Three-Dimensional Velocity Field for Blood Flow Using the Power-Law Viscosity Function

Abstract

The three-dimensional model associated with blood fl w where viscosity depends on shear-rate, such power-law type dependence, is a complex model to study in terms of computational optimization, which in many relevant situations becomes infeasible. In order to simplify the three-dimensional model and as an alternative to classic one-dimensional models, we will use the Cosserat theory related with flui dynamics to approximate the velocity fiel and thus obtain a one-dimensional system consisting of an ordinary differential equation depending only on time and on a single spatial variable, the fl w axis. From this reduce system, we obtain the unsteady equation for the mean pressure gradient depending on the volume fl w rate, Womersley number and the fl w index over a finit section of the tube geometry. Attention is focused on some numerical simulations for constant and non-constant mean pressure gradient using a Runge-Kutta method and on the analysis of perturbed fl ws. In particular, given a specifi data we can get information about the volume fl w rate and consequently we can illustrate the three-dimensional velocity fiel on the constant circular cross-section of the tube. Moreover, we compare the three-dimensional exact solution for steady volume fl w rate with the corresponding one-dimensional solution obtained by the Cosserat theory.