FLUID MECHANICS
EC 311
DEFINITION
• Fluid mechanics is the study of all aspects of the
behaviour of fluids, either at rest (fluid statics) or in motion
(fluid dynamics).
• Fluid mechanics is involved in nearly all areas of civil
engineering either directly or indirectly.
AREAS OF APPLICATIONS OF FLUID MECHANICS
 Aerospace engineering applications include the design
of aircraft, aerospace vehicles, rockets, missiles, and
propulsion systems.
 Biomedical applications include the study of blood flow
and breathing.
 Mechanical engineering applications include the design
of plumbing systems, heating ventilation and air
conditioning systems, lubrication systems, process-control
systems, pumps, fans, turbines, and engines.
AREAS OF APPLICATIONS OF FLUID MECHANICS
 Naval architecture applications include the design of
ships and submarines.
 Aside from engineering applications of fluid mechanics,
the earth sciences of hydrology, meteorology, and
oceanography are based largely on the principles of fluid
mechanics.
AREAS OF APPLICATIONS OF FLUID MECHANICS
• Fluid mechanics is also involved in nearly all areas of civil
engineering either directly or indirectly.
AREAS OF APPLICATIONS OF FLUID MECHANICS
 Direct application
o
o
o
o
o
o
o
o
o
Sea and river (flood) defences
Water distribution/sewerage (sanitation) networks
Hydraulic design of water/sewage treatment works
Oil and gas pipelines
Canals
Dams
Irrigation
Pumps and turbines
Water retaining structures
APPLICATIONS OF FLUID MECHANICS
 Indirect application
o Flow of air in or around buildings
 Determination of wind loads on buildings
o Bridge piers in rivers
o Groundwater flow
CHARACTERISTICS OF FLUIDS
• Three states of matter are recognized:
i. Solid
ii. Liquid
Fluids
iii. Gas
• These differ in the spacing and latitude of motion of their
molecules, these variables being large in a gas, smaller in a
liquid, and extremely small in a solid.
• Liquids and gases have a common characteristic in which they
differ from solids: they are fluids, lacking the ability of solids to
offer a permanent resistance to a deforming force.
CHARACTERISTICS OF FLUIDS
DEFINITION OF A FLUID
• A fluid is a substance that deforms continuously when
subjected to a shear stress, no matter how small that
shear stress maybe.
B
B’
C
φ
F
A
D
C’
F
DIFFERENCES BETWEEN LIQUIDS AND GASES
LIQUIDS
GASES
1
Molecular movement is
Molecular movement is
restricated by cohesive forces unrestricated by cohesive forces
2
Has a fixed volume
Incompressible
Has free surface
3
4
Has no shape or volume
Compressible
Has no free surface
THE CONTINUUM
• Continuous distribution of matter with no empty space.
• When the behaviour of a small element or particle of fluid
is studied, it will be assumed that it contains so many
molecules that it can be treated as part of this continuum.
• Quantities such as velocity and pressure can be
considered to be constant at any point, and changes due
to molecular motion may be ignored.
• Variations in such quantities can also be assumed to take
place smoothly, from point to point.
PROPERTIES OF FLUIDS
• Density, ρ
‒ defined as mass per unit volume
δm
ρ = δlim
V →ε δV
‒ In gases density is highly variable and it is nearly constant in
liquids
‒ In gases density increases proportionally to the pressure level
whereas in liquids it is considered independent to the pressure
level
‒ Typical values at atmospheric pressure: water, 1000 kg/m3; air,
1.23 kg/m3.
