Biomedical Engineering Reference
In-Depth Information
is sometimes more accurate to discuss the velocity as a property of the fluid as a whole. In
this case, the velocity vector of the fluid is defined first because it is difficult to describe
accurately each particle's position within the fluid. In this Eulerian type approach, the
velocity of the fluid becomes
-
5
v
ð
x
;
y
;
z
;
t
Þ 5
v
x
1
v
y
1
v
z
1
v
t
ð
2
:
18
Þ
The acceleration, which is the time derivative of the velocity, is defined as
d
-
-
@
-
@
-
@
-
@
-
@
-
@
-
@
-
@
dt
5
@
dt
dt
1
@
dx
dt
1
@
dy
dt
1
@
dz
dt
5
@
v
x
@
v
y
@
v
z
@
-
5
t
1
x
1
y
1
ð
2
:
19
Þ
t
x
y
z
z
-
@
@
which applies the differentiation chain rule to compute the time derivative.
t
is termed
the local acceleration of the fluid and the remaining terms are the convective acceleration
of the fluid. Steady flows have no local acceleration. Since acceleration is itself a vector, it
can be broken up into the following three terms (in Cartesian coordinates)
5
@
v
x
@
v
x
@
v
x
@
v
y
@
v
x
@
v
z
@
v
x
@
a
x
t
1
x
1
y
1
z
5
@
v
y
@
v
x
@
v
y
@
v
y
@
v
y
@
v
z
@
v
y
@
a
y
t
1
x
1
y
1
ð
2
:
20
Þ
z
a
z
5
@
v
z
@
v
x
@
v
z
@
v
y
@
v
z
@
v
z
@
v
z
@
t
1
x
1
y
1
z
As you will see later in the text, it is this formulation that is critical to the foundation of
many fluid mechanics problems.
Using the Eulerian approach to calculate velocity and acceleration, the main quantity of
interest is a particular volume within the system, not a particular particle within the fluid.
Again, this may make the computations significantly easier because we solve for the entire
fluid and not individual particles. Looking at the acceleration terms (
Equation 2.20
), it is
clear that temporal and spatial driving forces dictate the acceleration (same thought pro-
cess holds for the fluid velocity). Examples of temporal driving forces are pumps that may
work at different rates during the flow cycle (think of the heartbeat, which is not a steady
consistent pressure pulse,
Figure 2.17
). Spatial driving forces are changes in the geometry
of the tubing (narrowing or expanding of blood vessel,
Figure 2.17
). Velocity and accelera-
tion both are field quantities; that is, they are functions of the fluid itself.
To define the rotation of fluid particles, we must take a slightly different approach.
Figure 2.15
highlights the kinematic rotation of a fluid element at any time/space in the
flow field. This can be described by rotation vectors, namely, the angular velocity vector
(in three dimensions,
Equation 2.21
)
x
-
y
-
z
-
-
ω
5ω
1ω
1ω
ð
2
:
21
Þ
where each Cartesian component is the rotation of the element about that particular
Cartesian axis. For instance,
ω
x
is the angular velocity of the element about the x axis.
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