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 nondimensional 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 %.