Joined dissimilar thin elastic tubes containing steady perfect flow

Abstract

Two tubes of circular cross-section and of the same radius initially, but composed of different elastic materials, are joined together to form a single straight tube with discontinuous properties. Stretching in the axial direction causes radial displacement that varies along the length of the tube. In a perfect inviscid incompressible flow through the tube of variable cross-section the internal pressure varies as described by Bernoulli's equation, and the variable pressure also causes variation of the radial displacement. The equations of coupled finite deformation and fluid pressure problem can be integrated explicitly (using membrane theory for the tube) for arbitrary material properties, but determination of the integration constants is not trivial. The results are interpreted numerically for Mooney materials. Also considered in this context is the similar problem where two semi-infinite cylindrical membranes of the same material are separated by a cuff of different material. Numerical illustrations are obtained for various upstream velocities. The results obtained here thus solve the problem of steady internal pressure loading of this type in extended dissimilar thin isotropic tubes. The tube will become unstable if the fluid velocity is too large. Applications to engineering structures are possible.