Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

Abstract

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.

Modulation of nonlinear waves in a fluid‐filled elastic tube with stenosis

“…Most of the works on wave propagation in compliant tubes have considered small amplitude waves ignoring the nonlinear effects and focused on the dispersive character of waves (see Atabek and Lew [2], Rachev [13], and Demiray [5]). However, when the nonlinear terms arising from the constitutive equations and kinematical relations are introduced, one has to consider either finite amplitude or small-but-finite amplitude waves, depending on the order of nonlinearity.…”

The propagation of a liquid bolus along a liquid-lined flexible tube

“…The model we use for the elastic response of the wall was originally derived by Atabek & Lew (1966) and used by Halpern & Grotberg (1992) in modelling airway closure. The tube wall is characterized by a constant longitudinal tension T l and a circumferential tension that is constituted in terms of its Young's modulus, E, thickness, d, and Poisson's ratio, ν.…”

New analytic solutions for wave propagation in flexible, tapered vessels with reference to mammalian arteries

“…Witzig (1914) was the first to show the effects of fluid viscosity on propagation characteristics, impedance and velocity profiles. Significant developments of the theory took place in the 1950s and 1960s: Womersley (1955Womersley ( , 1957, Atabek & Lew (1961) and Atabek (1968), among † Email address for correspondence: g.papadakis@ic.ac.uk 466 G. Papadakis many others). A survey of the work in the area until the late 1960s is provided by Cox (1969).…”