Fluid mechanics is the study of forces and energy effects on fluids, subdivided into statics and dynamics, with fluids categorized as liquids and gases. Key concepts include fluid characteristics, types (ideal vs. real fluids), density, viscosity, compressibility, surface tension, and pressure measurement techniques like manometers. The document provides equations and examples related to fluid behavior and properties, such as pressure relation and Pascal's law.

1. FLUID MECHANICS

2. FLUID MECHANICS
- is the study of the effects of forces and energy on liquids and gases.
Like other branches of classical mechanics, the subject subdivides into statics
(often called hydrostatics) and dynamics (fluid dynamics, hydrodynamics, or
aerodynamics).

3. Fluids are classified into two, namely liquids and gases. The characteristics of fluids are as
follows:
CHARACTERISTICS LIQUIDS GASES
Shape and Volume
• Take the shape of their containers
• Have Fixed Volumes regardless of
their container volumes
• Fixed volumes are not greatly
affected by temperature and
pressure
• Take the shape of their containers
• Take the volume of their
containers
• Gas volume varies with
temperature and pressure
Resistance and Shear • Cannot support shear
• Will deform continuously to minimize
shear forces
• Cannot support shear
• Will deform continuously to
minimize shear forces
Compressibility • Slight Compressible
• Incompressible
• Highly Compressible
Resistance to motion • Resist instantaneous changes in
velocity due to viscosity
• Resistance to motion stops when the
liquid motion stops
• Cannot resist instantaneous
changes in velocity because of
very low viscosities

4. CHARACTERISTICS LIQUIDS GASES
Pressure • The same in all direction • The same in all direction
Molecular Spacing • Molecules are relatively
closer together
• Molecules are held
together by strong forces
of attraction
• Molecules have low kinetic
energy
• Distance each molecule
travels between collision is
small
• Molecules are relatively
far apart
• Molecules have weak
forces of attraction
• Molecules have high kinetic
energy
• Distance each molecule
travels between collision is
large

5. TYPES OF FLUIDS
Fluids are generally divided into two types, namely, IDEAL FLUIDS and REAL
FLUIDS. The characteristics of the two types of fluids are as tabulated:
IDEAL FLUIDS REAL FLUIDS
No viscosity(i.e. resistance to shear is
zero)
Viscous
Incompressible Compressible
Have uniform velocity distribution when
flowing
Non-uniform velocity distribution
No friction between moving layers of
fluids
Experience friction between moving
layers of fluids
No eddy currents Turbulent in flow
No eddy currents
No turbulence

6. Real fluids are further divide 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.
DENSITY OF FLUID
The density () is defined as mass (m) per unit volume (V).
𝜌 =
𝑚
𝑉

7. SPECIFIC VOLUME OF FLUID
Specific volume (v) is the volume (V) occupied per unit mass (m) of the fluid.
SPECIFIC WEIGHT OF FLUID
Specific weight () is defined as weight (W) per unit volume (V).
v=
𝑉
𝑚
=
1
𝜌
𝛾=
𝑊
𝑉
=𝜌 𝑔 =

8. SPECIFIC GRAVITY OF FLUID
Specific gravity (SG) is the ratio of the density of a substance to the density of
water at a reference temperature of 4. Sometime called relative density.
𝑆𝐺=
𝛾𝑓𝑙𝑢𝑖𝑑
𝛾𝑤𝑎𝑡𝑒𝑟
=
𝜌𝑓𝑙𝑢𝑖𝑑
𝜌𝑤𝑎𝑡𝑒𝑟
Food for the brain:
SG of water = 1
SG of mercury = 13.6

9. VISCOSITY OF A FLUID
The viscosity of a fluid is a measure of that fluid’s resistance to flow when
acted upon by an external force such as a pressure gradient or gravity.
v
F
h
fluid
stationary surface
A

10. FORMULA:
Where :
F = force on the plate
A = cross sectional area of the moving the plate
S = shear stress experienced by the fluid
= absolute viscosity of the fluid
v = velocity of the moving plate
h = distance between moving plate and stationary surface (height
of fluid)
𝐹=𝜇
v 𝐴
h
→𝑆=
𝐹
𝐴
=𝜇
v
h

