Fluid mechanics .........................

Reference Books:
1. Fluid Mechanics V. L Streeter,. and E. B Wylie,., , McGraw
Hill, 1985, New York
2. Theory and Applications of Fluid Mechanics,
K. Subramanya , Tata- McGraw Hill Publishing Co, 1993,
New Delhi.
3. Introduction to Fluid Mechanics E. J Shaughnessy,., I. M
Katz,. and J. P Schaffer,. SI edition, 2005, Oxford University
Press, New Delhi
4. Fluid Mechanics , F. M., White, 5th Edition, McGraw Hill,
New York.

5. Fluid mechanics By Dr. D. S. Kumar
6. Fluid mechanics & Hydraulic Machines By Dr. P. N. Modi
& Sheth
7. Fluid mechanics By Dr. A. K. Jain
8. Hydraulic Fluid mechanics & Fluid Machines By
S. Ramamurthan
9. Engineering Fluid Mechanic By R. J. Garde & A. C.
Mirajgaoker

Other books:
10. Fluid mechanics & Hydraulic Machines By R. K. Rajput
11. Fluid mechanics & Hydraulic Machines By R. K. Bansal

This chapter will begin with several concepts, definition,
terminologies and approaches which should be
understood by the students before continuing reading
the rest of this module.
Then, it introduces with typical properties of fluid and
their dimensions which are then being used extensively
in the next chapters and units like pressure, velocity,
density and viscosity. The handling of liquids is much
simpler, much cheaper, and much less troublesome than
handling solids.
Some of these can be used to classify type and
characteristic of fluid, such as whether a fluid is
incompressible or not or whether the fluid is Newtonian
or non-Newtonian.

At the end of this chapter, you should be able to :
Identify and describe typical fluid
properties and their units
and dimensions
Understand the basic concepts of
Fluid Mechanics.

Recognize the various types of fluid flow
problems encountered in practice.
Have a working knowledge of viscosity and the
consequences of the frictional effects it causes
in fluid flow.
Calculate the capillary rises and drops due to
the surface tension effects.

Definition:
Fluid mechanics is a physical science concerned with the
behavior of fluid (liquid or gas) at rest and in motion.
Fluid Mechanics is that section of applied mechanics, concerned
with the statics and dynamics of liquids and gases.
Fluid mechanics is basically study of the:
1. Physical behavior of fluid and fluid system, and a law
governing this behavior
2. Action of forces on fluids and the resulting flow pattern
Introduction to Fluid Mechanics

Fluid Mechanics
Statics
The study of fluid
at rest
Kinematics
The study of fluid
in motion without
pressure
consideration
Dynamics
The study of fluid in
motion with
pressure
consideration

E
Hooke law Newton law

y
u




Ct

  
p
t
F ,




• Distinction between solids and fluids:
A solid is “hard” and not easily deformed. A fluid is
“soft” and deforms easily.
Fluid is a substance that alters its shape in response to
any force however small, that tends to flow or to
conform to the outline of its container, and that
includes gases and liquids and mixtures of solids and
liquids capable of flow.
A fluid is defined as a substance that deforms
continuously when acted on by a shearing stress of any
magnitude.

The differences between the behaviors of solids and fluids
under an applied force are as follows:
For a solid, the strain is a function of the applied stress,
providing that the elastic limit is not exceeded. For a fluid,
the rate of strain is proportional to the applied stress.
The strain in a solid is independent of the time over
which the force is applied and, if the elastic limit is not
exceeded, the deformation disappears when the force is
removed. A fluid continues to flow as long as the force is
applied and will not recover its original form when the
force is removed.

With exception to solids, any other matters can be
categorised as fluid. In microscopic point of view, this
concept corresponds to loose or very loose bonding
between molecules of liquid or gas, respectively.
Examples of typical fluid used in engineering
applications are water, oil and air.
An analogy of how to understand different bonding
in solids and fluids is depicted in Fig.

Figure : Comparison Between Solids, Liquids and Gases
For solid, imagine that the molecules can be fictitiously
linked to each other with springs.
(a) Solid (b) Liquid (c) Gas
k
k
k
k
Free surface

In fluid, the molecules can move freely but are
constrained through a traction force called cohesion. This
force is interchangeable from one molecule to another.
For gases, it is very weak which enables the gas to
disintegrate and move away from its container.
For liquids, it is stronger which is sufficient enough to
hold the molecule together and can withstand high
compression, which is suitable for application as
hydraulic fluid such as oil. On the surface, the cohesion
forms a resultant force directed into the liquid region
and the combination of cohesion forces between adjacent
molecules from a tensioned membrane known as free
surface.

