ANALYSIS OF ONE-DIMENSIONAL NON-NEWTONIAN MODELS FOR SIMULATION OF BLOOD FLOW IN ARTERIES

Abstract

The paper is devoted to the theoretical analysis of one-dimensional (1D) models of blood flow. The non-Newtonian nature of blood is taken into account. The generalized Newtonian models, in which the dynamic viscosity is dependent only on the shear rate, are considered. The models are constructed by averaging the simplified Navier–Stokes system in cylindrical coordinates. The 1D models, corresponding to Newtonian, Power Law, Carreau, Carreau–Yasuda, Cross, Simplified Cross, Yeleswarapu, Modified Yeleswarapu, and Quemada models, are compared in this paper. The comparison is performed in order to estimate the influence of non-Newtonian frictional terms, velocity profiles and hematocrit values on the deviations of the solutions from the solution for the Newtonian case. For this purpose, two problems that can be solved analytically are considered: problems for the infinite interval and finite interval with periodic conditions. For the quantitative comparison of solutions, the non-Newtonian factor, which characterizes the relative deviations of solutions from the solution for the Newtonian model, is introduced. The analytical solutions of nonlinear problems are obtained by the perturbation method. As it is demonstrated for both solved problems, the relative deviations increase with the flattening of the velocity profile. For the models that depended on hematocrit, the non-Newtonian factor values increased with the increase in hematocrit value. The solutions obtained in the paper can be used as a tool for the comparison of different 1D models of blood flow and for the testing of programs that implement numerical algorithms.