Asian Power Systems Review Center
FLUID MECHANICS (HYDROSTATICS)
I. THEORY
Fluids – are substances capable of flowing having particles that easily move and change their relative
position without separation of mass. Examples of fluid are gases and liquids. Fluids are generally
divided into ideal and real fluids.
Real fluids are further divided into two, namely, Newtonian fluids and Non-Newtonian Fluids.
Newtonian fluids are fluids that exhibit constant or uniform viscosities while Non-Newtonian fluids
are those whose viscosities will vary with velocity.
Properties of a Fluid:
• Mass Density or Density (ρ) – is mass per unit volume.
ρ=
mass
volume
Units of density:
SI
English
kg/m3
slugs/ft3
Note: To convert pound mass to slugs, use the following formula:
ρ slugs =
ρlbm
32.2
Density of water = 1000 kg/m3
= 9810 N/m3 = 9.81 kN/m3
= 62.4 lb/ft3
= 1 gram/cm3
•
Density of Gases
ρ=
p
RT
Where: p = absolute pressure of gas in Pascal
R = specific gas constant in Joule/kg-K
T = absolute temperature
•
Specific Gravity (s) – is the ratio of the density of a fluid to the density of a certain standard
reference. Another term for specific gravity is relative density. The standard reference used
for liquids and solids is pure water while for gases, the standard reference is air.
For liquids,
sliquid =
ρ liquid
ρ water
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For gases,
s gas =
ρ gas
ρ air
If molecular weight or specific gas constant of a gas is given, use the following
formulas,
s gas =
s gas
•
MWair
R gas
=
=
Rair
=
MW gas
29
R gas
ft − lbf
53.3
lbm − ° R
Specific Volume (v) – is the volume occupied by a unit mass of fluid. It is the reciprocal of
density.
v=
•
MW gas
1
ρ
Unit Weight or Specific Weight (γ) – is the weight of a fluid per unit volume.
For SI measurements,
For English measurements,
 g 
γ = ρ  
 gc 
γ = ρg
Where: g = 9.81 m/s2 (SI)
= 32.2 ft/s2 (English)
gc = gravitational conversion {pound mass (lbm) to pound force (lbf)}
•
Viscosity – is the measure of the fluid’s resistance to flow when acted upon by an external
force.
For Newtonian fluids, the shear stress, τ is given as
τ=
µV
h
Where: μ = coefficient of viscosity or absolute viscosity
V = velocity of the upper plate (note lower plate is stationary)
h = distance between two plates
Units of shear stress:
SI
English
o
N/m2 or Pascal
pounds/in2 or psi
Dynamic or Absolute Viscosity (μ)
µ=
σ
dV / dy
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Units of Absolute Viscosity:
SI
Conventional metric
English
Note:
o
Pa-sec
dyne-sec/cm2 or poise
lbf-sec/ft2
1 poise = 10-1 Pa-sec
1 centipoise (cp) = 10-2 poise
Kinematic Viscosity (v) – is the ratio of the absolute viscosity to its mass.
For SI measurements,
v=
For English measurements,
µ
ρ
ν=
µ g 
 
ρ  gc 
Where: g = 32.2 ft/s2
gc = gravitational conversion (lbm to lbf)
Units of Kinematic Viscosity:
SI
Conventional metric
English
]
Note:
m2/sec
cm2/sec = stoke
ft2/sec
1 stoke = 0.0001 m2/sec
1 centistoke = 10-2 stoke
•
Surface Tension (σ) – is the property of the fluid which offers resistance at the membrane or
skin that forms on the free surface of the fluid which is due to the intermolecular cohesion.
o Cohesion – is the property of a liquid by virtue of which the molecules of the liquid
remain intact to each other.
o Adhesion – is the property of a liquid which enables it to adhere to another body with
which it comes in contact.
•
Capillarity – is the rise and fall of a fluid in a capillary tube which is caused by surface
tension and depends on the relative magnitudes of the cohesion of the liquid and the adhesion
of the liquid to the walls of the containing liquid.
