Traveling waves in a one-dimensional model of hemodynamics

We consider a one-dimensional model of hemodynamics—blood flow in the blood vessels—which is based on the Navier-Stokes equations averaged over the cross section of the vessel, and conjugate with a linear or nonlinear model for the elastic wall of the vessel. The objective is to study traveling wave solutions using this model. For such solutions, the system of partial differential equations reduces to an ordinary differential equation of the fourth order. The only singular point of the corresponding system of differential equations is found. It is established that at the singular point, the linearization matrix of the system has real or complex roots for different values of the parameters of the problem. With a special choice of the parameters, it has four complex conjugate roots with a nonzero real part or purely imaginary roots. For this case, the effect of the model parameter corresponding to the viscoelastic response of the vessel wall on the solution is investigated. Numerical experiments are performed to verify and analyze the results, and various modes of blood movement are discussed.