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 traveling 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 traveling 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.

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

“…It seems that none of the well-known constitutive assumptions guarantee that this is the case for all deformations, but it is known that under the membrane assumption ∂P/∂λ a is at least positive before the condition for localized bulging is satisfied (Fu & Il'ichev, 2015). In the present 3D setting, for each material model that we use the above assumption is checked numerically by inspecting the contour plots of ∂F/∂λ z = 0 and ∂P/∂λ a = 0 in the (λ a , λ z )-plane.…”

“…However, it was recognized by Fu et al (2008) and Pearce & Fu (2010) that it is precisely this zero mode number case that corresponds to localized bulging when nonlinear effects are brought into play. It was further shown in Fu & Il'ichev (2015) that in the case of fixed resultant axial force (hereafter simply referred to as axial force), the initiation pressure for localized bulging corresponds to the maximum pressure in uniform inflation, but this correspondence may no longer hold when other loading conditions are applied at the ends. In particular, when the axial stretch is fixed during inflation, localized bulging may occur even if the pressure in uniform inflation does not have a maximum.…”

Stability of an inflated hyperelastic membrane tube with localized wall thinning

“…In the context of blood flow in an artery, such a large amplitude bulge would modify the flow properties, which in turn would provide a possible mechanism triggering arterial wall growth and remodeling. Based on our recent study on the effects of a mean flow [6], we envisage that a mean flow may stabilize configurations corresponding to the upper branch. If this were proved to be the case, stable bulged configurations with even larger amplitude would become possible and their effects on the blood flow would be even more pronounced.…”

“…The most straightforward approach is the so-called determinant method, which determines α by solving the equation det (y 1 , y 2 , y 6) where the left hand side can be evaluated at any appropriate matching point on the real line. The method suffers from the "stiffness" problem in the sense that one column can get dominated by another column due to different exponential behaviour.…”

“…It was found that in the homogeneous case although internal fluid inertia would reduce the growth rate of the single unstable mode significantly, it alone cannot stabilize the unstable mode completely [13]. Stabilization of the exponential growth of the aneurysm solution takes place in the presence of a non-zero mean flow [6], but due to the translational symmetry of the problem the standing bulging configuration is only orbitally stable or stable in form [11]; it can still propagate axially with a non-zero speed under perturbations.…”