For SI measurements,
h=
For English measurements,
4σ cos β
ρd tube g
 4σ cos β
h = 
 ρd tub e
 g 
 
 gc 
Where: h = height at which capillarity can rise
σ = surface tension
β = angle of contact
ρ = density of fluid
Note:
For β < 90°, adhesive forces dominates
For β > 90°, cohesive forces dominates
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•
Bulk Modulus of Elasticity (E) of a fluid – is analogous to what is modulus of elasticity of a
solid. Bulk modulus is the reciprocal of compressibility.
E=
stress
1
Or E =
strain
β
Units of bulk modulus:
SI
kPa
English
psi
•
Compressibility (β) – is the fractional change in the volume of a fluid per unit change in
pressure at constant temperature process. It is the reciprocal of bulk modulus of elasticity of a
fluid.
HYDRAULICS
Hydraulics – is the branch of science which treats of water and other fluids.
Two Branches:
• Hydrostatics – is the branch of hydraulics which deals with pressure exerted by fluids at rest.
• Hydrodynamics – is the branch of hydraulics which relates to the forces exerted by or upon
fluids in motion.
HYDROSTATICS
A. PRESSURE
Pressure – is defined as the force per unit area.
Two pressure references used to measure fluid pressure:
• Gage pressure – when the pressure is measured with respect to the atmospheric pressure.
Pg = γh
Where: γ = density of fluid
h= pressure head
•
Absolute pressure – when the pressure is measured with the respect to the true zero pressure
reference. It is the sum of the gage pressure and the atmospheric pressure. Atmospheric
pressure is also known as barometric pressure. Also, absolute pressure may be taken as the
difference between atmospheric pressure and a vacuum.
Pabsolute = Pgage + Patmospheric
Pabsolute = Patmospheric − Pvacuum
Standard Atmospheric Pressure
Standard atmospheric pressure = 1.013 bar
= 1.01325 x 105 Pascal
= 101.325 kilopascals
= 14. 696 pounds per sq. inch (psi)
= 2116.2 pound per sq. foot
= 1.00 atmosphere
= 760 mm Hg
= 29.921 inches Hg
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= 407.1 inches H2O
= 33.93 feet H2O
= 760 torr
Variation of Pressure
The pressure at any point below the free surface of a liquid is equal to the product of the unit
weight of the liquid and the depth of the point in question.
dP = γdh
Where: dP = differential pressure
γ = specific weight
dh = differential elevation
Basic equation of fluid statics applied to liquids
P2 = P1 + γh
Where: P1 = γh1
P2 = γh2
Basic equation of fluid statics applied to gases
P2 = P1 − γh
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B. PASCAL’S LAW
“At any point in a fluid at rest the pressure is the same in all directions.”
Px = Py = Pz
C. MANOMETERS
Manometers
1. Simple U – Tube Manometer
PA + γ 1 h1 = γ 2 h2 + Patm
2. Differential Manometer
PB + γ 1 h1 = γ 2 h2 + γ 3 h3 + PA
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D. BOUYANCY (Archimedes’ Principle)
“A body submerged in a fluid is subjected by an unbalanced force called the buoyant force
equal to the weight of the displaced fluid.”
Fb = Wd
Where: Fb = buoyant force
Wd = weight of displaced fluid
Homogenous Solid Bodies Floating in Homogenous Fluid at Rest
Vs =
As =
Hs =
S .G.body
S .G.liquid
S .G.body
S .G.liquid
S .G.body
S .G.liquid
V
A
H
Where: Vs = submerged volume (equal to the displaced volume)
As = submerged area
Hs = submerged height (sometimes called draft)
E. HYDROSTATIC FORCE ON PLANE SURFACE
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Formulas
F = γ hA
Io = Ig + A( y ) 2
y=
Ig
Ay
Io
yp =
Ay
e=
h
sin θ
Where: F = hydrostatic force
γ = specific weight of fluid
h = distance of the centroid from the liquid surface
A = plane surface area
Ig = moment of inertia of the plane surface
y = distance of the centroid from 0 along the y-axis
A y = static moment
θ = angled of the plane with the horizontal
II. REVIEW PROBLEMS
1. A 4.9 ft3 of water are compressed to 5000 psig. What is the volume decrease if the temperature is
60°F? Assume the average bulk modulus of elasticity of 311000.