PROPERTIES OF FLUIDS
Variation of densities of pure water with temperatures between 0oC and
100oC
Density vs Temperature
1005
1000
995
990
Density (kg/m3)
•
985
980
975
970
965
960
955
0
20
40
60
Temperature
80
(oC)
100
120
PROPERTIES OF FLUIDS
Variation of densities of pure water with temperatures between 0oC and 10oC
Density vs Temperature (0 - 10oC)
1000.0
1000.0
999.9
Density (kg/m3)
•
999.9
999.8
999.8
999.7
999.7
999.6
0
2
4
6
Temperature (oC)
8
10
12
PROPERTIES OF FLUIDS
• Specific weight, w or γ
‒ defined as weight per unit volume
w = ρg
‒ Units: newtons per cubic metre (N/m3)
‒ Typical values: water, 9.81 x 103 N/m3 (9.81 kN/m3); air,
12.07 N/m3
PROPERTIES OF FLUIDS
• Specific gravity, s.g
‒ defined as the ratio of density of a fluid to the density of a
standard reference fluid, usually water at 4oC for liquids, and
air for gases.
‒ For liquids
ρ subs tan ce
density of substance
=
s.g =
o
density of water at 4 C ρ water at 4o C
PROPERTIES OF FLUIDS
• Specific gravity, s.g
Substance*
Specific Gravity (SG)
Substance*
Specific Gravity (SG)
Gold
19.3
Seawater
1.025
Uranium
18.7
Water
0.998
Mercury
13.6
SAE 10-W motor oil
0.92
Lead
11.4
Dense oak wood
0.93
Copper
8.91
Ice (0oC)
0.916
Steel
7.83
Benzene
0.879
Cast iron
7.08
Crude oil
0.87
Aluminium
2.64
Ethyl alcohol
0.790
Concrete (cured)
2.4
Gasoline
0.72
Blood (at 37oC)
1.06
Balsa wood
0.17
Example
• Mercury has a density of 13600 kg/m3. What are its
specific weight and specific gravity?
PROPERTIES OF FLUIDS
• Specific volume, v
‒ defined as the volume occupied by a unit mass of fluid.
‒ it is simply a reciprocal of density
V
1
=
ν =
m
ρ
PROPERTIES OF FLUIDS
• Temperature
‒ Temperature T is related to the internal energy level of a fluid.
Engineers often use Celsius or Fahrenheit scales for
convenience
‒ Many applications require absolute (Kelvin or Rankine)
temperature scales.
K = oC + 273.15
‒ Where K and oC are temperatures in Kelvin and degree Celsius.
PROPERTIES OF FLUIDS
• Vapour pressure
‒ Defined as the partial pressure exerted by the vapour
molecules on the liquid surface
‒ The vapour pressure of a given fluid depends on the
temperature, and increases with increasing temperature
‒ Boiling of liquids occurs when pressure above the liquid
equals the vapour pressure of the liquid
‒ In flowing liquids boiling would occur if the pressure of the
liquid is reduced to below the vapour pressure
PROPERTIES OF FLUIDS
• Vapour pressure
Vapour Pressure vs Temperature
Vapour Pressure (kN/m2)
100
10
1
0
0.1
10
20
30
40
50
Temperature (oC)
60
70
80
90
100
PROPERTIES OF FLUIDS
• Compressibility
‒ all matter is to some extent compressible
‒ compressibility of a perfect gas is described by the
perfect gas law
‒ compressibility of liquids is described by the bulk
modulus of elasticity, K
‒ In liquids, compressibility is only possible in situations
involving either sudden or great changes in pressure
PROPERTIES OF FLUIDS
• Compressibility
‒ For liquids, bulk modulus of elasticity is given by
‒
δP
K =−
δV
or
K =
V
dP
dρ
ρ
‒ where δp = increase in pressure
δV = decrease in volume
‒ Compressibility is the reciprocal of bulk modulus of elasticity
PROPERTIES OF FLUIDS
• Surface tension, σ
‒ Caused by the force of cohesion at the free surface
‒ At the free surface a thin layer of molecules is formed
‒ It is because of this film that a small needle can float
on the free surface.
‒ Surface tension is usually expressed in N/m.
‒ Formation of bubbles, droplets and free jets are due to
the surface tension of the liquid.
