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Simple Calculation of the Velocity Profiles for Pulsatile Flow in a Blood Vessel Using Mathematica

X. He
D. Ku
J. Moore
Journal / Anthology

Annals of Biomedical Engineering
Year: 1993
Volume: 21
Page range: 45-49

The solution for a homogeneous, incompressible, Newtonian fluid flow in a rigid, cylindrical tube was proposed by Hale, McDonald and Womersley in 1955 and has provided important information to students of Biofluid mechanics for more than three decades. The original velocity profiles were plotted in increments of \delta omega t = 15 for a sinusoidally oscillating flow in a pipe with a pressure gradient varying like cos \omega t. With the rise in availability of personal computers, the Womersley calculation can now be performed without such tedium. Since the analysis is suitable for an arbitrary periodic waveform, the calculation of the solution for a range of physiology waveforms can provide important physical insight as to the fluid mechanics of unsteady flows. This article outlines a computer program which can be implemented by a graduate student as a homework assignment. The program can be used to perform a series of computer experiments to compare the effects of different waveform shapes on the detailed velocity profiles for unsteady flow. As an extension of the analysis, the pulsatile centerline velocity and wall shear stress rates can be calculated. Similarly, one can use any one of the time-dependent variables of flow, pressure gradient, wall shear, or centerline velocity to calculate the other three. As a byproduct, the student gains valuable experience with the use of numerical solutions to analytical equations.

*Science > Biology
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