Biomedical Engineering Reference
In-Depth Information
dx
dt
5
2
x
L
u
x
5
v
i
1
1
To solve this equation, first we separate the variables:
dx
v
i
dt
5
2
x
L
1
1
Integrate this equation with the bounds that at time zero the particle is located at zero and at
some time
t
the particle is located at some point
x
:
ð
ð
t
x
dx
v
i
dt
5
2
x
L
1
1
0
0
v
i
t
L
2
ln
1
2
x
L
t
0
5
x
0
1
L
2
ln
1
2
x
L
v
i
t
5
1
5
2
x
L
2
v
i
t
L
ln
1
1
2
x
L
5
e
2
v
i
t
1
1
L
L
2
e
2
v
i
t
x
ð
t
Þ
5
1
2
L
This example illustrates the use of an exact solution for a tapering blood vessel.
As stated above, most blood vessels are actually tapered and curved in the body.
Curving blood vessels do not add much complexity to the problems that we have dis-
cussed above. Similar to the previous examples and the examples in Chapter 3, continuity,
momentum conservation, and energy conservation can be used to calculate the velocity
profile, the forces needed to hold a vessel in place, the energy needed to maintain the
flow, the pressure within the fluid, acceleration, shear stress, and others fluid properties.
The Navier-Stokes equations can be used to determine the velocity profile as a function of
geometric changes. The following example illustrates the use of the Navier-Stokes equa-
tions to solve for the velocity profile in a curved tapering vessel.
Example
Determine the velocity profile in the three sections of the blood vessel shown in Figure 5.21.
Assume that the only portion that is tapering is within the bend. Also assume that gravity only
acts in the y-direction and that this is pressure-driven flow.
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