Biomedical Engineering Reference
In-Depth Information
enough (e.g., aorta or vena cava), the Bernoulli equations can be applied, but as the vessel
diameter reduces, the viscous forces play a more critical role in the flow. Therefore, the
Bernoulli equations cannot be used in these situations and the Navier-Stokes equations,
the Conservation of Momentum, and the Conservation of Mass should be applied.
END OF CHAPTER SUMMARY
3.1
The body forces acting on a differential fluid element are
F
-
-
dm
-
ρ
dxdydz
. The surface
5
5
forces acting on a differential fluid element are
dF
-
-
dxdydz
. For static fluids, where all
acceleration terms are zero, the pressure gradient is equal to the gravitational acceleration
multiplied by the fluid density. In Cartesian components, this is
2
@
52
r
p
x
1
ρ
g
x
5
0
@
2
@
p
y
1
ρ
g
y
5
0
@
2
@
p
z
1
ρ
g
z
5
0
@
Most pressures that are recorded in biofluids are gauge pressures, which can be defined as
p
gauge
5
p
absolute
2
p
atmospheric
3.2
Buoyancy is the net vertical force that acts on a floating or an immersed object. The buoyancy
forces can be defined as
ð
ð
V
ρ
F
z
5
dF
z
5
gdV
5
ρ
gV
V
3.3
A generalized formulation for the time rate of change of a system property can be repre-
sented as
ð
ð
dW
dt
5
@
d
-
-
U
w
ρ
dV
w
ρ
1
@
t
area
V
Applying this formulation to the Conservation of Mass, we would get
system
5
@
ð
V
ρ
ð
area
ρ
dm
dt
d
-
-
dV
1
U
5
0
@
t
Depending on the particular flow conditions, the conservation of mass formula can be sim-
plified in various ways.
3.4
The Conservation of Momentum can be represented as
ð
ð
dP
dt
5
-
s
5
@
@
-
-
b
1
d
-
-
ρ
-
ρ
-
U
dV
5
1
t
area
V
Again, this can be simplified depending on the particular flow conditions.
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