Abstract
We first characterize strain solitary waves propagating in a fluid-filled membrane tube when the fluid is stationary prior to wave propagation and the tube is also subjected to a finite stretch. We consider the parameter regime where all traveling waves admitted by the linearized governing equations have nonzero speed. Solitary waves are viewed as waves of finite amplitude that bifurcate from the quiescent state of the system with the wave speed playing the role of the bifurcation parameter. Evolution of the bifurcation diagram with respect to the pre-stretch is clarified. We then study the stability of solitary waves for a representative case that is likely of most interest in applications, the case in which solitary waves exist with speed c lying in the interval [0, c_1) where c_1 is the bifurcation value of c, and the wave amplitude is a decreasing function of speed. It is shown that there exists an intermediate value c_0 in the above interval such that solitary waves are spectrally stable if their speed is greater than c_0 and unstable otherwise.
Highlights
Understanding pulse wave propagation and reflection in distensible fluid-filled tubes has a range of applications such as the detection of the site of obstruction and diagnosis of the health status of arteries [1]
We examine the problem of stability of solitary waves, propagating with a nonzero velocity in a fluid-filled axisymmetric membrane tube
With the aid of the software package Mathematica [24], we find that for each fixed value of bf there exists at least one unstable eigenvalue on the real η-axis for c less than a threshold value c0 = c0(bf, r∞, λ2∞)
Summary
Understanding pulse wave propagation and reflection in distensible fluid-filled tubes has a range of applications such as the detection of the site of obstruction and diagnosis of the health status of arteries [1]. In [17] a stability analysis of the aneurysm solutions in the presence of a mean flow was undertaken and it was found that if the speed of the mean flow is large enough, the aneurysm solutions may be spectrally stable It was found in [18] that for membrane tubes with localized wall thinning there exist two families of bulging solitary waves, and the lower family with amplitudes increasing with the inflation pressure is spectrally stable. In the latter case, we can speak about the standing wave (not orbital) stability, because the problem has no translational invariance any more.
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