1. CHAPTER 2 Fluid Statics and Its Applications Nature of fluids
Hydrostatic Equilibrium
Applications of fluid statics

2. Nature of fluids
A fluid is a substance that does not permanently resist distortion.
During the change in shape, shear stresses exist, the magnitudes of which depend upon the viscosity of the fluid and the rate of sliding

3. Fluids include liquid , gas
and solid particles suspended in liquid and gas or slurry

4. Fluids also can be divided as
Incompressible——the density changes only slightly with moderate changes in temperature and pressure
Compressible——the changes in density caused by temperature and pressure are significant
(Pressure concept : the pressure at any point in the fluid is independent of direction)

5. Hydrostatic Equilibrium
There is a vertical column of fluid shown in Fig.2.1
Three vertical forces are acting on this volume:
(1)the force from pressure p acting in an upward direction , which is pS;
(2) the force from pressure p+dp acting in a downward direction , which is (p+dp)S;
(3)the force of gravity acting downward, which is gρsdz
Figure Hydrostatic equilibrium
p +dp p g

6. Then
(2.1)
After simplification and division by S,Eq.(2.1) becomes
(2.2)
Integration of Eq.(2.2) on the assumption that
density is constant gives
(2.3)

7. Between the two definite heights Za and Zb shown in Fig.2.1,
(2.4)
Equation (2.3) expresses mathematically the condition of hydrostatic equilibrium.

8. Gauge pressure, absolute pressure and vacuum
The relationship between gauge pressure and absolute pressure
P(gauge)=P(absolute)-P(atmosphere)
The relationship between vacuum and absolute pressure
P(vacuum)=P(atmosphere)-P(absolute)
Or P(vacuum)=- P(gauge)

9. The reading in the gauge is 1. 5 kgf /cm2 = =(
The reading in the gauge is 1.5 kgf /cm2 = =(?)N/m2, and the reading of the vacuum gauge is 736 mmHg = ( )m H2O .If the atmospheric pressure is 1 atm, what happens to the above cases in absolute pressure?

10. Barometric equation
For an ideal gas , the density and pressure are related by the equation
(2.5)
Substitution from Eq.(2.5)intoEq.(2.2)gives
(2.6) or

11. (2.7) Equation(2.7)is known as the barometric equation.or

12. Hydrostatic equilibrium in a centrifugal field
In a rotating centrifuge a layer of liquid is
thrown outward from the axis of rotation and
is held against the wall by centrifugal force.
The free surface of the liquid takes the shape of a paraboloid of revolution.

13. The rotational speed is so high and the
centrifugal force is so much greater than the
force of gravity that the liquid surface is
virtually cylindrical and coaxial with the
rotation.

14. The situation in shown in Fig.r dr b r1 r2

15. The entire mass of liquid indicated in Figure
is rotating as a rigid body, with no sliding of
layer of liquid over another.

16. Under these conditions the pressure distribution in the liquid may be found from the principles of fluid static.
The pressure drop over any ring of rotating liquid is calculated as follows.

17. The volume element of thickness dr at a
radium r.
If ρ is the density of the liquid and b the breadth of the ring.

18. Eliminating dm gives
The change in pressure over the element is the force exerted by the element of liquid, divided by the area of the ring.

19. The pressure drop over the entire ring is
Assuming the density is constant and integration gives
(2.8)

20. Applications of fluid statics
Manometer (pressure gauge)
The manometer is an important device for measuring pressure differences.
U tube manometer (or reverse U tube)
Inclined manometer
Differential manometer

21. U tube manometer It is the simplest form of manometer.
A pressure pa is exerted in one
arm of U tube and a pressure pb
in the other.
As a result of the difference in
pressure, the meniscus in one
branch of the tube is higher than
that in the other.
Vertical distance between the
two meniscuses Rm may be
used to measure the difference
in pressure. pa pb zm 4 ρB Rm 1 3 2 ρA

22. The pressure at the point 1 is
p1 is equal to p2 for the continuous fluid at the same level, thus

23. Simplification of this equation gives
Note that this relationship is independent of the distance zm, and of the dimensions of the tube, provided that pressure pa and pb are measured in the same horizontal plane.
If fluid B is a gas, ρB is usually negligible compared to ρA and may be omitted from Eq. (2.10)
(2.10)

24. Inclined manometer
Used for measuring small differences in pressure.

25. By making α small, the magnitude of Rm is multiplied into a long distance R1, and large reading becomes equivalent to a small pressure difference
(2.11)

26. Differential manometer

27. Example: H2O flows through the pipe as shown in Fig
Example: H2O flows through the pipe as shown in Fig. A U-tube manometer is used to measure the pressure P in the pipe. If the atmosphere pressure pa is 1 atm, R and h of mercury and water columns are 0.1 and 0.5 m, respectively, what is pressure P in the pipe, N/m2?