11. F A S v h
N pa N-s/
or pa-s
m/s m
lb psi lb-s/
or psi-s
in/s in
dyne dyne/ dyne-s/
or poise
cm/s cm

12. KINEMATIC VISCOSITY
The kinematic viscosity () is the ratio of the absolute viscosity () of a fluid to
its mass density ().
𝑣 =
𝜇
𝜌
or stoke or stoke
or Pa-s
Food for the brain:
1 poise = 0.1 Pa-s
1 centipoise = 0.01 poise

13. Ex. A sliding-plate viscometer (50 cm x 25 cm) is used to measure
the viscosity of a Newtonian fluid. A force of 25 N is required to
keep the top plate moving against 1 mm height of fluid at a
constant velocity of 5 m/s. What is the viscosity of the fluid? Ans.
0.04 Pa-s

14. Given: A = 50cm x 25cm F = 25 N h = 1
mm
v = 5 m/s
Solution:

15. Ex. The viscosity of a fluid with specific gravity 1.3 is measured to
be 0.0034 N-s/. Solve its kinematic viscosity in /s. Ans. 2.615 x

16. Given: sg = 1.3
Solution:
*

17. Ex. A given oil has a kinematic viscosity of 0.000125 /s
and a specific gravity of 0.8. What is the dynamic
viscosity? Ans. 0.1 Pa-s

18. Given: sg = 0.8
Solution:
*

19. Ex. A given oil () is sheared between two parallel plates
0.005 inch apart with the lower plate fixed and the upper
plate moving at 13 ft/s. Compute the shear stress oil. Ans.
287

20. Given:
v = 13 ft/s
Solution:

21. COMPRESSIBILITY OF A FLUID (B)
- is the fractional change in the volume, of a fluid per unit change in
pressure, at constant temperature process.
BULK MODULUS (E)
- of a fluid is similar to the modulus of elasticity of a solid. Bulk
modulus is the reciprocal of compressibility. Bulk modulus is sometimes
called coefficient of compressibility.
B=
|∆ V
V |
∆ P
E=
1
B
=
∆ P
|∆ V
V |
Food for the brain: If not given, the bulk modulus
of water at standard condition is 2.1 GPa

22. Ex. A 0.1 volume of water is observed to be 0.0982 after a
pressure is applied. What is that pressure? Bulk modulus of
elasticity = 2.1 GPa.
Ans. 37.8 Mpa

23. Given:
Solution:

24. Ex. The compressibility of water is 5 x /N. Find the decrease
in volume of 100 mL of water when subjected to a pressure
of 15 Mpa. Ans. 0.75 mL

25. Given:
Solution:

26. Ex. Water in the hydraulic press, initially at 20 psia, is
subjected to a pressure of 17 000 psia. Determine the
percentage decrease in its volume if the average bulk
modulus of elasticity is 365 000 psi. Ans. 4.65 %

27. Given:
Solution:

28. SURFACE TENSION OF A FLUID
The membrane or “skin” that seems to form on free surface of a fluid is caused
by intermolecular cohesive forces and is known as surface tension. Surface tension is
the reason why insects can sit on a pond or a needle can float on the surface of a
glass of water.
Where:
= surface tension (N/m or lb/ft)
= change in pressure
r = radius of droplet or bubble
for droplets
for bubbles
Food for the brain: If not given, the
surface tension of water in air is
approximately 0.0756 N/m or
0.00518 lb/ft

29. Ex. A soap bubble has a radius of 5 cm. If the soap solution
has a surface tension of 0.03 N/m. What is the gage
pressure within the bubble? Ans. 2.4 Pa.

30. Given:
Solution:

31. Ex. The surface tension force of a water in air is
approximately 0.00518 lb/ft. If the atmospheric pressure is
14.7 psia, what is the pressure inside a droplet 0.01 inch in
diameter? Ans. 14.87 psia

32. Given:
Solution:

33. Ex. A soap bubble 50 mm in diameter contains a pressure
(in excess of atmospheric pressure) of 2 bars. Find the
surface tension in the soap film?
Ans. 1250 N/m2.

