Nonlinear Analysis of Oscillatory Flow, With a Nonzero Mean, in an Elastic Tube (Artery)

Abstract

Oscillatory flow of a Newtonian fluid in an elastic tube, which is a model of blood flow in arteries, is analyzed in this paper. For a rigid tube, the steady flow field can be described by Poiseuille’s law and the unsteady flow field by Womersley’s solution. These are the linearized solutions for flow in elastic tubes. To evaluate the importance of nonlinear effects, a perturbation solution is developed realizing that the amplitude of arterial wall movement is small (typically 5–10 percent of the diameter). The nonlinear effects on the amplitude of the wall shear rate, on the amplitude of the pressure gradient, and on the mean velocity profile have been considered. Nonlinear effects on the oscillatory components depend on Womersley’s unsteadiness parameter (α), the ratio between the mean and amplitude of the flow rate, the diameter variation, and the phase difference between the diameter variation and the flow rate (φ) which is indicative of the degree of wave reflection. On the other hand, the mean velocity profile is found to be dependent on the steady-streaming Reynolds number, R s . When R s is small, the mean velocity profile is parabolic (1 − ξ2 ); however, when R s is large, the velocity profile is distorted by the nonlinear effect and can be described by sin(πξ2 ). The increase of the amplitude and reduction of the mean of wall shear rate as π changes from 0 to −90 deg suggests an indirect mechanism for the role of hypertension in arterial disease: hypertension → increased wave reflection → wall shear stress is reduced and more oscillatory.