Basics of Fluid Mechanics, 2014a

74 CHAPTER 4. FLUIDS STATICS
A tank with opening at the top to the
atmosphere contains two immiscible liquids
one heavy and one light as depicted
in Figure 4.5 (the light liquid is on the
h 1
top of the heavy liquid). Which piezometric
tube will be higher? why? and
h
h L
h H
2
how much higher? What is the pressure
at the bottom of the tank?
Fig. -4.5. Tank and the effects different
liquids.
Solution
The common instinct is to find that the lower tube will contain the higher liquids. For
the case, the lighter liquid is on the top the heavier liquid the the top tube is the same
as the surface. However, the lower tube will raise only to (notice that g is canceled)
h L = ρ 1 h 1 + ρ 2 h 2
ρ 2
(4.II.a)
Since ρ 1 >ρ 1 the mathematics dictate that the height of the second is lower. The
difference is
h H − h L
= h ( )
H ρ1 h 1 + ρ 2 h 2
−
h 2 h 2 h r 21 ρ 2
(4.II.b)
It can be noticed that h H = h 1 + h − 2 hence,
h H − h L
= h ( )
1 + h 2 ρ1 h 1 + ρ 2 h 2
−
= h (
1
1 − ρ )
1
h 2 h 2
h 2 ρ 2 h 2 ρ 2
(4.II.c)
or
(
h H − h L = h 1 1 − ρ )
1
ρ 2
(4.II.d)
The only way the h L to be higher of h H is if the heavy liquid is on the top if the stability
allow it. The pressure at the bottom is
P = P atmos + g (ρ 1 h 1 + ρ 2 h 2 ) (4.16)
End Solution
Example 4.3:
The effect of the water in the car tank is more than the possibility that water freeze
in fuel lines. The water also can change measurement of fuel gage. The way the
interpretation of an automobile fuel gage is proportional to the pressure at the bottom
of the fuel tank. Part of the tank height is filled with the water at the bottom (due to
the larger density). Calculate the error for a give ratio between the fuel density to the
water.
Solution