Differences between liquids and gases:
liquids and gases both share the common characteristics of
fluids.
• A liquid is difficult to compress and, for many
purposes, may be regarded as incompressible.
• A given mass of liquid occupies a fixed volume,
irrespective of the size or shape of its container,
and a free surface is formed if the volume of the
container is greater than that of the liquid.
Liquid
• A gas is comparatively easy to compress.
• Changes of volume with pressure are large, cannot
normally be neglected and are related to changes
of temperature.
• A given mass of gas has no fixed volume and will
expand continuously unless restrained by a
containing vessel. It will completely fill any vessel
in which it is placed and, therefore, does not form
a free surface.
Gas

Fluids:
A fluid is a substance which deforms continuously
under the action of shearing forces at any magnitude,
however small they may be.
In other words, it can flow continuously as a result of
shearing action. This includes any liquid or gas.
If a fluid is at rest, there can be no shearing forces
acting and, therefore, all forces in the fluid must be
perpendicular to the planes upon which they act.

• Incompressible and having no
viscosity
• Only an imaginary fluid
Ideal fluid
• Which possesses viscosity
• All the fluid in actual practice is
real fluid
Real Fluid
• A fluid in which shear stress is
more than the yield value and
shear stress is proportional to
the rate of shear strain is known
as ideal plastic fluid
Ideal plastic
fluid

• which obey the Newton's law of
viscosity are called as Newtonian
fluids.
• A real fluid In which shear stress
is directly proportional to the
rate of shear strain
• water, benzene, ethyl alcohol,
CCl4, hexane
Newtonian
fluid
• which do not obey the Newton's
law of viscosity are called as non-
Newtonian fluids.
• A real fluid In which shear stress
is not directly proportional to the
rate of shear strain
• slurries, pastes, gels, polymer
solutions etc
Non
Newtonian
fluid


du
dy
Dilatant
Newtonian fluid
Pseudo
plastic
Bingham
Plastic
Ideal fluid   0
Types of Fluids

Density or Mass Density:
Density of a fluid; 
Definition: The density ρ of a fluid is defined as its mass per
unit volume and indicates its inertia or resistance to an
accelerating force.
 Slightly affected by changes in temperature and pressure.
Units: kg/m3
Typical values:
Water = 1000 kg/m3; Air = 1.23 kg/m3
A medium characterized by a density is called a continuum, and
follows the classical laws of mechanics— including Newton’s law
of motion

Specific Weight or Weight Density (unit wt.):
Specific weight of a fluid, 
Definition: It is the ratio of weight of the fluid to its unit
volume
 = W/V = mg/V
Arising from the existence of a gravitational force.
The relationship  and g can be found using the following:
Since  = m/V
therefore  = g
Units: N/m3
Typical values:
Water = 9814 N/m3; Air = 12.07 N/m3

Definition: It is defined as the volume of a fluid occupied
by a unit mass or volume per unit mass.
Specific volume of a fluid = Volume of fluid = 1/ 
Mass of fluid
Thus it is reciprocal of mass density.
Unit: m3/kg
It is commonly applied to gases.

The specific gravity (or relative density) can be defined in
two ways:
Definition 1: A ratio of the density of a liquid to the density
of water at standard temperature and pressure (STP) (20C,
1 atm), or
Definition 2: A ratio of the specific weight of a liquid to the
specific weight of water at standard temperature and
pressure (STP) (20C, 1 atm),
Unit: dimensionless.
STP
water
liquid
STP
water
liquid
SG
@
@ 






A reservoir of oil has a mass of 825 kg. The reservoir
has a volume of 0.917 m3. Compute the density, specific
weight, and specific gravity of the oil.
Solution:
3
/
900
917
.
0
825
m
kg
m
volume
mass
oil 





3
oil m
/
N
8829
81
.
9
x
900
g
mg
volume
weight








9
.
0
998
900
@



STP
w
oil
oil
SG



Definition: It is defined as the property of a fluid which
offers resistance to the movement of one layer of fluid
over another adjacent layer of the fluid. Or The viscosity
(m) of a fluid measures its resistance to flow under an
applied shear stress.
Viscosity is a measure of resistance to fluid flow as a result
of intermolecular cohesion. In other words, viscosity can
be seen as internal friction to fluid motion which can then
lead to energy loss.
Different fluids deform at different rates under the same
shear stress. The ease with which a fluid pours is an
indication of its viscosity. Fluid with a high viscosity such
as syrup deforms more slowly than fluid with a low
viscosity such as water. The viscosity is also known as
dynamic viscosity.
Viscosity, ,

