Fluid Mechanics Unit-1 (vk-ssm)

1
Dr V.KANDAVEL Asp/Mech. SSMIET, DGL-2

CE8394-FLUID MECHANICS
AND MACHINERY
Regulation 2017
SEM:3rd
2Dr V.KANDAVEL Asp/Mech. SSMIET, DGL-2

OBJECTIVES
• The properties of fluids and concept of control
volume are studied
• The applications of the conservation laws to flow
through pipes are studied.
• To understand the importance of dimensional
analysis
• To understand the importance of various types of
flow in pumps.
• To understand the importance of various types of
flow in turbines.
3Dr V.KANDAVEL Asp/Mech. SSMIET, DGL-2

4
Contd.,
• Understand the basic concepts of Fluid Mechanics.
• Recognize the various types of fluid flow problems
encountered in practice.
• Model engineering problems and solve them in a
systematic manner.
• Have a working knowledge of accuracy, precision, and
significant digits, and recognize the importance of
dimensional homogeneity in engineering calculations.

UNIT I - FLUID PROPERTIES AND FLOW
CHARACTERISTICS
• Units and dimensions-
• Properties of fluids-
– mass density, specific weight,
– specific volume, specific gravity,
– viscosity, compressibility, vapor pressure,
– surface tension and capillarity.
• Flow characteristics – concept of control volume
• application of
– continuity equation,
– energy equation and
– momentum equation.
Dr V.KANDAVEL Asp/Mech. SSMIET, DGL-2 5

Fluid Mechanics Unit-1 (vk-ssm)

8
INTRODUCTION
Fluid mechanics deals with
liquids and gases in motion
or at rest.
Mechanics: The oldest physical science that
deals with both stationary and moving
bodies under the influence of forces.
Statics: The branch of mechanics that
deals with bodies at rest.
Dynamics: The branch that deals with
bodies in motion.
Fluid mechanics: The science that deals
with the behavior of fluids at rest (fluid
statics) or in motion (fluid dynamics), and
the interaction of fluids with solids or other
fluids at the boundaries.
Fluid dynamics: Fluid mechanics is also
referred to as fluid dynamics by considering
fluids at rest as a special case of motion
with zero velocity.

9
Hydrodynamics: The study of the motion of fluids that can be
approximated as incompressible (such as liquids, especially
water, and gases at low speeds).
Hydraulics: A subcategory of hydrodynamics, which deals with
liquid flows in pipes and open channels.
Gas dynamics: Deals with the flow of fluids that undergo
significant density changes, such as the flow of gases through
nozzles at high speeds.
Aerodynamics: Deals with the flow of gases (especially air) over
bodies such as aircraft, rockets, and automobiles at high or low
speeds.
Meteorology, oceanography, and hydrology: Deal with naturally
occurring flows.

• Fluid mechanics is a study of the behavior of fluids,
either at rest (fluid statics) or in motion (fluid dynamics).
• The analysis is based on the fundamental laws of
mechanics, which relate continuity of mass and energy
with force and momentum.
• An understanding of the properties and behavior of fluids
at rest and in motion is of great importance in
engineering.
10

1.1 Definition of Fluid
• Fluid mechanics is a division in applied mechanics related to
the behaviour of liquid or gas which is either in rest or in
motion.
• The study related to a fluid in rest or stationary is referred
to fluid static, otherwise it is referred to as fluid dynamic.
• Fluid can be defined as a substance which can deform
continuously when being subjected to shear stress at any
magnitude. In other words, it can flow continuously as a
result of shearing action. This includes any liquid or gas.
11

Definition of Fluid
 A fluid is a substance, which deforms continuously, or
flows, when subjected to shearing force
 In fact if a shear stress is acting on a fluid it will flow
and if a fluid is at rest there is no shear stress acting on
it.
Fluid Flow Shear stress – Yes
Fluid Rest Shear stress – No
12

Contd.,
• Thus, 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.
Deformation of a rubber block placed
between two parallel plates under
the influence of a shear force. The
shear stress shown is that on the
rubber—an equal but opposite shear
stress acts on the upper plate.
13

Fluid Concept
 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.
14

Definition of Fluid
Figure 1.1 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
kk
k
Free surface
15

