Biomedical Engineering Reference
In-Depth Information
4.3
Nonidealities in Biological Systems
4.3.1 Oscillatory and Pulsating Flows
Majority of the flow systems in the body including blood flow and gas exchange
vary with time. This is referred to as
unsteady
flow. If the flow has a periodic
behavior and a net directional motion throughout the cycle, it is called
pulsatile
flow. Furthermore, if the fluid flow has a periodic behavior and oscillates back and
forth without a net forward output, it is called
oscillatory
flow. Body fluid flow is
either oscillatory or pulsatile. This would be in contrast to many manmade piping
systems in which fluid flow is continuous. Both oscillatory and pulsatile flows oc-
cur naturally in respiratory, vascular, and acoustic flows in the ear. For example,
blood is ejected from the heart during systole due to which a pressure pulse propa-
gates along the arterial tree. The pulse propagation is wavelike in character with a
variation in velocity (or volumetric flow rate), unlike steady state flow (Figure 4.4).
Hence,
N
Re
varies as a function of time and can
be higher than 4,000 despite the
absence of turbulance. The shape of the pulse wave also changes as it propagates
through the arterial system.
Similar to the Reynolds number, British physiologist J. R. Womersley devel-
oped a dimensionless number to characterize the dominant features of pulsatile
flow. The Womersley number (
N
) is the ratio of transient inertial forces associated
with the local acceleration to the viscous forces per unit mass. In other words,
α
Nl
α
=
ωρ
/
μ
(4.15)
c
where
ω
is the angular velocity of the applied pressure gradient.
ω
is related to the
frequency (
f
) of the pulsatile wave by
f
and the units are cycles per second
or hertz (1 Hz is one cycle per second). While calculating
N
Re
, one needs to use the
characteristic length as the radius of the tube rather than the tube diameter if
l
is the
radius of the tube in the
N
ω
=
2
π
definition. However, if diameter is used in calculating
α
N
, then
N
Re
is calculated using the diameter.
If
α
is less than 1, then viscous forces dominate. The velocity profile will be
similar to a steady state condition, and flow is said to be in a quasi-steady state.
When
α
is greater than 1, the importance of the transient inertial force relative to
the viscous force is greater. Hence
N
α
is used to assess the occurrence of turbulence
in a pulsatile flow. Condition in the human aorta can result in an
N
α
of 10 or more;
α
Figure 4.4
Pulsatile fl uid fl ow.
