Biomedical Engineering Reference
In-Depth Information
where
L
is the tube length and
r
i
is the blood vessel radius at the inlet. By using
Equation 5.27
in the Conservation of Mass, Momentum, or Energy formulation, quanti-
ties of interest can easily be calculated. In a similar situation, the velocity of a fluid ele-
ment at a particular location along the blood vessel length can be given as a function of
position and the other fluid parameters can be calculated.
Example
The velocity of a fluid element along the tapered channels centerline (see Figure 5.20) is given
by
-
5
L
i
. Calculate the acceleration of any particle along the centerline (i.e., as a func-
tion of
x
) and the position of a particle (as a function of time) that is located at
x
2
x
v
i
1
1
0 at time zero.
5
Solution
To find the acceleration of a particle, one must use the total derivative of velocity, developed
in Chapter 2 (for the x-direction):
D
-
Dt
5
@
u
@
u
@
u
v
@
u
w
@
u
t
1
x
1
y
1
@
@
@
z
For this situation,
u
is only a function of
x
and
y
, and
v
and
w
are equal to zero.
0
@
1
A
@
@
2
4
0
@
0
@
1
A
1
A
3
5
5
0
@
1
A
@
@
2
4
0
@
1
A
3
5
D
-
Dt
5
u
@
u
2
x
L
2
x
L
2
x
L
2
v
i
x
L
v
i
1
v
i
1
v
i
1
x
5
1
1
1
@
x
x
0
@
1
A
0
@
1
A
5
0
@
1
A
v
i
L
2
x
L
2
v
i
L
4
x
L
v
i
1
2
5
1
1
The acceleration of any particle along the centerline is
v
i
L
4
x
L
a
x
ð
x
Þ
5
2
1
To determine the location of a particle, we first know that a time 0, the particle is located at 0
and has a velocity of
v
i
. At some later time, the particle will be located at
L
and will have a veloc-
ity of 3
v
i
. From Chapter 2, we learned that the velocity of a particle is the time derivative of its
position. Using this relationship, we can develop a formula for the location of fluid element as a
function of time.
FIGURE 5.20
Model of a continuous taper for the
example problem.
x
v
x
=
L
x
=
0
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