Page 96 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

PHYSICAL AND FLOW PROPERTIES OF BLOOD 73

3.3 BLOOD FLOW IN ARTERIES

3.3.1 Introduction

The aorta and arteries have a low resistance to blood flow compared with the arterioles and capil-
laries. When the ventricle contracts, a volume of blood is rapidly ejected into the arterial vessels.
Since the outflow to the arteriole is relatively slow because of their high resistance to flow, the arter-
ies are inflated to accommodate the extra blood volume. During diastole, the elastic recoil of the
arteries forces the blood out into the arterioles. Thus, the elastic properties of the arteries help to
convert the pulsatile flow of blood from the heart into a more continuous flow through the rest of the
circulation. Hemodynamics is a term used to describe the mechanisms that affect the dynamics of
blood circulation.
An accurate model of blood flow in the arteries would include the following realistic
features:
1. The flow is pulsatile, with a time history containing major frequency components up to the eighth
harmonic of the heart period.
2. The arteries are elastic and tapered tubes.
3. The geometry of the arteries is complex and includes tapered, curved, and branching tubes.
4. In small arteries, the viscosity depends upon vessel radius and shear rate.
Such a complex model has never been accomplished. But each of the features above has been
“isolated,” and qualitative if not quantitative models have been derived. As is so often the case in the
engineering analysis of a complex system, the model derived is a function of the major phenomena
one wishes to illustrate.
The general time-dependent governing equations of fluid flow in a straight cylindrical tube are
given by the continuity and the Navier-Stokes equations in cylindrical coordinates,
∂v v ∂u
+ + = 0 (3.4)
∂r r ∂z

2
2
∂u ∂u ∂u 1 ∂P μ ⎛ ∂ u 1 ∂u ∂ ∂ ⎞ u
+ v + u = F z − + ⎜ + + ⎟ (3.5)
2
∂t ∂r ∂z ρ ∂z ρ ⎝ ∂r 2 r ∂r ∂ ⎠
z
2
∂v ∂v ∂v 1 ∂P μ ⎛ ∂ v 1 ∂v v v ∂ 2 v⎞
+ v + u = F r − + ⎜ 2 + − 2 + 2 ⎟ (3.6)
∂t ∂r ∂z ρ ∂r ρ ⎝ ∂r r ∂r r z ∂ ⎠
Here, u and v are the axial and radial components of the fluid velocity, r and z are the radial and axial
coordinates, and r and m are the fluid density and viscosity, respectively. Equations (3.5) and (3.6) are
the momentum balance equations in the z and r directions.

3.3.2 Steady Flow
The simplest model of steady laminar flow in a uniform circular cylinder is known as the Hagen-
Poiseuille flow. For axisymmetric flow in a circular tube of internal radius R and length l, the boundary
0
conditions are
u ∂
ur = R ) = 0 and r ( = 0 ) = 0 (3.7)
(
0
r ∂