34. Given:
Solution:

35. CAPILLARITY OF A FLUID
Capillarity action is the name given to the behaviour of a liquid in a thin
bore tube. Capillary action is caused by surface tension between the liquid
and a vertical solid surface. The liquid tends to climb the wall.
±h=
4 𝜎 𝑐𝑜𝑠𝜃
𝛾 𝑑
Where:
h = height of rise or
fall in the tube
𝜃 = contact angle
= surface tension
d = diameter of the
tube

36. CONTACT ANGLES
The angle of contact, ,
𝜃 indicates whether adhesive or cohesive forces
dominate. For angles of contact less than 90º, adhesive forces dominates while
for angles more than 90º, cohesive forces dominates
MATERIALS ANGLE, 𝜃
Mercury-glass 140º
Water-paraffin 107º
Water-silver 90º
Kerosene-glass 26º
Glycerine-glass 19º
Water-glass 0º
Ethyl alcohol-glass 0º

37. Ex. Water at 20 ºC ( N/m) will rise in a clean 1 mm
diameter glass tube a distance of ? Ans. 3 cm

38. Given:
Solution:

39. Ex. Mercury makes an angle of 130º with respect to the
vertical when in contact with clean glass. How far will
mercury depress in a clean, 10 m diameter glass tube. If
N/m for mercury. Ans. – 0.9 m

40. Given:
Solution:
*

41. Ex. A given liquid has a surface tension of 0.4 N/m. In a 3
mm diameter vertical tube if the liquids rises 6 mm above
the liquid outside the tube, calculate the contact angle.
Ans. 83.66º

42. Given:
Solution:

43. PRESSURE OF A FLUID
Pressure is the force per unit area exerted by a liquid or gas on a body or
surface, with the forces acting right angles to the surface uniformly in all
directions. Fluid pressure are measured in respect to two pressure references:
atmospheric pressure and zero pressure.
Atmospheric Pressure is the pressure caused by a gases which composes
in the atmosphere.
Gage pressure is the pressure measured with respect to the atmospheric
pressure or barometric pressure. It is measured using pressure gages and
manometers.
Absolute pressure is the pressure measured with respect to the true zero
pressure reference.

44. Negative gage
pressure
Positive gage
pressure
(Absolute Vacuum)

45. Food for the brain: Unless otherwise specified, the term pressure, means
gage pressure.
Where:
= absolute pressure
= atmospheric
pressure
= gage pressure
Standard Atmospheric Pressures:
= 14.7 psi
= 101.325 Kpa
= 1 atm or 1
atmosphere
= 760 mm Hg
= 29.92 in Hg
= 760 torr

46. Ex. The pressure 8 ft below the free surface of a liquid is
2.4 psi. What is the specific gravity of the liquid? Ans.
0.692

47. Given:
Solution:

48. Ex. The gage on a condenser shows a vacuum of 24
inches of mercury. What is the absolute pressure in psi?
Ans. 2.91 psi

49. Given:
Solution:

50. Ex. What is the gage pressure in the tank of water 2 m
below the water surface? Ans. 19.62 KPa

51. Given:
Solution:

52. Ex. How high does a mercury barometer stand on a day
when atmospheric pressure is 98.6 KPa? Ans. 739 mm

53. Given:
Solution:
*

54. VARIATION OF PRESSURE WITH DEPTH IN A FLUID

55. Note: FFS stands for Free Fluid Surface which refers to fluid surface subject
to zero gauge pressure.
𝑝2=𝑝1 +𝛾 h

56. PRESSURE RELATION DUE TO DIFFERENT LEVELS OF FLUID
hc
hb
ha
P1
P2
Fluid a
Fluid b
Fluid c
+
Where:
= pressure at level 1 (top of fluid)
= pressure at level 2 (bottom of fluid)
= specific weight of the given fluid
h = head of the given fluid

