Basics of Fluid Mechanics, 2014a

88 CHAPTER 4. FLUIDS STATICS
The second approximation for small C x is
(
P
= lim 1 − C ) g
RCx
x
h
P 0 C x −>0 T 0
= e − gh
RT 0
− gh2 gh
C x RT
2 T 2 e− 0
− ... (4.59)
0 R
Equation (4.59) shows that the correction factor (lapse coefficient), C x , influences at
only large values of height. It has to be noted that these equations (4.58) and (4.59)
are not properly represented without the characteristic height. It has to be inserted to
make the physical significance clearer.
Equation (4.57) represents only the pressure ratio. For engineering purposes, it
is sometimes important to obtain the density ratio. This relationship can be obtained
from combining equations (4.57) and (4.52). The simplest assumption to combine these
equations is by assuming the ideal gas model, equation (2.25), to yield
ρ
ρ 0
= PT 0
P 0 T =
P
P 0
T
{ }} {
0
(
1 − C )
x h (
g {
T
) ( }} {
RCx
1+ C )
x h
T 0
T
Advance material can be skipped
(4.60)
4.3.4.2 The Stability Analysis
It is interesting to study whether
this solution (4.57) is stable and if so
h + dh
under what conditions. Suppose that
for some reason, a small slab of material
moves from a layer at height, h,
h
to layer at height h + dh (see Figure
4.11) What could happen? There are Fig. -4.11. Two adjoin layers for stability analysis.
two main possibilities one: the slab
could return to the original layer or two: stay at the new layer (or even move further,
higher heights). The first case is referred to as the stable condition and the second
case referred to as the unstable condition. The whole system falls apart and does not
stay if the analysis predicts unstable conditions. A weak wind or other disturbances can
make the unstable system to move to a new condition.
This question is determined by the net forces acting on the slab. Whether these
forces are toward the original layer or not. The two forces that act on the slab are
the gravity force and the surroundings pressure (buoyant forces). Clearly, the slab
is in equilibrium with its surroundings before the movement (not necessarily stable).
Under equilibrium, the body forces that acting on the slab are equal to zero. That is,
the surroundings “pressure” forces (buoyancy forces) are equal to gravity forces. The
buoyancy forces are proportional to the ratio of the density of the slab to surrounding
layer density. Thus, the stability question is whether the slab density from layer h, ρ ′ (h)
undergoing a free expansion is higher or lower than the density of the layer h + dh. If
ρ ′ (h) >ρ(h + dh) then the situation is stable. The term ρ ′ (h) is slab from layer h that
had undergone the free expansion.