CE 473/471L

MECHANICS OF FLUIDS
Chapter 1

INTRODUCTION/
PROPERTIES OF FLUIDS
 CHAPTER 1 – INTRODUCTION/PROPERTIES OF FLUIDS

INTRODUCTION

DEFINITION OF FLUID MECHANICS

Fluid Mechanics is the section of Applied Mechanics concerned with the statics and dynamics of
liquids and gases. It is the general title given to the study of all aspects of the behavior of fluids
which are relevant to engineers.

Engineering Mechanics

I. Mechanics of Solids

A. Rigid Bodies
1. Statics
2. Dynamics
2.1 Kinematics
2.2 Kinetics

B. Deformable Bodies
1. Strength of Materials
2. Theory of Elasticity
3. Theory of Plasticity

II. Mechanics of Fluids

A. Ideal Fluids
B. Viscous Fluids

SOME APPLICATIONS OF FLUID MECHANICS

 breathing  airplane  icebergs

 blood flow  ships  engines
 swimming  rivers  filters
 pumps  windmills  jets
 fans  pipes  sprinklers
 turbines  missiles

DEFINITION OF FLUID

Fluid - a substance which can readily flow, i.e. in which there can be a continuous relative motion
between one particle and another; a substance that conform to the shape of containing vessel.

Classifications of fluid:

a. Liquid
b. Gas
Characteristics of a fluid:

a. Fluid can offer no resistance to any force causing change of shape

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b. Fluid flows under its own weight and takes the shape of any solid body with which
they are in contact

c. Fluids have some degree of compressibility

d. Fluids cannot sustain tangential or shear forces when in equilibrium

Differences between Liquid and Gas as Fluids

Liquid:
1. Practically incompressible
2. Occupies definite volumes and have free surface

Gas:
1. Compressible
2. Expands to fill any vessel in which it is contained and does not have free surface

Distinctions between a Solid and a Fluid

The strain of a solid is independent of the time of application of the force and if the elastic limit
is not exceeded the deformation disappears when the stress is removed, but a fluid continues
to flow as long as the stress is applied and does not recover its original form when the stress
is removed.

SYSTEMS OF UNITS

Table 1.1

British Gravitational (BG) International System English Engineering (EE)

Dimension
System (SI) System
Length Foot, ft Meter, m Foot, ft
Time Seconds, s Seconds, s Seconds, s
Force Pound, lb Newton, N Pound, lb or lbf
Temperature Fahrenheit, F Celcius, C Fahrenheit, F
Absolute
Rankine, R Kelvin, K Rankine, R
Temperature
Mass Slug Kilogram, kg Pound mass, lbm

NOTE: This learning guide will use BG System and SI.

PROPERTIES OF FLUIDS

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1. Density - the ratio of mass of a given quantity of a substance to the volume occupied by
that quantity.
m
 1.1
V

where:  = density (kg/m3, slugs/ft3)

m = mass (kg, slugs)
V = volume (m3, ft3)

2. Specific Weight - the ratio of a given quantity of a substance to the volume occupied by
that quantity. An alternative definition is that specific weight equals the product of density
and gravitational acceleration.

W
  g 1.2
V

where:  = specific weight (kN/m3, lb/ft3)

W = weight (kN, lb)
V = volume (m3, ft3)

3. Relative Density - (also called 'specific gravity') the ratio of the density of a substance to
some standard density

substan ce  substan ce
SG   1.3
 water  water

4. Specific Volume – the volume per unit mass and is therefore the reciprocal of the density.

1 1.4
V


where: v = specific volume (m3/kg, ft3/slug)

 = density (kg/m3, slugs/ft3)

This property is not commonly used in fluid mechanics but is used in thermodynamics.

5. Ideal Gas Law (Perfect Gas Law or Equation of State for an Ideal Gas)

p  RT 1.5
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where: p = absolute pressure (Pa, kPa, psi, psf)

 = density (kg/m3, slugs/ft3)
R = gas constant (J/kg-K, ft-lb/slug-R)
T = absolute temperature (K, R)

Absolute temperatures:

K  C  273.15 1.6

R  F  459.67 1.7

6. Viscosity - internal friction between the layers of fluid and represents the susceptibility of
a given fluid to shear deformation. It is the ratio of the applied shear stress to rate of shear
strain.

 Coefficient of Dynamic Viscosity - the shear force per unit area required to drag
one layer of fluid with unit velocity past another layer unit distance away from it
in the fluid.


 1.8
dV
dy

where:  = absolute (dynamic) viscosity (N-s/m2, lb-s/ft2)

 = shear stress (Pa, kPa, MPa, psi, psf)
dV/dy = rate of shear strain (s-1)

 Kinematic Viscosity - the ratio of dynamic viscosity to mass density


v 1.9


where:  = kinematic viscosity (m2/s, ft2/s)

 = absolute viscosity (N-s/m2, lb-s/ft2)
 = mass density (kg/m3, slugs/ft3)

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Figure 1.1

7. Surface Tension - the tensile force per unit length at the free surface of a liquid. It is
responsible for capillary action which causes a liquid to rise or to fall in porous media.
Liquids rise in tubes they wet (adhesion > cohesion, Figure1.1) and fall in tubes they do
not wet (cohesion > adhesion, Figure1.2).