PROPERTIES OF FLUIDS
• Surface tension, σ
PROPERTIES OF FLUIDS
• Pressure inside a water droplet
‒ Liquids have a tendency of minimizing their surface area
‒ As such drops of liquid tend to take a spherical shape in
order to minimize surface area
PROPERTIES OF FLUIDS
• Pressure inside a water droplet
‒ From free body diagram, we have
π 2
i. Pressure force, F = P × d
4
ii. Surface tension force acting around the circumference
= σ × πd
‒ Under equilibrium conditions these two forces will be equal
and opposite i.e.
P×
π
4
d = σ × πd
2
4σ
⇒P=
d
Example
• In order to form a stream of bubbles, air is introduced
through a nozzle into a tank of water at 20oC. If the
process requires 3.0 mm diameter bubbles to be formed,
by how much the air pressure at the nozzle must exceed
that of the surrounding water?
Take surface tension of water at 20oC = 0.0735 N/m.
PROPERTIES OF FLUIDS
• Capillarity
‒ Rise or fall of a liquid in a capillary tube
Causes of rise and fall
 cohesion
‒ when cohesion is greater than adhesion liquid level in a
tube will fall
 adhesion
‒ when adhesion is greater than cohesion liquid level in a
tube will rise
PROPERTIES OF FLUIDS
• Capillarity
• Liquids rise in tubes they wet and fall in tubes they do not wet .
PROPERTIES OF FLUIDS
• Capillarity - wetting and contact angle
 This represents the case of a liquid which wets a
solid surface well.
 The angle θ shown is the contact angle between
the edge of the liquid surface and the solid surface.
 It measures the quality of wetting.
 For perfect wetting, in which the liquid spreads as
a thin film over the surface of the solid, θ is zero.
PROPERTIES OF FLUIDS
• Capillarity - wetting and
contact angle
 This represents the case of no wetting.
 If there was exactly zero wetting, θ would be 180o.
PROPERTIES OF FLUIDS
• Capillarity - height of capillary rise
h=
4σ cos θ
ρgd
• Example
Water has a surface tension of 0.4 N/m. In a 3 mm diameter
vertical tube a liquid rises 6 mm above the liquid outside the
tube, calculate the contact angle.
PROPERTIES OF FLUIDS
• Viscosity, μ
‒ property of a fluid which offers resistance to shear
deformation
‒ resistance to shear deformation is achieved by cohesion
and interaction between molecules
‒ the resisting forces are referred to as shear forces and
these forces induce shear stresses in the fluid as a result
of particle movement
PROPERTIES OF FLUIDS
• Viscosity, μ
For a pipe in which water is flowing
i. At the pipe wall the velocity of fluid particles is zero
ii. At the centre of the pipe the velocity is maximum
‒ This way adjacent fluid particles will have different
velocities
• particles closer to the pipe boundary will move slower
than the particles closer to the centre
PROPERTIES OF FLUIDS
• Viscosity, μ
Velocity profile
PROPERTIES OF FLUIDS
• Viscosity, μ - Newton's law of viscosity
PROPERTIES OF FLUIDS
• Viscosity, μ - Newton's law
of viscosity
• F = shear force
F
F
=τ
• shear sress = =
A δz × δx
• Shear stress is measured by
the deformation angle Φ, the
shear strain
x
shear strain, φ =
y
PROPERTIES OF FLUIDS
• Viscosity, μ - Newton's law
of viscosity
• Experimentally, shear stress is
directly proportional to rate of
shear strain
rate of shear strain =
u
⇒ τ = constant ×
y
x u
=
ty y
PROPERTIES OF FLUIDS
• Viscosity, μ - Newton's law
of viscosity
• Proportionality constant is the
viscosity, μ
• In differential form
du
τ =µ
dy
τ = shear stress in N / m 2
du
= velocity gradient
dy
µ = coefficient of dynamic viscosity
PROPERTIES OF FLUIDS
• COEFFICIENT OF DYNAMIC VISCOSITY
• defined as the shear force per unit area, required to drag one
layer of fluid with unit velocity past another layer a unit distance
away
du
µ =τ /
dy
2
• Units: Ns/m or Pa.s
• Also Poise,P, 10P = 1Ns/m2
• Typical values: water, 1.14 x 10-3 Ns/m2, Air, 1.78 x 10-5 Ns/m2
PROPERTIES OF FLUIDS
• KINEMATIC VISCOSITY
• Kinematic viscosity, ν, is defined as the ratio of dynamic viscosity
to mass density.
