Biomedical Engineering Reference
In-Depth Information
ð
P
1
P
2
Þ
Q
¼
2pR
4
ð
4
Þ
8ll
where Q is the flow rate, R is the radius of the chamber, l is the viscosity and l is
the length of the vessel. Or for a rectangular chamber such as the parallel plate
flow chamber:
Q
¼
h
3
w
12l
ð
P
1
P
2
Þ
ð
5
Þ
l
where h is the height of the channel and w is the width. Shear stress is then
calculated for a rectangular channel using the equation s
¼
l
ð
dV
=
dx
Þ
producing
the equation for steady shear stress:
s
¼
6Ql
wh
2
:
ð
6
Þ
These flow equations are the most basic measurement for flow velocity, with
the assumptions that the flow is laminar and does not vary with time. Poisuille's
equation predicts fully developed viscous flow with a parabolic profile (Fig.
5
). If
flow in arteries is not steady, it will display a pulsatile behavior. Steady flow can
only be applied to extremely small vessels where steady flow dominates. Vessels
beyond the size of arterioles will still experience pulsatile flow. Pulsatile behavior
is a function of the pumping of the heart, the pulse flow behavior changes in the
arterial system due to extensive branching and vessel diameter.
5.2 Pulsatile Flow Behavior
The Womersley parameter is a measure of the degree of departure from normal
parabolic flow characteristic of pulsatile flow. The Womersley number is given by:
a
¼
R
xq
=
p
:
Where R is the radius of the vessel, x is the frequency, q is the
density, and l is the viscosity. This equation exemplifies that the degree of departure
from parabolic form increases with the Womersley number and frequency.
The equation to describe the flow of a pulsatile fluid was first derived by
Womersley in 1955. The equation of motion of a liquid is:
2
w
o
or
2
þ
1
o
w
r
1
ow
ot
¼
1
o
P
oz
:
ð
7
Þ
r
t
l
oz
¼
P
1
P
2
The pressure gradient
oP
can be represented by a simple harmonic
L
function:
ow
2
or
2
þ
1
o
w
or
þ
i
3
n
t
¼
A
l
e
ixt
:
ð
8
Þ
r
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