F
 1.10
L

where:  = surface tension of a liquid (N/m, lb/ft)

F = the elastic force transverse to any length element L (N, lb)

Figure 1.2

adhesion cohesion

 tube
tube

h
h 

liquid liquid

d d

Figure 1.3 Figure 1.4

capillary rise capillary depression

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2 cos 
h 1.11
r

where: h = height of capillary rise (or depression) (mm, in), see Figures 1.1 and 1.2
 = surface tension (N/m, lb/ft)
 = wetting angle (degrees)
 = specific weight of liquid (kN/m3, lb/ft3)
r = radius of tube (mm, in)

8. Vapor Pressure

It is a common observation that liquids such as water and gasoline will evaporate if they
are simply placed in a container open to the atmosphere. Evaporation takes place because
some liquid molecules at the surface have sufficient momentum to overcome the
intermolecular cohesive forces and escape into the atmosphere. If the container is closed
with a small air space above the surface, and this space evacuated to form a vacuum, a
pressure will develop in the space as a result of the vapor that is formed by the escaping
molecules. When an equilibrium condition is reached so that the number of molecules
leaving the surface is equal the number entering, the vapor is said to be saturated and the
pressure that the vapor exerts on the liquid surface is termed the vapor pressure.

Figure 1.5 Figure 1.6

Figure 1.3 shows that when there is no lid, liquid molecules can escape until there is no
more liquid in the container. Figure 1.4 shows that with a lid, the molecules are trapped
in the container so there is no net loss of liquid in the container.

9. Bulk Modulus - the measure of the compressibility of a liquid, and is the ratio of the change
in pressure to the volumetric strain caused by the pressure change.

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dp
E 1.12
dv
v

where: E = bulk modulus of elasticity (GPa, psi)

dp = change in unit pressure (Pa, psi)
dv/v = volume change per unit of volume (m3/m3, in3/in3)

Note:
Because a pressure increase, dp, results in a decrease in fractional volume, dv/v, the minus
is inserted to render E positive.

Figure 1.7

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Table 1.2

Table 1.3

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CONVERSION FACTORS

1 cubic foot = 7.48 U.S. gallons = 28.32 liters

1 U.S. gallon = 8.338 pounds of water at 60F
1 cubic foot per second (ft3/s) = 0.646 million gallons per day = 448.8 gallons per minute
1 pound-second per square foot (lb-s/ft2),  = 478.7 poises
1 square foot per second (ft2/s)  = 0.0929 square meter per second (m2/s)
1 horsepower (hp) = 550 foot-pounds per second (ft-lb/s) = 0.746 kilowatt (kW)
30 inches of mercury = 34 feet of water = 14.7 pounds per square inch (lb/in2 or psi)
762 millimeters of mercury = 10.4 meters of water = 101.3 kiloPascals (kPa)

SYMBOLS AND ABBREVIATIONS

British Engineering System to International System to British Engineering
Parameter
International System System
Length 1 in = 0.0254 m 1 m = 39.37 in
1 ft = 0.3048 m 1 m = 3.281 ft
Mass 1 slug = 14.49 kg 1 kg = 0.06854 slug
Time 1 sec = 1s 1 s = 1 sec
Specific weight 1 lb/ft3 = 157.1 N/m3 1 N/m3 = 0.006366 lb/ft3
Mass density 1 slug/ft3 = 515.2 kg/m3 1 kg/m3 = 0.001941 slug/ft3
Specific gravity Dimensionless Dimensionless
Dynamic viscosity 1 lb-sec/ft2 = 47.88 N-s/m2 1 N-s/m2 = 0.02089 lb-sec/ft2
Kinematic viscosity 1 ft2/sec = 0.09290 m2/s 1 m2/s = 10 76 ft2/sec
Pressure 1 lb/ft2 = 47.88 Pa 1 Pa = 0.02089 lb/ft2
1 lb/in2 = 6.895 kPa 1 kPa = 0.1450 lb/in2
Surface tension 1 lb/ft = 14.59 N/m 1 N/m = 0.06853 lb/ft

*Slug = lb-s2/ft

Absolute and kinematic viscosities are converted according to:

(a) for t ≤ 100,  in poises = (0.00226t - 1.95/t) x sp gr

for t > 100,  in poises = (0.00220t - 1.35/t) x sp gr

(b) for t ≤ 100,  in stokeses = (0.00226t - 1.95/t)

for t > 100,  in stokeses = (0.00220t - 1.35/t)

where: t = Saybolt seconds

stoke = cm2/s

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IMPORTANT: Standard Values of the Properties of Water

