Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow

Abstract

We first give a complete analysis of the dispersion relation for travelling waves propagating in a pre-stressed hyperelastic membrane tube containing a uniform flow. We present an exact formula for the so-called pulse wave velocity, and demonstrate that as any pre-stress parameter is increased gradually, localized bulging would always occur before a superimposed small-amplitude travelling wave starts to grow exponentially. We then study the stability of weakly and fully nonlinear localized bulging solutions that may exist in such a fluid-filled hyperelastic membrane tube. Previous studies have shown that such localized standing waves are unstable under pressure control in the absence of a mean-flow, whether the fluid inertia is taken into account or not. Stability of such localized aneurysm-type solutions is desired when aneurysm formation in human arteries is modelled as a bifurcation phenomenon. It is shown that in the near-critical regime axisymmetric perturbations are governed by the Korteweg–de Vries equation, and so the associated (weakly nonlinear) aneurysm solutions are (orbitally) stable with respect to axisymmetric perturbations. Stability of the fully nonlinear aneurysm solutions are studied numerically using the Evans function method. It is found that for each wall–fluid density ratio there exists a critical mean-flow speed above which no axisymmetric unstable modes can be found, which implies that a fully nonlinear aneurysm solution may be completely stabilized by a mean flow.