The top layer causes a shear stress on the adjacent lower
layer while the lower layer causes a shear stress on the
adjacent top layer. This shear stress is proportional to the
rate of change of velocity with respect to Y.
τ α du/dy
Where, µ is the constant known as Coefficient of dynamic
viscosity
µ = τ/ (du/dy)
Hence viscosity is defined as the shear stress required to
produce unit rate of shear strain.
dy
du

 
 dynamic viscosity
kg/m  s or N  s/m2 or Pa  s
1 poise = 0.1 Pa  s

Units:
µ = Shear stress
Change of velocity/ Change of distance
= Force/ area
(Length/time)* (1/length)
Area = length2
= Force * time
Length2
Unit: N.s/m2 or kg/m/s
Unit of viscosity in CGS is called Poise which is equal to
dyne . Sec / cm2
Typical values:
Water = 1.14x10-3 kg/m/s; Air = 1.78x10-5 kg/m/s

Kinematic Viscosity:
The kinematic viscosity (ν) is the ratio of the
viscosity to the density of fluid.
ν (nu) = µ/ρ
Unit : length2/time = m2/sec
Kinematic viscosity is also known as stoke.
One stoke = cm2/sec = 10-4 m2/sec
and will be found to be important in cases in which
significant viscous and gravitational forces exist.
Typical values:
Water = 1.14x10-6 m2/s; Air = 1.46x10-5 m2/s;

Newton’s Law of Viscosity:
It state that the shear stress on a fluid element layer
is directly proportional to the rate of shear strain.
dy
du



 = shear stress
 = viscosity of fluid
du/dy = shear rate, rate of strain or velocity gradient
The viscosity  is a function only of the condition of the
fluid, particularly its temperature.
The magnitude of the velocity gradient (du/dy) has no
effect on the magnitude of .

Example:
Air, Water, Oil, Gasoline
Alcohol, Kerosene, Benzene
Glycerine
Fluid Newton’s law
of viscosity
Newtonian fluids
obey refer
Newtonian and Non-Newtonian Fluid
Fluid Newton’s law
of viscosity
Non- Newtonian
fluids
Do not
obey refer

•The viscosity of the non-Newtonian fluid is dependent
on the velocity gradient as well as the condition of the
fluid.
Newtonian Fluids
a linear relationship between shear stress and the
velocity gradient (rate of shear),
the slope is constant
the viscosity is constant
Non-Newtonian fluids
slope of the curves for non-Newtonian fluids varies
Newtonian and Non-Newtonian Fluid

If the gradient m is constant, the fluid is termed as
Newtonian fluid. Otherwise, it is known as non-
Newtonian fluid. Fig. 1.5 shows several Newtonian and
non-Newtonian fluids.

Viscosity of liquids:
Viscosity of liquids in general, decreases with increasing
temperature.
The viscosities (m) of liquids generally vary approximately
with absolute temperature T according to:
ln µ = a - b ln T
Viscosity of gases:
Viscosity of gases increases with increase in temperature.
The viscosity (m) of many gases is approximated by the
formula:
µ= µo(T/To)n
in which T is the absolute temperature, µo is the viscosity at
an absolute reference temperature To, and n is an empirical
exponent that best fits the experimental data.

Viscosity of liquids are generally two orders of magnitude
greater than gases at atmospheric pressure.
For example, at 25oC, µ water = 1 centipoise and
µ air = 1 x 10-2centipoise.
In general,
viscosity of liquids with of temperature, whereas
viscosity of gases with of temperature.
The viscosity of an ideal gas is independent of pressure,
but the viscosities of real gases and liquids usually
increase with pressure.

The viscosity of liquids decreases and the
viscosity of gases increases with temperature

A fluid moving relative to a
body exerts a drag force on
the body, partly because of
friction caused by viscosity.
Drag force: The force a flowing fluid exerts on
a body in the flow direction. The magnitude of
this force depends, in part, on viscosity

Fluid consist of liquid or gases. Gases are
compressible fluid hence thermodynamic propertes play
an important role. With thw change in a pressure and
temperature, the gases undergo large variation in density.
the relationship between pressure, specific volume
and temperature of a gas is given by,
p V =R T
where, p = Absolute pressure of a gas in N/m2
V = Specific volume = 1/ρ
R = gas constant, depend on particular gas in J/kg-K
T = Absolute temperature in °K
ρ = Density of gas
Thermodynamic Properties:

It can be made universal, applicable to all gases if it is
expressed in mole-basis.
One kilogram mole is defined as the product of one kilogram
mass of the gas and its molecular weight.