57. Ex. An open tank contains 9.4 ft of water beneath 1.8 ft of oil (SG
= 0.85). Find the gage pressure at the bottom of the tank. Ans.
4.736 psig
Solution:
* Note: The oil on the top is understood to be at
open atmosphere, thus
zero gauge

58. Ex. A pressure gage 19 ft above the bottom of a tank containing a
liquid reads 13.19 psi, another gage at height 14 ft reads 15.12
psi. Compute the specific gravity of liquid. Ans. 0.891
Given:
Solution:

59. PASCAL’S LAW
Pascal's law is a principle in fluid mechanics given by Blaise Pascal
that states “that a pressure change at any point in a confined incompressible
fluid is transmitted throughout the fluid such that the same change occurs
everywhere”.

60. PRESSURE HEAD
Pressure head is the height “h” of a column of homogeneous liquid of
unit weight “” that will produce an intensity of pressure “p”.
To convert Pressure Head (height) of liquid A to B
or or
h=
𝑝
𝛾
h𝐵=h𝐴
𝑆𝐺 𝐴
𝑆𝐺𝐵
h𝐵 =h 𝐴
𝜌𝐴
𝜌𝐵
h𝐵 =h 𝐴
𝛾 𝐴
𝛾𝐵

61. To Convert Pressure Head (height) of any liquid to water, just multiply its
height by its specific gravity
MANOMETER
A manometer is a tube, usually bent in a from of U, containing a liquid of
known specific gravity, the surface of which moves proportionally to changes of
pressure. It is used to measure pressure.
TYPES OF MANOMETER
• OPEN TYPE – Open manometer is a tube bent into a U-shape to contain one or
more fluids of different specific gravities. It is used to measure gage pressure.
h𝑤𝑎𝑡𝑒𝑟=h𝑙𝑖𝑞𝑢𝑖𝑑 ∙𝑆𝐺𝑙𝑖𝑞𝑢𝑖𝑑

62. • DIFFERENTIAL TYPE – without an atmospheric surface and capable of
measuring only differences of pressure.
• PIEZOMETER – The simplest form of open manometer. It is a tube tapped
into a wall of a container or conduit for the purpose of measuring
pressure. The fluid in the container or conduit rises in this tube to form
a free surface.
LIMITATIONS OF PIEZOMETER
• Large pressures in the lighter liquids require long tubes
• Gas pressure can not be measured because gas can not form free
surface.

63. (a) Open Manometer (b) Differential Manometer (c) Piezometer

64. STEPS IN SOLVING MANOMETER PROBLEMS
Ordinarily, it is easier to work in units of pressure head rather than pressure for solving
any manometer problem.
1. Draw a sketch of the manometer approximately to scale.
2. Decide on the fluid of which head are to be expressed. Water is more desirable. In
most cases, we suggest to use head in water even if there is no water in the system.
3. Starting at a point of know pressure head, number in order the levels of contact of
fluids of different specific gravities.
4. Proceed from level to level, add pressure head in going down and subtract pressure
head in going up with due regard to the specific gravity of the fluids.

65. Ex. For the tank shown in the Fig., h1 = 3 m and h3 = 4 m. Determine the value
of h2.
OIL
SG = 0.84
WATER
WATER
h3
h1
h2
Ans.
= 1.19 m

66. Solution:

67. Ex. In the figure shown what is the static pressure in KPa in the air chamber.
OIL
SG = 0.80
WATER
Ans.
AIR 2 m
3 m
4 m
2 m

68. Solution:

69. Ex. Determine the value of y in the manometer shown in the fig.

70. Solution:

71. Ex. For the manometer setup shown, determine the difference in
pressure between A and B.

72. Solution:

73. HYDRAULIC PRESSURE
In a hydraulic pressure at both ends are equal, thus its force is directly
proportional the cross sectional area of its piston.

74. Ex. What is the radius of the small piston of a hydraulic
press when a force of 20 lbs on it produces a force of
8,000 lbs on the large piston whose diameter is 20
inches? Neglect friction

75. Note: In a hydraulic press, the pressure under the large
piston is equal to the pressure under small piston
Solution: