Biomedical Engineering Reference
In-Depth Information
rotation. Particles along the point of rotation (axis of rotation in three dimensions) have a
zero displacement, while those at the periphery of the body have the largest displacement.
In real fluid mechanics problems, fluid elements do not experience pure translation or
rotation. However, in some cases, this can be used as an assumption to make the calcula-
tions simpler. For most fluid mechanics problems, when
m moves throughout the fluid,
the internal organization of the particles are not conserved. If this is the case, the kine-
matic representation of the element will be a combination of extension and pure shear
(plus translation/rotation). If the element is subjected to pure extension, there are changes
in the boundary lengths of
δ
m, inducing some internal reorganization of the fluid parti-
cles. This is similar to compression or elongation in mechanics of materials problems, in
which bars are subjected to some loading condition. Pure shear is when only the internal
angles along the boundary of
δ
m are altered. This occurs if the element is under a pure
shearing force and there are no normal forces acting on the element. Please note that,
δ
α
and
do not need to be equal to each other when the body is under pure shear.
Figure 2.15
illustrates the kinematic movement of a mass in two dimension. These move-
ments can be extended to three dimensions to describe any motion that a volume
β
δ
x
δ
y
δ
z
would experience.
Similar to previous courses in mechanics, velocity is defined as the time rate of change
of the position vector of any particle. Here we will define a particle's position vector in
three-dimensional Cartesian coordinates as
Þ
j
-
Þ
-
Þ
i
-
-
ð
t
Þ 5
u
x
ð
t
1
u
y
ð
t
1
u
z
ð
t
ð
2
:
15
Þ
which is relative to the origin of a fixed coordinate system. This particle has a position of
(x
i
, y
i
, z
i
) at time t. Using this formulation, the velocity vector would be defined as
d
-
ð
du
y
ð
t
Þ
t
Þ
du
x
ð
t
Þ
du
z
ð
t
Þ
i
-
j
-
-
Þ
i
-
Þ
j
-
Þ
-
-
ð
t
Þ 5
5
1
1
5
v
x
ð
t
1
v
y
ð
t
1
v
z
ð
t
ð
2
:
16
Þ
dt
dt
dt
dt
As in mechanics, acceleration is defined as the time rate of change of the velocity vector
of the particle in three-dimensional space. Similarly, it takes the form of
d
-
ð
d
2
-
ð
t
Þ
t
Þ
dv
x
ð
t
Þ
dv
y
ð
t
Þ
dv
z
ð
t
Þ
j
-
-
i
-
-
ð
t
Þ 5
5
5
1
1
dt
dt
2
dt
dt
dt
d
2
u
y
ð
d
2
u
x
t
Þ
d
2
u
z
ð
t
Þ
ð
t
Þ
i
-
j
-
-
Þ
i
-
Þ
j
-
Þ
-
5
1
1
5
a
x
ð
t
1
a
y
ð
t
1
a
z
ð
t
ð
2
:
17
Þ
dt
2
dt
2
dt
2
Example
Given the following position vector, calculate the velocity and acceleration of a fluid particle
as a function of time. This particle starts at the origin of the coordinate axis at time zero. Plot the
fluid particles position and velocity for times 0s, 1s, 2s and 4s.
2
s
2
1
t
2
i
-
5
t j
-
-
3ms
2
1
2ms
2
1
ð
t
Þ 5 ð
Þ
t
1
1 ð
Þ
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