µ
ν =
ρ
• Units: square metres per second, m2/s
• (Also expressed in stokes, st, where 104 st = 1 m2/s)
• Typical values: water = 1.14 x 10-6 m2/s, Air = 1.46 x 10-5 m2/s,
mercury = 1.145 x 10-4 m2/s, paraffin oil = 2.375 x 10-3 m2/s.
Example 1
• The velocity distribution for flow over a plate is given by u
= 2y – y2 where u is the velocity in m/s at a distance y
metres above the plate. Determine the velocity gradient
and shear stress at the boundary and 0.15m from it. Take
dynamic viscosity of fluid as 0.9 Ns/m2.
Example 2
• A hydraulic lift used for lifting automobiles has a 25 cm
diameter ram which slides in a 25.018 cm diameter
cylinder, the annular space being filled with oil having a
kinematic viscosity of 3.7 cm2/s and relative density of
0.85. If the rate of travel of the ram is 15 cm/s, find the
frictional resistance when 3.3 m of ram is engaged in the
cylinder.
NEWTONIAN AND NON-NEWTONIAN FLUIDS
• NEWTONIAN FLUIDS
‒ Fluids which obey Newton’s law of viscosity.
‒ All gases and most liquids which have simpler molecular
formula and low molecular weight such as water,
benzene, kerosene and most solutions of simple
molecules are Newtonian fluids.
NEWTONIAN AND NON-NEWTONIAN FLUIDS
• NON-NEWTONIAN FLUIDS
• Fluids which do not obey Newton’s law of viscosity.
• These include slurries, mud flows, polymer solutions,
blood etc.
NEWTONIAN AND NON-NEWTONIAN FLUIDS
NEWTONIAN AND NON-NEWTONIAN FLUIDS
Newtonian and nonNewtonian fluids can be
represented by the
equation
 du 
τ = A + B 
 dy 
n
NEWTONIAN AND NON-NEWTONIAN FLUIDS
Examples of Non-Newtonian fluids
1. Pseudoplastic or Thixotropic: Examples of such fluids
are blood plasma, syrups, adhesives, molasses, and
inks.
2. Dilatant fluids: Examples are slurries with high
concentrations of solids such as corn starch, starch in
water etc.
3. Bingham fluids: Examples of Bingham fluids are
chocolate, mayonnaise, toothpaste and sewage sludge.
HYDROSTATICS
• Hydrostatics is a branch of fluid mechanics which is
concerned with fluid at rest
• The fluid is not subjected to any tangential force or shear
stress
• In hydrostatics all forces act normally to a boundary
surface and are independent of viscosity
• Viscosity is a measure of fluid resistance to tangential
force or shear stress
HYDROSTATICS
INTRODUCTION TO PRESSURE
• Atmospheric pressure, gauge pressure and absolute
pressure
Absolute Pressure
Gauge pressure
(+ve pressure)
Atmospheric Pressure (Patm = 0)
Vacuum pressure
(-ve pressure)
Absolute zero
Pabs = Pgauge + Patm
INTRODUCTION TO PRESSURE
• For example, if a pressure of 50 kN/m2 is measured with a
gauge referenced to the atmosphere and the atmospheric
pressure is 100 kN/m2, then the pressure can be
expressed as either P = 50 kN/m2 (gauge) or P = 150
kN/m2 (absolute).
• If on the other hand, a gauge indicates a vacuum
pressure of 31 kN/m2, then this can be stated as 70 kN/m2
(absolute), or -31 kN/m2 (gauge), assuming that the
atmospheric pressure is 101 kN/m2 (absolute).
