Biomedical Engineering Reference
In-Depth Information
FIGURE 5.22
A schematic of the
velocity waveform in the large arteries,
showing that the velocity changes tem-
porally and is pulsatile. As we can see,
this mimics the pressure pulse of the
left ventricle.
one sense, the calculations of the fluid properties never describe the actual condition
within the body. However, by restricting yourself to one instant in time, the calculations
can be carried out by ignoring the temporal derivatives. The effect of this is that one
loses the transmittance of the fluid throughout the vessel, but the solution of the pro-
blems can be carried out (i.e., you lose some information but gain some other informa-
tion). By restricting the calculation to one instant in time, the assumption is made that
the speed of blood everywhere along the blood vessel is the same at that time. However,
we have already discussed that due to the fluid flow and the elastic properties of the
blood vessel, there is wave propagation within the walls of the blood vessel and within
the fluid itself. Therefore, when looking at one section of a blood vessel, the velocity pro-
file will not be constant in magnitude and direction (at least in larger vessels). By consid-
ering the pulsatile nature of blood velocity, it more than likely requires the use of
numerical methods to calculate flow properties (see Chapter 13 for more details) unless
the velocity waveform is simplified. The way to work around this is to define the inlet
velocity as a function of time at discrete time points. Then solve for the flow properties
atthosediscretetimepoints.
Another factor that we have largely ignored is that blood flow can become turbulent,
especially during peak systole, when the blood velocity is the greatest. When flow
becomes turbulent, there are random fluctuations within the fluid velocity and pressure.
By nature of being random, they cannot be calculated or predicted, but they can be
observed. For turbulent flows, we would discuss the velocity as a mean velocity plus
some random velocity fluctuation term, because it is not realistic to be able to define the
entire flow field as a function of time and space. Using this approach, the mean velocity
and the fluctuation in the x-component of velocity would be defined as
ð
t
0
1
T
1
T
u
udt
5
ð
5
:
28
Þ
t
0
u
0
5
u
u
2
where any fluid quantity with a prime (
0
) superscript denotes that we are talking about the
fluctuating (random) component of that variable. For completeness, we include the turbu-
lent Navier-Stokes equation, which is used extensively in computational fluid dynamics:
5
ρ
D
-
Dt
1
ρ
@
@
-
2
r
-
2
-
u
i
u
j
ρ
1
μr
x
j
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