Biomedical Engineering Reference
In-Depth Information
The normal force, acting on a differential element would be defined as
d
-
N
d
-
5
σ
To define the time rate of change of work,
-
d
-
Δ
0
@
W
Δ
U
N
d
-
-
W
n
5
lim
Δ
t
5
lim
Δ
5
σ
U
t
t
t
0
-
-
because
d
-
-
. Therefore, the total work done by normal forces is
dt
5
ð
area
σ
d
-
N
-
W
n
52
U
ð
3
:
46
Þ
where the negative sign is needed to quantify the work done on the control volume
instead of by the control volume.
The work of a shear force is defined in a similar way. Remember that shear force is
defined as
d
-
5
-
dA
In this formulation, the shear stress is the vector quantity (to provide directionality of
the stress which is different from the area vectors direction) not the area normal vector.
Using the same process as above, the work of shear becomes
ð
area
τ
-
dA
W
sh
-
52
U
ð
3
:
47
Þ
The work done by a shaft is not applicable to many biofluid mechanics problems, but
would be defined as the negative of work input into the shaft to move the fluid. Similarly,
other work would need to be defined by the type of work that is being done. For instance,
if energy from an x-ray is being absorbed into the fluid, this can be considered as work
being absorbed by the system. Using these definitions,
Equation 3.45
becomes
ð
area
σ
N
-
U
ð
area
-
U
d
-
-
dA
Q
2
W
shaft
2
W
other
1
1
ð
ð
v
2
2
1
v
2
2
1
5
@
@
d
-
-
u
gz
ρ
dV
u
gz
ρ
1
1
1
U
t
area
V
ð
area
-
U
ð
v
2
2
1
2
W
shaft
2
W
other
5
@
@
Q
-
dA
u
gz
ρ
dV
1
1
t
V
ð
ð
area
σ
v
2
2
1
d
-
d
-
-
N
-
u
gz
ρ
1
1
U
2
U
area
We make use of the definition of specific volume (
ν
):
1
ν
ρ
5
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