Basics of Fluid Mechanics, 2014a

90 CHAPTER 4. FLUIDS STATICS
The comparison of the right hand terms of equations (4.66) and (4.65) provides
the conditions to determine the stability. From a mathematical point of view, to keep
the inequality for a small dh only the first term need to be compared as
gρ
Pk > gρ
P − C x
T
(4.67)
After rearrangement of the inequality (4.67) and using the ideal gas identity, it transformed
to
C x
T
(k − 1) gρ
>
kP
C x < k − 1
k
g
R
(4.68)
The analysis shows that the maximum amount depends on the gravity and gas
properties. It should be noted that this value should be changed a bit since the k should
be replaced by polytropic expansion n. When lapse rate C x is equal to the right hand
side of the inequality, it is said that situation is neutral. However, one has to bear in
mind that this analysis only provides a range and isn’t exact. Thus, around this value
additional analysis is needed 9 .
One of the common question this author has been asked is about the forces
of continuation. What is the source of the force(s) that make this situation when
unstable continue to be unstable? Supposed that the situation became unstable and
the layers have been exchanged, would the situation become stable now? One has to
remember that temperature gradient forces continuous heat transfer which the source
temperature change after the movement to the new layer. Thus, the unstable situation
is continuously unstable.
4.3.5 Gravity Variations Effects on Pressure and Density
Until now the study focus on the change of density
and pressure of the fluid. Equation (4.11) has two
terms on the right hand side, the density, ρ and
the body force, g. The body force was assumed
until now to be constant. This assumption must be
deviated when the distance from the body source
is significantly change. At first glance, the body
force is independent of the fluid. The source of
the gravity force in gas is another body, while the
gravity force source in liquid can be the liquid itself.
Thus, the discussion is separated into two different
issues. The issues of magnetohydrodynamics are
r b
P b ρ b
g ∝ r 2
Fig. -4.12. The varying gravity effects
on density and pressure.
too advance for undergraduate student and therefore,will not be introduced here.
r
9 The same issue of the floating ice. See example for the floating ice in cup.