INTRODUCTION TO PRESSURE
• Pressure intensity
‒ defined as the pressure force per unit area.
P
P
‒ Mathematically,
P=
F
A
h
INTRODUCTION TO PRESSURE
• Pressure intensity
‒ Or P = W = mg = ρVg = ρAhg = ρgh
A
A
A
A
‒ where
ρ = density of the liquid
g = acceleration of gravity =
9.81 m/s2
h = depth below the free
surface or pressure head
W
P
P
h
INTRODUCTION TO PRESSURE
• Pressure intensity
‒ Units of pressure
i. N/m2, kN/m2
ii. Pascal, abreviated Pa
1 Pa = 1 N/m2
iii. bar
1bar = 1 x 105 N/m2
W
P
P
h
INTRODUCTION TO PRESSURE
• Pascal’s law for pressure at a point
• Pascal’s law states as follows:
“The intensity of pressure at any point in a liquid at rest
is the same in all directions.”
INTRODUCTION TO PRESSURE
• Proof of Pascal’s law
Force due to Px = Px × Area ABEF = Pxδyδz
Horizontal component of force
due to Ps = −(Ps × Area ABCD )sin θ
= − Psδsδz δy / δs = − Psδyδz
INTRODUCTION TO PRESSURE
• Proof of Pascal’s law
• Equating Force due to Px to
Horizontal component of force
due to Ps , we get
Pxδyδz = Psδyδz
⇒ Px = Ps
INTRODUCTION TO PRESSURE
• Proof of Pascal’s law
• Similarly in the y-direction
Force due to Py = Py × Area CDEF = Pyδxδz
Vertical component of force
due to Ps = −(Ps × Area ABCD ) cos θ
= − Psδsδz δx / δs = − Psδxδz
force due to weight of element = − ρ (δxδyδz / 2 )g
INTRODUCTION TO PRESSURE
• Proof of Pascal’s law
• Equating Force due to Px to
Horizontal component of force
due to Ps , we get
Pxδyδz = Psδyδz
⇒ Px = Ps
INTRODUCTION TO PRESSURE
• Proof of Pascal’s law
• δxδyδz is negligible, so weight
of element is neglected
• Equating Force due to Py to
vertical component of force
due to Ps , we get
⇒
Py = Ps
• Hence, Px = Py = Ps (proving
pascal's law)
PRESSURE VARIATION IN A STATIC FLUID
• Fundamental equation of fluid statics relates pressure, specific weight and
vertical distance.
• This equation may be derived by considering the static equilibrium of a
typical differential element of fluid in the figure below
PRESSURE VARIATION IN A STATIC FLUID
• For equilibrium, the algebraic sum
of the forces in any direction must
be zero. Resolving in the direction
QP
( p + δp )δA − pδA + ρgδAδl cos θ = 0
• But δl cos θ = δz
⇒
or
δp + ρgδz = 0
δp
= − ρg
δz
• The minus sign indicates that the
pressure decreases upwards.
PRESSURE VARIATION IN A STATIC FLUID
• Integrating the equation
∫
p2
p1
i.e.
z2
δp = − ρg ∫ δz
z1
p2 − p1 = − ρg ( z 2 − z1 )
PRESSURE VARIATION IN A STATIC FLUID
• Variation of pressure vertically in a
fluid under gravity
p2 − p1 = − ρgh
• where h = z2 - z1
PRESSURE VARIATION IN A STATIC FLUID
• Equality of pressure at the same
level in a static fluid
p2 − p1 = − ρg ( z 2 − z1 )
• But
z1 = z2
⇒
p2 − p1 = 0
or
p1 = p2
PRESSURE VARIATION IN A STATIC FLUID
• Equality of pressure at the same
level in a continuous body of fluid
• In a continuous body of liquid,
pressure at the same level is the
same i.e
PP = PQ
PR = PS