1. Chapter 2 FLUID STATICS

2. The science of fluid statics :
the study of pressure and its variation throughout a fluid
the study of pressure forces on finite surfaces
Special cases of fluids moving as solids are included in the treatment of statics because of the similarity of forces involved.
Since there is no motion of a fluid layer relative to an adjacent layer, there are no shear stresses in the fluid
 all free bodies in fluid statics have only normal pressure forces acting on their surfaces

3. 2.1 PRESSURE AT A POINT
Average pressure: dividing the normal force pushing against a plane area by the area.
Pressure at a point: the limit of the ratio of normal force to area as the area approaches zero size at the point.
At a point: a fluid at rest has the same pressure in all directions  an element δA of very small area, free to rotate about its center when submerged in a fluid at rest, will have a force of constant magnitude acting on either side of it, regardless of its orientation.
To demonstrate this, a small wedge-shaped free body of unit width is taken at the point (x, y) in a fluid at rest (Fig.2.1)

4. Figure 2.1 Free-body diagram of wedge-shaped particle

5. There can be no shear forces  the only forces are the normal surface forces and gravity  the equations of motion in the x and y directions
px, py, ps are the average pressures on the three faces, γ is the unit gravity force of the fluid, ρ is its density, and ax, ay are the accelerations
When the limit is taken as the free body is reduced to zero size by allowing the inclined face to approach (x, y) while maintaining the same angle θ, and using
the equations simplify to
Last term of the second equation – infinitestimal of higher of smallness, may be neglected

6. When divided by δy and δx, respectively, the equations can be combined
θ is any arbitrary angle  this equation proves that the pressure is the same in all directions at a point in a static fluid
Although the proof was carried out for a two-dimensional case, it may be demonstrated for the three-dimensional case with the equilibrium equations for a small tetrahedron of fluid with three faces in the coordinate planes and the fourth face inclined arbitrarily.
If the fluid is in motion (one layer moves relative lo an adjacent layer), shear stresses occur and the normal stresses are no longer the same in all directions at a point  the pressure is defined as the average of any three mutually perpendicular normal compressive stresses at a point,
Fictitious fluid of zero viscosity (frictionless fluid): no shear stresses can occur  at a point the pressure is the same in all directions

7. 2.2 BASIC EQUATION OF FLUID STATICS
Pressure Variation in a Static Fluid
Force balance:
The forces acting on an element of fluid at rest (Fig. 2.2): surface forces and body forces.
With gravity the only body force acting, and by taking the y axis vertically upward, it is -γ δx δy δz in the y direction
With pressure p at its center (x, y, z) the approximate force exerted on the side normal to the y axis closest to the origin and the opposite e side are approximately
δy/2 – the distance from center to a face normal to y

8. Figure 2.2
Rectangular parallelepiped element of fluid at rest

9. Summing the forces acting on the element in the y direction
For the x and z directions, since no body forces act,
The elemental force vector δF
If the element is reduced to zero size, alter dividing through by δx δy δz = δV, the expression becomes exact.
This is the resultant force per unit volume at a point, which must be equated to zero for a fluid at rest.
The gradient ∇ is

10. -∇p is the vector field f or the surface pressure force per unit volume
The fluid static law or variation of pressure is then
For an inviscid fluid in motion, or a fluid so moving that the shear stress is everywhere zero, Newton's second law takes the form
a is the acceleration of the fluid element, f - jγ is the resultant fluid force when gravity is the only body force acting

11. In component form, Eq. (2.2.4) becomes
The partials, for variation in horizontal directions, are one form of Pascal's law; they state that two points at the same elevation in the same continuous mass or fluid at rest have the same pressure.
Since p is a function of y only,
relates the change of pressure to unit gravity force and change of elevation and holds for both compressible and incompressible fluids
For fluids that may be considered homogeneous and incompressible, γ is constant, and the above equation, when integrated, becomes
in which c is the constant of integration. The hydrostatic law of variation of pressure is frequently written in the form
h = -y, p is the increase in pressure from that at the free surface

12. Example 2.1 An oceanographer is to design a sea lab 5 m high to withstand submersion to 100 m, measured from sea level to the top of the sea lab. Find the pressure variation on a side of the container and the pressure on the top if the relative density of salt water is
At the top, h = 100 m, and
If y is measured from the top of the sea lab downward, the pressure variation is #

