Hydrostatic Pressure in Compressible Fluids (Gases – pg. 4546)
We know that in the case of a fluid at rest (or a fluid moving as a solid body at constant
velocity) with gravity being the only body force (acting in the –
z
direction), Newton’s
second law reduces to
g
dz
dp
ρ
−
=
.
(1)
Further, if our fluid of interest is an ideal gas, we can use the ideal gas law
p=
ρ
RT
to
rewrite Eq. (1) as
g
RT
P
dz
dp
−
=
or
dz
RT
g
p
dp
−
=
.
(2)
Therefore, computing the hydrostatic pressure in a gas reduces to integrating Eq. (2) for
different temperature variations, i.e.,
T
=
T
(
z
). Although gravity does actually vary as
(1/
r
2
), this effect is negligible for most engineering problems so we assume (for this
class) that
g
is constant. We then have two cases of interest.
Isothermal Atmosphere:
T
=
T
0
For the case of an isothermal atmosphere, the temperature is constant. Notice that this
does not imply the density is constant only that the pressure and density are linearly
related. We integrate between
z
1
, where the pressure is
p
1
, and
z
2
, where the pressure
p
2
,
to obtain
∫
∫
−
=
2
1
0
2
1
dz
RT
g
p
dp
(
)
1
2
0
1
2
ln
z
z
RT
g
p
p
−
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
(
)
1
2
0
1
2
z
z
RT
g
e
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 Spring '08
 Thompson
 Fluid Mechanics, Thermodynamics, linear temperature variation

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