1. SPECIFIC or UNIT WEIGHT OF WATER (for ordinary temperature variations)

WATER= 62.4 lbs/ft3 (English units)

WATER= 9.81 kN/m3 (S.I. units)

NOTE: The standard reference temperature for water is 4C

2. MASS DENSITY OF WATER (at 4C)

WATER = 1.94 slugs/ft3 (English units)

WATER = 1000 kg/m3 (S.I. units)

3. SPECIFIC GRAVITY OF A BODY (dimensionless)

Solids and liquids are referred to Water (at 68F = 20C) as standard.
Gases are referred to air free of carbon dioxide or hydrogen
(at 32F = 0C and 1 atmosphere = 14.7 lbs/in2 = 101.3 kPa pressure)

4. SURFACE TENSION OF WATER (surface tension for a water-air surface)

WATER = 0.00518 lbs/ft (at 32F)

WATER = 0.0756 N/m (at 0C)

5. ELASTICITY OF WATER

EWATER  2.26 N/m2 = 0.05 % change in volume for a change of 1 MN/m2 in pressure

6. VAPOR PRESSURE OF WATER

Water boils at 212F (100C) at sea level atmospheric pressure (14.7 psi = 101.3 kPa)

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SAMPLE PROBLEMS

*PROBLEM 1 (density, unit weight, specific volume, specific gravity)

A reservoir of carbon tetrachloride (CCl4) has a mass of 500 kg and a volume of 0.315 m3. Find the carbon
tetrachloride’s
(a) mass density,
(b) specific volume,
(c) specific weight, and
(d) specific gravity.

*PROBLEM 2 (density, unit weight, specific volume, specific gravity)

A certain gasoline weighs 46.5 lb/ft3. What are its
(a) mass density,
(b) specific volume, and
(c) specific gravity?

PROBLEM 3 (ideal gas law)

The density of oxygen contained in a tank is 2.0 kg/m3 when the temperature is 25C. Determine the gage pressure
(kPa) of the gas if the atmospheric pressure is 97 kPa.

*PROBLEM 4 (ideal gas law)

A closed tank having a volume of 2 ft3 is filled with 0.30 lb of a gas. A pressure gage attached to the tank reads 12 psi
when the gas temperature is 80F. There is some question as to whether the gas in the tank is oxygen or helium.
Which do you think it is?

PROBLEM 5 (viscosity)
A liquid has a specific weight of 59 lb/ft3 and a dynamic viscosity of 2.75 lb-s/ft2. Determine its kinematic viscosity
(ft2/s).

PROBLEM 6 (viscosity)
Carbon tetrachloride at 20C has a viscosity of 0.000967 N-s/m2. What shear stress is required to deform this fluid at
a rate strain of 5000 s-1?

PROBLEM 6 (viscosity)
A large movable plate is located between two large fixed plates as shown. Two Newtonian fluids having the viscosities
indicated are contained between the plates. Determine the magnitude and direction of the shearing stresses that act
on the fixed walls when the moving plate has a velocity of 4 m/s as shown. Assume that the velocity distribution
between the plates is linear.

Fixed plate

6 mm  = 0.02 N-s/m2

4 m/s
3 mm  = 0.01 N-s/m2
Fixed plate

PROBLEM 7 (viscosity)
If the fluid is glycerin at 20C and the width between plates is 6 mm, what shear stress is required to move the upper
plate at 2.5 m/s?

PROBLEM 8 (vapor pressure)

Early mountaineers used boiling water to estimate their height. If they reach the top, and find that water boils at 80C,
approximately how high is the mountain?

PROBLEM 9 (vapor pressure)

At approximately what temperature will water boil at an elevation of 10,000 ft?
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PROBLEM 10 (surface tension)

A needle 35 mm long rests on water surface at 20C. What force over and above the needle’s weight is required to lift
the needle from contact with the water surface? Neglect the weight of the needle.

PROBLEM 11 (surface tension)

Estimate the height to which water at 70F ( = 0.00497 lb/ft) will rise in a capillary tube of diameter 0.12 in. For
water, it may be assumed that  = 0.

*PROBLEM 12 (surface tension)

Calculate the approximate depression of mercury at 20C in a capillary tube of radius 1.5 mm. Surface tension for
mercury is 0.514 N/m at 20C, and its specific weight is 133.1 kN/m3. For clean tube, Hg = 140.

PROBLEM 13 (bulk modulus)

(a) From the following test data, determine the bulk modulus of elasticity of water:
At 400 psi, the volume was 10 ft3, and at 3500 psi, the volume was 9.90 ft3.

(b) Find the change in volume of 10 ft3 of water at 80F when subjected to a pressure increase of 500 psi.
Water’s bulk modulus of elasticity at this temperature is approximately 325,000 psi.

PROBLEM 14 (bulk modulus)

Approximately what pressure must be applied to water to reduce its volume by 1.25% if the bulk modulus of elasticity
is 2.19 GPa?

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