13. Pressure Variation in a Compressible Fluid
When the fluid is a perfect gas at rest at constant temperature
When the value of γ in Eq. (2.2.7) is replaced by ρg and ρ is eliminated between Eqs. (2.2.7) and (2.2.9),
If P = P0 when ρ = ρ0, integration between limits
- the equation for variation of pressure with elevation in an isothermal gas
- constant temperature gradient of atmosphere 

14. Example 2.2 Assuming isothermal conditions to prevail in the atmosphere, compute the pressure and density at 2000 m elevation if P = 105Pa, ρ = 1.24 kg/m3 at sea level.
From Eq. (2.2.12)
Then, from Eq. (2.2.9) #

15. 2.3 UNITS AND SCALES OF PRESSURE MEASUREMENT
Pressure may be expressed with reference to any arbitrary datum
absolute zero
local atmospheric pressure
Absolute pressure: difference between its value and a complete vacuum
Gage pressure: difference between its value and the local atmospheric pressure

16. Figure 2.3 Bourdon gage.

17. The bourdon gage (Fig. 2.3): typical of the devices used for measuring gage pressures
pressure element is a hollow, curved, flat metallic tube closed at one end; the other end is connected to the pressure to be measured
when the internal pressure is increased, the tube tends to straighten, pulling on a linkage to which is attached a pointer and causing the pointer to move
the dial reads zero when the inside and outside of the tube are at the same pressure, regardless of its particular value
the gage measures pressure relative to the pressure of the medium surrounding the tube, which is the local atmosphere

18. Figure 2.4 Units and scales for pressure measurement

19. Figure 2.4: the data and the relations of the common units of pressure measurement
Standard atmospheric pressure is the mean pressure at sea level, 760 mm Hg.
A pressure expressed in terms of the length of a column of liquid is equivalent to the force per unit area at the base of the column. The  relation for variation of pressure with altitude in a liquid p = γh [Eq. (2.2.8)] (p is in pascals, γ in newtons per cubic metre, and h in metres)
With the unit gravity force of any liquid expressed as its relative density S times the unit gravity force of water:
Water: γ may be taken as 9806 N/m3.

20. Local atmospheric pressure is measured by
mercury barometer
aneroid barometer (measures the difference in pressure between the atmosphere and an evacuated box or tube in a manner analogous to the bourdon gage except that the tube is evacuated and sealed)
Mercury barometer: glass tube closed at one end, filled with mercury, and inverted so that the open end is submerged in mercury.
It has a scale: the height of column R can be determined
The space above the mercury contains mercury vapor. If the pressure of the mercury vapor hv is given in millimetres of mercury and R is measured in the same units, the pressure at A may be expressed as (mm Hg)
Figure 2.5 Mercury barometer

21. Figure 2.4: a pressure may be located vertically on the chart, which indicates its relation to absolute zero and to local atmospheric pressure.
If the point is below the local-atmospheric-pressure line and is referred to gage datum, it is called negative, suction, or vacuum.
Example: the pressure 460 mm Hg abs, as at 1, with barometer reading 720 mm, may be expressed as -260 mm Hg, 260 mm Hg suction, or 260 mm Hg vacuum.
Note:
Pabs = pbar + pgage
Absolute pressures : P, gage pressures : p.

22. Example 2.3 The rate of temperature change in the atmosphere with change in elevation is called its lapse rate. The motion of a parcel of air depends on the density of the parcel relative to the density of the surrounding (ambient) air. However, as the parcel ascends through the atmosphere, the air pressure decreases, the parcel expands, and its temperature decreases at a rate known as the dry adiabatic lapse rate. A firm wants lo burn a large quantity of refuse. It is estimated that the temperature of the smoke plume at 10 m above the ground will be 11oC greater than that of the ambient air. For the following conditions determine what will happen to the smoke.
(a) At standard atmospheric lapse rate β = oC per meter and t0 =20oC.
(b) At an inverted lapse rate β = oC per meter.

23. By combining Eqs. (2.2.7) and (2.2.14),
The relation between pressure and temperature for a mass of gas expanding without heat transfer (isentropic relation, Sec. 6.1) is
in which T1 is the initial smoke absolute temperature and P0 the initial absolute pressure; k is the specific heat ratio, 1.4 for air and other diatomic gases.
Eliminating P/P0 in the last two equations
Since the gas will rise until its temperature is equal to the ambient temperature,
the last two equations may be solved for y. Let
Then
For β = oC per metre, R = 287 m·N/(kg·K), a = 2.002, and y = 3201 m. For the atmospheric temperature inversion β = oC per metre, a = , and y = m. #