Shear stress in moving fluid
• If fluid is in motion, shear stress are developed if the
particles of the fluid move relative to each other. Adjacent
particles have different velocities, causing the shape of the
fluid to become distorted
• On the other hand, the velocity of the fluid is the same at
every point, no shear stress will be produced, the fluid
particles are at rest relative to each other.
Moving plate Shear force
Fluid particles New particle position
Fixed surface
16

Differences between liquid and gases
Liquid Gases
Difficult to compress and often
regarded as incompressible
Easily to compress – changes of volume is
large, cannot normally be neglected and
are related to temperature
Occupies a fixed volume and will
take the shape of the container
No fixed volume, it changes volume to
expand to fill the containing vessels
A free surface is formed if the
volume of container is greater than
the liquid.
Completely fill the vessel so that no free
surface is formed.
Unlike a liquid, a gas does
not form a free surface, and
it expands to fill the entire
available space.
17

Example:
Air
Water
Oil
Gasoline
Alcohol
Kerosene
Benzene
Glycerine
Fluid Newton’s law
of viscosity
Newtonian fluids
obey refer
Newton’s’ law of viscosity is given by;
dy
du
 (1.1)
• 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 .
 = shear stress
 = viscosity of fluid
du/dy = shear rate, rate of strain or velocity gradient
Newtonian and Non-Newtonian Fluid
18

Fluid Newton’s law
of viscosity
Non- Newtonian
fluids
Do not obey
•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
19

Shear stress vs.
velocity gradient
Bingham plastic : resist a small shear stress but flow easily under large shear
stresses, e.g. sewage sludge, toothpaste, and jellies.
Pseudo plastic : most non-Newtonian fluids fall under this group. Viscosity
decreases with increasing velocity gradient, e.g. colloidal
substances like clay, milk, and cement.
Dilatants : viscosity decreases with increasing velocity gradient, e.g.
quicksand.
20

Units and Dimensions
• The primary quantities which are also referred to as basic
dimensions, such as L for length, T for time, M for mass and
Q for temperature.
• This dimension system is known as the MLT system where it
can be used to provide qualitative description for secondary
quantities, or derived dimensions, such as area (L), velocity
(LT-1) and density (ML-3).
• In some countries, the FLT system is also used, where the
quantity F stands for force.
21

1.2 Units and Dimensions
• An example is a kinematic equation for the velocity V of a
uniformly accelerated body,
V = V0 + at
where V0 is the initial velocity, a the acceleration and t the
time interval. In terms for dimensions of the equation, we
can expand that
LT-1 = LT -1 + LT-2 • T
22

Example
 The free vibration of a particle can be simulated by the
following differential equation:
where m is mass, u is velocity, t is time and x is
displacement. Determine the dimension for the stiffness
variable k.
0 kx
dt
du
m
23

Example
 By making the dimension of the first term equal to the
second term:
[m] • = [k]•[x]
Hence,
[k] = =
= MT-2
[ u ]
[ t ]
[ m ] • [ u ]
[ t ] • [ x ]
M • LT-1
LT
24

Primary Units
In fluid mechanics we are generally only interested in the top four units from this
table.
1.2 Engineering Units
Quantity SI Unit
Length Metre, m
Mass Kilogram, kg
Time Seconds, s
Temperature Kelvin, K
Current Ampere, A
Luminosity Candela
25

Derived Units
Quantity SI Unit
velocity m/s -
acceleration m/s2 -
force Newton (N) N = kg.m/s2
energy (or work) Joule (J) J = N.m = kg.m2/s2
power Watt (W) W = N.m/s = kg.m2/s3
pressure (or stress) Pascal (P) P = N/m2 = kg/m/s2
density kg/m3 -
specific weight N/m3 = kg/m2/s2 N/m3 = kg/m2/s2
relative density a ratio (no units) dimensionless
viscosity N.s/m2 N.s/m2 = kg/m/s
surface tension N/m N/m = kg/s2
26

Unit Cancellation Procedure
1. Solve the equation algebraically for the desired terms.
2. Decide on the proper units of the result.
3. Substitute known values, including units.
4. Cancel units that appear in both the numerator and
denominator of any term.
5. Use correct conversion factors to eliminate unwanted units
and obtain the proper units as described in Step 2.
6. Perform the calculations.
27

