Biomedical Engineering Reference
In-Depth Information
where density multiplied by the differential volume has been substituted for the differen-
tial mass. The significance of this equation is that the gravitational force must be balanced
by the pressure force at each individual point within the fluid. Remember that this is only
true if the fluid does not experience acceleration. In terms of the vector component equa-
tions, which must independently summate to zero, for fluid static problems,
Equation 3.7
can be represented as
2
@
p
x
1
ρ
g
x
5
0
@
2
@
p
y
1
ρ
g
y
5
0
ð
3
:
8
Þ
@
2
@
p
@
z
1
ρ
g
z
5
0
It is conventional to choose a coordinate system in a particular way, so that the gravita-
tional force acts in only one direction. Typically, for these types of fluids problems, the
gravitational force acts in the z-direction of the Cartesian coordinate system. With this def-
inition,
Equation 3.8
simplifies to
@
p
0
x
5
@
@
p
0
y
5
ð
3
:
9
Þ
@
@
p
@
z
52
ρ
g
z
5
ρ
g
because
g
x
5
g
. Using these assumptions, the pressure is only a function
of one coordinate variable (
z
) and it is independent of the other two coordinate variables
(
x
and
y
). Note that pressure can act in the x/y directions; however, the pressure must be
constant. The assumptions made in this analysis are that the fluid is under static flow con-
ditions (has no acceleration term), the only body force is the gravitational force, and that
gravity is only aligned with the
z
-axis (using the Cartesian coordinate system). Combined,
this allows the use of a total derivative instead of a partial derivative in
Equation 3.9
.
Therefore,
g
y
5
0 and
g
z
52
dp
dz
52
ρ
g
z
ð
3
:
10
Þ
Equation 3.10
relates the pressure within a fluid to the vertical height of the fluid, if
the assumptions made are valid or are within a reasonable estimate of the flow condi-
tions. The previous equation can be integrated to calculate the pressure distribution
throughout a static fluid, if the correct boundary conditions are applied. In general, you
would need to know if the fluid's density or if gravity varies with changes in vertical
distances.
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