b. 0.079 ft3
c. 0.099 ft3
d. 0.069 ft3
a. 0.089 ft3
2. An ice berg has a density of 57.1 lb/ft3. If it floats on fresh water, what percent of he iceberg’s
volume will be visible?
a. 10.5%
b. 7.5%
c. 8.5%
d. 5.5%
3. If the atmospheric air 14.7 psia and 60°F at sea level, what is the pressure at 12000 ft altitude if
the air is compresses isothermally?
a. 19.54 psia
b. 9.53 psia
c. 11.64 psia
d. 11.64 psia
4. A 2000 – lbm satellite with an average density of 27 lbm/ft3 is in orbit where the local acceleration
due to gravity is 30.6 ft/s2. If the density of the atmosphere is 0.011 lbm/ft3, what is the buoyant
force of the satellite?
a. 0.774 lbf
b. 0.874 lbf
c. 0.576 lbf
d. 0.234 lbf
5. 100 g of water are mixed with 150 g of alcohol (ρ = 790 kg/cu. m). What is the specific volume
of the resulting mixtures, assuming that the fluids mix completely?
a. 0.88 cu. cm/g
c. 0.82 cu. cm/g
b. 1.20 cu. cm/g
d. 0.63 cu. cm/g
6. What is the sea level (g= 32.2 ft/s2) specific weight (lb/ft3) of liquid with density of 58.3 lb/ft3?
a. 64.3
b. 25.3
c. 58.3
d. 43.3
7. What is the pressure of 8000 ft below the ocean surface?
a. 514,176 lbf/ft2 b. 312,000 lbf/ft2
c. 312,000 lbf/ft2
d. 548,000 lbf/ft2
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8. A water-tight, cubical, 12 inches outside dimensions, is made from ¼ inch iron plate. If SG = 7.7
for the iron, will the box float?
a. The box will sink
c. The iron will float
b. The box will not float
d. The box will float
9. What is the atmospheric pressure on a planet where the absolute pressure is 100 kPa and the gage
pressure is 10 kPa?
a. 90 kPa
b. 80 kPa
c. 100 kPa
d. 10 kPa
10. Calculate the pressure at a depth of 100 meters in seawater, for which the specific weight is 10.1
kN/m3.
a. 1212 kPa
b. 1414 kPa
c. 1616 kPa
d. 1010 kPa
11. What is the resulting pressure when one pound of air at 15 psia and 200°F is heated at constant
volume to 800°F?
a. 28.6 psia
b. 52.1 psia
c. 36.4 psia
d. 15 psia
12. A scuba diver at a depth of 200 feet notices the bubbles he exhales have diameters of ½ inch.
Assume the surface tension and water temperature remains relatively constant. What is the depth
if the diameter is ¼ inch?
a. 46.8 psi
b. 48.6 psi
c. 84.6 psi
d. 64.8 psi
13. At STP the density of chlorine is 3.22 kg/m3. What is the weight of this gas is contained in a flask
of 100 cubic centimeters at 24°C and 100 kPa?
a. 0.563 g
b. 0.420 g
c. 0.292 g
d. 0.321 g
14. A cube of wood floating in water supports a 200 g mass. When the mass is removed the cube
rises 2 cm. Determine the size of the cube.
a. 20 cm
b. 15 cm
c. 10 cm
d. 25 cm
15. A gold ring set with a diamond has a mass of 4 grams and when immersed in water, it weighs
3.72 grams. Find the mass of diamond of specific gravity of gold is 19.3 and that of diamond is
3.6.
a. 0.831 gram
b. 0.311 gram
c. 0.511 gram
d. 0.711 gram
16. Find the total water pressure on a vertical circular gate, 2 meters in diameter, with its top 3.5
meters below the water surface.
a. 138.7 kN
b. 107.9 kN
c. 169.5 kN
d. 186.5 kN
17. A horizontal capillary tube closed at one end contains a column of air imprisoned by means of a
small volume of water. At 7 degrees C and a barometric pressure of 76 cm of mercury, the length
of the air column is 15 cm. What is the length at 17 deg C if the saturation pressures of water
vapor at 7 deg C and 17 deg C are 0.75 cm and 1.42 cm of mercury respectively?