Example
Given m = 80 kg and a=10 m/s2. Find the force
Solution
 F = ma
 F = 80 kg x 10 m/s2 = 800 kg.m/s2
 F= 800N
28

Fluid Properties
Density
Density of a fluid, ,
Definition: mass per unit volume,
• slightly affected by changes in temperature and pressure.
 = mass/volume = m/ (1.2)
Units: kg/m3
Typical values:
Water = 1000 kg/m3; Air = 1.23 kg/m3
29

Fluid Properties (Continue)
Specific weight
Specific weight of a fluid, 
• Definition: weight of the fluid per unit volume
• Arising from the existence of a gravitational force
• The relationship  and g can be found using the following:
Since  = m/
therefore  = g (1.3)
Units: N/m3
Typical values:
Water = 9814 N/m3; Air = 12.07 N/m3
30

Specific gravity
The specific gravity (or relative density) can be defined in two ways:
Definition 1: A ratio of the density of a substance to the density
of water at standard temperature (4C) and
atmospheric pressure, or
Definition 2: A ratio of the specific weight of a substance to the
specific weight of water at standard temperature
(4C) and atmospheric pressure.
(1.4)
Unit: dimensionless.
Cw
s
Cw
s
SG
 





44 @@
31

Example
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
mkg
m
volume
mass
oil 


3
oil m/N882981.9x900g
mg
volume
weight



9.0
998
900
@

STPw
oil
oilSG


32

Viscosity
• Viscosity, , is the property of a fluid, due to cohesion and
interaction between molecules, which offers resistance to shear
deformation.
• 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.
Units: N.s/m2 or kg/m/s
Typical values:
Water = 1.14x10-3 kg/m/s; Air = 1.78x10-5 kg/m/s
33

Kinematic viscosity, 
Definition: is the ratio of the viscosity to the density;
• will be found to be important in cases in which significant
viscous and gravitational forces exist.
Units: m2/s
Typical values:
Water = 1.14x10-6 m2/s; Air = 1.46x10-5 m2/s;
In general,
viscosity of liquids with temperature, whereas
viscosity of gases with in temperature.
 /
34

Bulk Modulus
 All fluids are compressible under the application of an external
force and when the force is removed they expand back to their
original volume.
 The compressibility of a fluid is expressed by its bulk modulus of
elasticity, K, which describes the variation of volume with change
of pressure, i.e.
 Thus, if the pressure intensity of a volume of fluid, , is increased
by Δp and the volume is changed by Δ, then
Typical values:Water = 2.05x109 N/m2; Oil = 1.62x109 N/m2
strainvolumetric
pressureinchange
K 



/
p
K



p
K
35

Vapor Pressure
 A liquid in a closed container is subjected to a partial
vapor pressure in the space above the liquid due to the
escaping molecules from the surface;
 It reaches a stage of equilibrium when this pressure
reaches saturated vapor pressure.
 Since this depends upon molecular activity, which is a
function of temperature, the vapor pressure of a fluid
also depends on its temperature and increases with it.
 If the pressure above a liquid reaches the vapor pressure
of the liquid, boiling occurs; for example if the pressure
is reduced sufficiently boiling may occur at room
temperature.
36

Engineering significance of vapor pressure
 In a closed hydraulic system, Ex. in pipelines or pumps, water vaporizes
rapidly in regions where the pressure drops below the vapor pressure.
 There will be local boiling and a cloud of vapor bubbles will form.
 This phenomenon is known as cavitations, and can cause serious
problems, since the flow of fluid can sweep this cloud of bubbles on
into an area of higher pressure where the bubbles will collapse
suddenly.
 If this should occur in contact with a solid surface, very serious
damage can result due to the very large force with which the liquid hits
the surface.
 Cavitations can affect the performance of hydraulic machinery such as
pumps, turbines and propellers, and the impact of collapsing bubbles
can cause local erosion of metal surface.
 Cavitations in a closed hydraulic system can be avoided by
maintaining the pressure above the vapor pressure everywhere in the
system.
37