Now m = n X M
Where,
n = number of moles in volume of a gas
M = mass of the gas molecules/mass of a hydrogen
atom
m = mass of a gas in kg.
Hence , p V = n M R T
The product M X R is called universal gas constant and
is equal to 848 kgf- m/kg-mole °K in MKS units and
8314 J/kg mole °K in SI units

Bulk modulus (K) = (change in pressure)
(volumetric strain)
Compressibility and the Bulk modulus:

Volumetric strain is the change in volume divided by the
original volume.
Therefore,
(change in volume) = (change in pressure)
(original volume) (bulk modulus)
i.e., -dV/V = dp/K
Negative sign for dV indicates the volume as pressure
Typical values of Bulk Modulus:
K = 2.05 x 109 N/m2 for water
K = 1.62 x 109 N/m2 for oil.
The concept of the bulk modulus is mainly applied
to liquids, since for gases the compressibility is so great
that the value of K is not a constant.

50
Some consequences of surface
tension.
Liquid droplets behave like small
balloons filled with the liquid on a
solid surface, and the surface of the
liquid acts like a stretched elastic
membrane under tension.
The pulling force that causes this
tension acts parallel to the surface
and is due to the attractive forces
between the molecules of the liquid.
The magnitude of this force per unit
length is called surface tension (or
coefficient of surface tension) and is
usually expressed in the unit N/m.

This effect is also called surface energy [per unit area]
and is expressed in the equivalent unit of N  m/m2.
Surface tension coefficient s can be defined as the
intensity of intermolecular traction per unit length
along the free surface of a fluid, and its SI unit is N/m.
The surface tension effect is caused by unbalanced
cohesion forces at fluid surfaces which produce a
downward resultant force which can physically seen as
a membrane.
The coefficient is inversely proportional to
temperature and is also dependent on the type of the
solid interface.
For example, a drop of water on a glass surface will
have a different coefficient from the similar amount of
water on a wood surface.

Attractive forces acting on a liquid molecule at the surface
and deep inside the liquid.
Free surface of liquid
act as very thin film
under tension and act
as elastic
membrance.
Surface tension: The work done per unit increase in the surface area of the
liquid.

A molecule in the interior of a liquid is under attractive
forces in all directions and the vector sum of these forces is
zero. But a molecule at the surface of a liquid is acted by a
net inward cohesive force that is perpendicular to the
surface. Hence it requires work to move molecules to the
surface against this opposing force, and surface molecules
have more energy than interior ones.
The surface tension (s sigma) of a liquid is the work that
must be done to bring enough molecules from inside the
liquid to the surface to form one new unit area of that
surface (J/m2 = N/m).
Historically surface tensions have been reported in
handbooks in dynes per centimeter (1 dyn/cm = 0.001
N/m).

Surface tension is the tendency of the surface of a
liquid to behave like a stretched elastic
membrane.
There is a natural tendency for liquids to
minimize their surface area.
For this reason, drops of liquid tend to take a
spherical shape in order to minimize surface area.
For such a small droplet, surface tension will
cause an increase of internal pressure p in order
to balance the surface force.

The obvious case is that of a liquid droplet on a
horizontal surface that is not wetted by the liquid—
mercury on glass, or water on a surface that also has
a thin oil film on it. For small droplets, such as those
on the left of Fig., the droplet adopts a shape that is
almost perfectly spherical, because in this
configuration there is the least surface area for a
given volume.

Surface tension prevents the paper clip from submerging.

 The pressure inside a drop of fluid can be
calculated using a free-body diagram of a
spherical shape of radius R cut in half, as
shown in Fig,
 The force developed around the edge of
the cut sphere is tensile force due to surface
tension acting around the circumference of
the cut portion = 2R.
 This force must be balance with the difference between the
internal pressure pi and the external pressure pe acting on
the circular area of the cut R2 Thus,
2R = pR2
p = pi –pe = =
2
R
Force acting on one half
of the liquid drop
4
d

The free-body diagram of half a droplet
or air bubble and half a soap bubble.
A hollow bubble like a soap bubble in air has two surface s
in contact with air, one inside and other outside. Thus two
surface area are subjected to surface tension.
4R = pR2
p = pi –pe = = 8
d
4
R

