Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membran
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O R I G I NA L PA P E R
A. T. Il’íchev · V. A. Shargatov · Y. B. Fu
Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube
Received: 27 March 2020 / Revised: 26 May 2020 © The Author(s) 2020
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, c1 ) where c1 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 c0 in the above interval such that solitary waves are spectrally stable if their speed is greater than c0 and unstable otherwise.
1 Introduction 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]. A complete study would need to take into account the fact that blood is non-Newtonian, and the arterial wall is dissipative, dispersive and nonlinear. However, very often only some of these features are considered depending on the particular emphasis of the study. For instance, most of the early literature is based on a linear, long wavelength (non-dispersive) theory, first put forward in [2] and refined in [3]. A linear theory taking into account the leading-order dispersive effect was derived in [4] and was used in [5] to explain the dispersive effect observed in [6]. Weakly nonlinear solitary waves based on various approximate models have been studied in [7–13]. The development of nonlinear elasticity theory makes it possible to describe the constitutive behavior of distensible tubes such as arteries in a more precise manner [14]. Under this theory, the dispersion relation can be derived for a general material model and a finite deformation prior to wave propagation may be taken into account. Figure 1 shows a typical dispersion curve for the Ogden material model when the tube is stretched by 20% prior to wave propagation and the inertia of fluid is neglected. The character of the two branches can be understood by noting that in the limit k → ∞ the ρc2 associated with the upper and lower branches tend to, A. T. Ilíchev Steklov Mathematical Institute, Gubkina Str. 8, Moscow, Russia 119991 V. A. Shargatov National Research Nuclear University “MePHI”, Kashirskoye sh. 31, Mosc
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