Surface Tension
 Liquids possess the properties of cohesion and adhesion due to molecular
attraction.
 Due to the property of cohesion, liquids can resist small tensile forces at
the interface between the liquid and air, known as surface tension, .
 Surface tension is defined as force per unit length, and its unit is N/m.
 The reason for the existence of this force arises from intermolecular
attraction. In the body of the liquid (Fig. 1.2a), a molecule is surrounded
by other molecules and intermolecular forces are symmetrical and in
equilibrium.
 At the surface of the liquid (Fig. 1.2b), a molecule has this force acting only
through 180.
 This imbalance forces means that the molecules at the surface tend to be
drawn together, and they act rather like a very thin membrane under
tension.
 This causes a slight deformation at the surface of the liquid (the meniscus
effect).
Figure 1.2: Surface Tension
38

 A steel needle floating on water, the spherical shape of
dewdrops, and the rise or fall of liquid in capillary tubes is
the results of the surface tension.
 Surface tension is usually very small compared with other
forces in fluid flows (e.g. surface tension for water at 20C is
0.0728 N/m).
 Surface tension,, increases the pressure within a droplet of
liquid. The internal pressure, P, balancing the surface
tensional force of a spherical droplet of radius r, is given by
r
2
P

 (1.7)
2R = pR2
39

Capillarity
• The surface tension leads to the phenomenon known as capillarity
• where a column of liquid in a tube is supported in the absence of
an externally applied pressure.
• Rise or fall of a liquid in a capillary tube is caused by surface
tension and depends on the relative magnitude of cohesion of the
liquid and the adhesion of the liquid to the walls of the containing
vessels.
• Liquid rise in tubes if they wet a surface (adhesion > cohesion),
such as water, and fall in tubes that do not wet (cohesion >
adhesion), such as mercury.
• Capillarity is important when using tubes smaller than 10 mm (3/8
in.).
• For tube larger than 12 mm (1/2 in.) capillarity effects are
negligible.
40

Figure 1.3
Capillary actions
r
cos2
h


 (1.8)
where h = height of capillary rise (or depression)
 = surface tension
 = wetting (contact) angle
 = specific weight of liquid
r = radius of tube
41

Water has a surface tension of 0.4 N/m. In a 3-mm diameter
vertical tube, if the liquid rises 6 mm above the liquid outside the
tube, calculate the wetting angle.
Solution
Capillary rise due to surface tension is given by;
r
cos2
h



 = 83.7
4.0x2
006.0x0015.0x9810
2
rh
cos 



Example
42

Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering and
Technical Studies

Momentum Equation
Applications momentum equation:
1. Force due to the flow of fluid round a pipe bend.
2. Force on a nozzle at the outlet of a pipe.
3. Impact of a jet on a plane surface.

Equations in Fluid Mechanics
Commonly used equations in fluid
mechanics - Bernoulli, conservation of
energy, conservation of mass, pressure,
Navier-Stokes, ideal gas law, Euler equations,
Laplace equations, Darcy-Weisbach Equation
and more

The Bernoulli Equation
The Bernoulli Equation - A statement of the conservation of energy in a form
useful for solving problems involving fluids. For a non-viscous, incompressible
fluid in steady flow, the sum of pressure, potential and kinetic energies per
unit volume is constant at any point.
Conservation laws
The conservation laws states that particular measurable properties of an
isolated physical system does not change as the system evolves.
Conservation of energy (including mass)
Fluid Mechanics and Conservation of Mass - The law of conservation of mass
states that mass can neither be created or destroyed.
The Continuity Equation - The Continuity Equation is a statement that mass is
conserved.
Darcy-Weisbach Equation
Pressure Loss and Head Loss due to Friction in Ducts and Tubes - Major loss -
head loss or pressure loss - due to friction in pipes and ducts.
as time-averaged values.

