Fluid flow in a tube with an elastic membrane insertion

The unsteady flow of a viscous incompressible fluid in a circular tube with an elastic insertion is studied numerically. The deformation of the elastic membrane is obtained by the theory of finite elasticity whose equations are solved simultaneously with the fluid equations in the axisymmetric approximation. The elastic wall expands outwards due to the positive transmural pressure and represents an idealized model for the response of pathologies in large arteries. It is found that if either the fluid discharge or the reference pressure are imposed downstream of the insertion, the fluid-wall interaction develops travelling waves along the membrane whose period depends on membrane elasticity; these are unstable in a perfectly elastic membrane and are stabilized by viscoelasticity. In the reversed system, when the fluid discharge is imposed on the opposite side, the stable propagation phenomenon remains the same because of symmetry arguments. Such arguments do not apply to the originally unstable behaviour. In this case, even when the membrane is perfectly elastic, propagation is damped and two natural fluctuations appear in the form of stationary waves. In all cases the resonance of the fluid-wall interaction has been analysed. Comparisons with previously observed phenomena and with results of analogous studies are discussed.