Consider a liquid jet of diameter d and length L,
p = pressure intensity inside the liquid jet above the outside
pressure
 = Surface tension of the liquid
Force due to pressure = p X area of semi jet
= p X L X d
Force due to surface tension =  X 2L
Hence,
p X L X d =  X 2L
p = = 4
R
2
d
L
d

Capillary effect:
Capillaries:
Meniscus:

The effect may be becoming significant for small fluid system
such as liquid level in a capillary, as depicted in Fig. where it
will decide whether the interaction form by the fluid and the
solid surface is wetted or non-wetted.
If the adhesion of fluid molecules to the adjacent solid
surface is stronger than the intermolecular cohesion, the
fluid is said to wet on the surface. Otherwise, it is a non-
wetted interaction.

The capillary rise of water and
the capillary fall of mercury in a
small-diameter glass tube.
The forces acting on a liquid
column that has risen in a tube
due to the capillary effect.
Capillary rise is inversely proportional to the radius of the
tube and density of the liquid.

We normally say that a liquid wets a surface if Φ is less
than 90o and does not wet if Φ is more than 90o. Values
of Φ less than 20o are considered strong wetting, and
values of Φ greater than 140o are strong nonwetting.
The angle Φ is the angle between the edge of the liquid
surface and the solid surface, measured inside the liquid.
This angle is called the contact angle and is a measure of
the quality of wetting.
Wetting and contact angle
Fluids wet some solids and do not others.

For perfectly wetting, in which the liquid spreads as
a thin film over the surface of the solid, Φ is zero.
In case of no wetting. If there were exactly zero
wetting, Φ would be 180o. However, the gravity
force on the drop flattens the drop, so that
180o angle is never observed. This might represent
water on teflon or mercury on clean glass.

This is opposed by the gravity force on the column of
fluid, which is equal to the height of the liquid which is
above (or below) the free surface and which equals
Fg = (Π/4)d2hgρ
where ρ is the density of liquid.
Vertical component of the surface tensile force=
= σ Π d cosΦ
Equating these forces and solving for Capillary rise (or
depression),
we find
h = 4scos(Φ)
ρ g d

This pressure is a function of temperature (vapor
pressure increases with temperature). In this context we
usually think about the temperature at which boiling
occurs.
For example, water boils at 100oC at sea-level atmospheric
pressure (1 atm abs). However, in terms of vapor pressure,
we can say that by increasing the temperature of water at
sea level to 100 oC, we increase the vapor pressure to the
point at which it is equal to the atmospheric pressure (1
atm abs), so that boiling occurs. It is easy to visualize that
boiling can also occur in water at temperatures much below
100oC if the pressure in the water is reduced to its vapor
pressure.

For example, the vapor pressure of water at 10oC is 0.01
atm. Therefore, if the pressure within water at that
temperature is reduced to that value, the water boils.
Such boiling often occurs in flowing liquids, such as on
the suction side of a pump. When such boiling does
occur in the flowing liquids, vapor bubbles start growing
in local regions of very low pressure and then collapse
in regions of high downstream pressure. This
phenomenon is called as cavitation

Saturation temperature Tsat:
The temperature at which a
pure substance changes
phase at a given pressure.
Saturation pressure Psat: The
pressure at which a pure
substance changes phase at a
given temperature.

Vapor pressure (Pv): The pressure exerted by its
vapor in phase equilibrium with its liquid at a given
temperature. It is identical to the saturation
pressure Psat of the liquid (Pv = Psat).
Partial pressure: The pressure of a gas or vapor in a
mixture with other gases. For example, atmospheric
air is a mixture of dry air and water vapor, and
atmospheric pressure is the sum of the partial
pressure of dry air and the partial pressure of
water vapor.

73
There is a possibility of the liquid pressure in liquid-flow
systems dropping below the vapor pressure at some
locations, and the resulting unplanned vaporization.
The vapor bubbles (called cavitation bubbles since they
form “cavities” in the liquid) collapse as they are swept
away from the low-pressure regions, generating highly
destructive, extremely high-pressure waves.
This phenomenon, which is a common cause for drop in
performance and even the erosion of impeller blades, is
called cavitation, and it is an important consideration in
the design of hydraulic turbines and pumps.
Cavitation damage on a 16-mm by
23-mm aluminum sample tested at
60 m/s for 2.5 h. The sample was
located at the cavity collapse region
downstream of a cavity generator
specifically designed to produce high
damage potential.

Fluid mechanics .........................

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