the tube. It is shown that velocity of the flow on the tube depends continuously and compactly on the function describing the shape. The pressure field on the ring, depending on U, P, and the velocity field on the tube, is substituted into the equilibrium equations for the tube, yielding a system of ordinary functional-differential equations. These are converted into a fixed-point form, which is analyzed by a global implicit function theorem. Refined results from conformal mapping theory are used to handle serious technical difficulties with regularity, which apparently do not arise in the study of flows past rigid obstacles.