The Effects of Non-Linearities on Wave Propagation and Time-Averaged Flow in Elastic Axi-Symmetric Vessels

In this paper, a power series and a Fourier series approach is used to solve the governing equations of motion in an elastic axi-symmetric vessel, assuming that blood is an incompressible Newtonian fluid. For vessels with wall stiffness in the arterial range, the viscosity reduces the wave speed by approximately 10 % and the non-linear terms increases it by approximately 5 % from that predicted by linear wave theory for inviscid fluids. When considering time-averaged flow, spatial perturbations in the flow field were observed, the amplitude being strongly dependent on the amplitude of the temporal perturbations, but only weakly dependent upon the non-dimensional groups governing the equations of motion. This variation was strongly non-linear, increasing rapidly at large amplitudes of perturbation. A 10 % change in radius about its steady state value resulted in spatial perturbations of approximately 4 %.