Comparison of inviscid and viscid one-dimensional models of blood flow in arteries

Abstract

The paper is devoted to the comparison of one-dimensional blood flow models in application to the solution of model problems. In the viscid case, the non-Newtonian properties of blood are considered. Original one-dimensional models, based on the Carreau, Carreau–Yasuda, Cross, and Powell–Eyring rheological models, are constructed by the averaging of the Navier–Stokes equations.The approach to the analytical solution of problems in the inviscid case is proposed. The originality of the method is based on the small perturbation of the initial rest state, corresponding to zero velocity. It leads to the solution of the linear wave equations. The solutions of three problems — for the infinite, semi-infinite, and finite intervals are obtained. The examples are presented for the small parameter value ∼10−2. Analytical solutions are used for the comparison of different one-dimensional models of blood flow, where the viscosity (Newtonian and non-Newtonian) is considered. The problems for the viscid models are solved numerically by the third-order WENO scheme. As the results of the comparison of models, the effects of the viscosity and velocity profile are analyzed.From a practical viewpoint, the solutions obtained by the perturbation method can be used for the testing of programs for numerical simulations and for the comparison of different blood flow models.