Biomedical Engineering Reference
In-Depth Information
FIGURE 5.12
Wave propagation within a
deformable homogenous artery. With a deformable
boundary, one would need to take into account the
change in area, with time, of the fluid element. The
material properties of the arterial wall need to be
included within this formulation as well.
∂
A
∂
t
dx
∂
u
∂
x
dx
u(x)
+
u(x)
∂
p
∂
x
dx
p(x)
+
p(x)
A
∂
A
∂
x
dx
A
+
dx
therefore, they are not considered on this figure. In the x-direction, the summation of the
pressure forces is
X
F
x
5
p
inlet
A
inlet
2
p
outlet
A
outlet
1
p
walls
A
walls
A
Þ
1
@
p
1
@
A
@
Þ
@
A
@
A
@
p
p
ð
x
Þ
A
p
ð
x
x
dx
x
dx
p
ð
x
x
dx
x
dx
2
1
52
@
@
if the second order differential terms are ignored. Using Newton's second law of motion,
the net force acting on a differential element is equal to the mass multiplied by the accelera-
tion of the object. Writing this in equation form, we get (using density/volume)
A
@
p
@
u
@
u
@
u
x
dx
5
ðρ
Adx
Þ
2
t
1
@
@
x
1
ρ
@
p
x
1
@
u
@
u
@
u
@
0
ð
5
:
10
Þ
5
t
1
@
x
which is a simplified form of the Navier-Stokes equation. The Conservation of Mass on
this element must take into account the increase in area of the element during time.
Assuming the density remains constant within the entire element (the fluid is incompress-
ible), Conservation of Mass states (neglecting second-order terms)
A
1
@
u
1
@
A
@
1
@
A
@
uA
u
x
dx
x
dx
t
dx
5
@
u
@
A
@
A
@
u
1
@
A
@
uA
uA
x
dx
x
dx
t
dx
5
1
1
@
u
@
A
@
A
@
u
x
1
@
A
@
t
5
@
A
@
t
1
@
x
ð
uA
Þ
5
0
ð
5
:
11
Þ
x
1
@
@
To determine the wave propagation throughout the blood vessel, the material properties
of the vessel wall must be known and related to the fluid properties. In some instances, we
can make the assumption that the changes in the blood vessel radius (
r
i
) is linearly propor-
tional to the blood pressure acting on the vessel wall (
p
i
), using the following relationship:
r
i
5
r
i;
0
1
α
p
ð
5
:
12
Þ
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