A one-dimensional model of blood flow in arteries with friction, convection and unsteady Taylor diffusion based on the Womersley velocity profile

Abstract

In this thesis, we present a one-dimensional model for blood flow in arteries, without assuming an a priori shape for the velocity profile across an artery. We combine the one-dimensional equations for conservation of mass and momentum with the Womersley model for the velocity profile in an iterative way. The pressure gradient of the one-dimensional model drives the Womersley equations, and the velocity profiles calculated then feed back into both the friction and nonlinear parts of the one-dimensional model. Besides enabling us to evaluate the friction correctly and also use the velocity profile to correct the nonlinear terms, the velocity profiles play a central role in the calculation of the effective diffusion coefficient, and convection coefficient, in the theory of Taylor diffusion. We present flow simulations using both structured tree and pure resistance models for the small arteries, and compare the resulting flow and pressure waves under various friction models. Moreover, we present several simulations where some arterial tree characteristics were altered to model two disease conditions, namely hypertension and atherosclerosis. We consider next the problem of calculating the axial concentration profile of a solute transported by a time-dependent flow in a rigid straight pipe. This generalizes the result that Taylor derived in 1953 for calculating the axial concentration profile of a solute in a steady flow. Using asymptotic analysis, we derive a time-dependent diffusion equation for the mean concentration profile along the axial-direction in a pipe. In the special case of time-independent flow, our result reduces to that of Taylor. Finally, we show how to couple the one-dimensional model with the unsteady Taylor diffusion limit. We use the velocity profiles calculated using the one-dimensional model to drive the unsteady Taylor diffusion equation. We present several parameter studies to show the influence of the effective diffusion coefficient on the solution of the convection-diffusion equation under study.