a. 12.55 cm
b. 19.68 cm
c. 15.68 cm
d. 18.50 cm
18. A tank having a capacity of 2 cu. ft is filled with oxygen which has a gauge pressure of 60 psi
when the temp. is 47 deg C. At a later time it is found that because of a leak the pressure has
dropped to 50 psi and the temperature has decreased to 27 deg C. What is the amount of oxygen
that has leak out?
a. 0.0583 lb
b. 0.0915 lb
c. 0.153 lb
d. none of these
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19. A wooden spherical ball with sp. gr. of 0.42 and a diameter of 30 cm is dropped from a height of
4.2 m above the surface of the water in a pool. If the ball barely touched the bottom of the pool
before it began to float, how deep is the water in the pool at that point?
a. 3.91 m
b. 3.40 m
c. 3.19 m
d. 3.04 m
20. Viscosity for a fluid is define as the constant of proportionality between shear stress and what
other variable?
a. The spatial derivative of velocity
c. The time derivative of density
b. The time derivative of pressure
d. The spatial derivative of density
21. An iceberg having specific gravity of 0.92 is floating on salt water sp. gr. 1.03. If the volume of
ice above the water surface is 1000 cu. m., what is the total volume of the ice?
b. 7862 m3
c. 9324 m3
d. 6325 m3
a. 8523 m3
22. A block of wood requires a force of 40 N to keep it immersed in water and a force of 100 N to
keep it immersed in glycerin (sp. gr. = 1.3). Find the weight and sp. gr. of the wood.
a. 0.7
b. 0.6
c. 0.9
d. 0.8
III. SELF-TEST
1. A mass of copper suspected of being hollow weigh 523 grams in air and 447.5 grams in water. If
specific gravity of copper is 8.92, the volume of the cavity in cc is:
a. 53.63
b. 16.87
c. 17.50
d. 21.35
2. A cylindrical water tank tower 12 m in diameter contains water to a depth of 20 m. The pressure
on the base is nearest to:
a. 196 kPa
b. 196 Pa
c. 2 MPa
d. 2 kPa
3. An open top cylindrical water tank 10 ft in diameter contains water at a height of 10 ft. The total
pressure on the side is nearest to:
a. 28.5 tons
b. 62.7 tons
c. 20 tons
d. none of these
4. A block of ice 12 inches in height with specific gravity of 0.917 floats on fresh water. What must
its surface area be to support a 180 lb man if the ice is to submerge so that its top is coincident
with the water surface?
a. 34.75 ft2
b. 34.75 in2
c. 54.25 ft2
d. 75.51 ft2
5. A cube of wood floating in water supports a 500 g mass. When the mass is removed the cube rises
5 cm. The size of the cube is:
a. 5 cm
b. 10 cm
c. 15 cm
d. 6 cm
6. An object weighs 1,600 grams in air, 1, 350 grams in water and 1,450 grams in an unknown
liquid. (1) The specific gravity of the object and (2) specific gravity of the unknown liquid is:
a. 10, 5
b. 6.4, 0.6
c. 8, 2
d. 8.4 , 10
7. A block of wood has a vertical projection of inches when placed in water and 4 inches when
placed in alcohol. If the specific gravity of alcohol is 0.82, the specific gravity of the wood is:
a. 10
b. 0.8
c. 0.6
d. 6
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8. A floating cylinder 8 cm in diameter and weighing 950 grams is placed in a cylinder container 20
cm in diameter partially full of water. The increase if depth of water in the container due to
placing the float is it is
a. 2 cm
b. 3.02 cm
c. 1.38 cm
d. 4.16 cm
9. A gas bubble rising from the ocean floor is 1 inch in diameter at a depth of 50 ft. Given that the
specific gravity of sea water is 1.03, the buoyant force in lbs being exerted on the bubble at this
instant is nearest to:
a. 0.014
b. 0.020
c. 0.076
d. 0.147
10. The SI unit of atmospheric pressure is the
b. psi
a. N/m2
c. Atmosphere
d. Torr
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