Euler Equations
In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid
fluid. They correspond to the Navier-Stokes equations with zero viscosity, although
they are usually written in the form shown here because this emphasizes the fact
that they directly represent conservation of mass, momentum, and energy.
Laplace's Equation
The Laplace Equations describes the behavior of gravitational, electric, and fluid
potentials.
Ideal Gas Law
The Ideal Gas Law - For a perfect or ideal gas the change in density is directly related
to the change in temperature and pressure as expressed in the Ideal Gas Law.
Properties of Gas Mixtures - Special care must be taken for gas mixtures when using
the ideal gas law, calculating the mass, the individual gas constant or the density.
The Individual and Universal Gas Constant - The Individual and Universal Gas
Constant is common in fluid mechanics and thermodynamics.
Navier-Stokes Equations
The motion of a non-turbulent, Newtonian fluid is governed by the Navier-Stokes
equations. The equation can be used to model turbulent flow, where the fluid
parameters are interpreted

Pressure calculation
Pressure due to the weight of a liquid of constant density is given by
p=ρgh, where p is the pressure, h is the depth of the liquid, ρ is the
density of the liquid, and g is the acceleration due to gravity.

Applications of Pascal’s Law:
1. Hydraulic Lift:
2. Hydraulic Jack

Manometer Pressure
Manometers measure a pressure difference by balancing the weight
of a fluid column between the two pressures of interest.
Large pressure differences are measured with heavy fluids, such as
mercury (e.g. 760 mm Hg = 1 atmosphere).

1) a.) What is the correct formula for absolute pressure?
a. Pabs = Patm – Pgauge
b. Pabs = Pvacuum – Patm
c. Pabs = Pvacuum + Patm
d. Pabs = Patm+ Pgauge
Ans . : d
b.) According to Archimede's principle, if a body is immersed partially or fully in a fluid
then the buoyancy force is _______ the weight of fluid displaced by the body.
a. equal to
b. less than
c. more than
d. unpredictable
Ans . : a
c.) The sum of components of shear forces in the direction of flow of fluid is called
as
a. shear drag
b. friction drag
c. skin drag
d. all of the above
Ans . : d

1. The gauge pressure in a water main is 50 kN/m2, what is the
pressure head ?
P=pgh
h=p/pg
Formula:
= 50X103 / 1000X9.81
= 5.1mAns.

2. Define Piezometer with the neat sketch
Ans: This is the simplest gauge. A small vertical tube is
connected to the pipe and its top is left open to the
atmosphere, as shown.
The pressure at A is equal to the
pressure due to the column of liquid
of height h1:
PA=pgh1
PB=pgh2

3. State the assumptions used in deriving Bernoulli’s equation
Ans:
 Flow is steady;
 Flow is laminar;
 Flow is ir-rotational;
 Flow is incompressible;
 Fluid is ideal.
4. List the instruments works on the basis of Bernoulli’s equation

5. Calculate the capillary rise in a glass tube of 2.5 mm diameter when immersed
vertically in (a) water and (b) mercury. Take surface tensions σ = 𝟎. 𝟎𝟕𝟐𝟓 N/m
for water and σ = 𝟎. 𝟓𝟐 N/m for mercury in content with air. The specific
gravity for mercury is given as 13.6 and angle of content =1300
AU Nov / Dec, 2016

6. The dynamic viscosity of an oil used for lubrication between a shaft and sleeve is 6 poise.
The shaft is of diameter 0.4 m and rotates at 190 r.p.m. Calculate the power lost in the
bearing for a sleeve length of 90 mm. The thickness of the oil film is 1.5 mm.
AU Nov / Dec, 2016

OBJECTIVE QUESTION WITH ANSWERS
Ans: a
Ans: d

Ans: a
Ans: cd) less than that of a real fluid

FILL IN THE BLANK QUESTIONS WITH ANSWERS
1. The value of the viscosity of an ideal fluid is ----------------
2. The value of the surface tension of an ideal fluid is ---------
3. Dimension of mass density -----------------
4. Dimension of specific gravity of a liquid --------------------
5. Dimension of specific volume of a liquid ----------------------
6. Which one of the following is the dimension of specific
weight of a liquid --------------
zero
zero
[M1 L-3 T0].
[M0 L0 T0].
[M-1 L3 T0].
[M L-2 T-2].

7. Two fluids 1 and 2 have mass densities of p1 and p2 respectively. If
p1 > p2, which one of the following expressions will represent the
relation between their specific volumes v1 and v2 :
8. A beaker is filled with a liquid up to the mark of one litre and
weighed. The weight of the liquid is found to be 6.5 N. The specific
weight of the liquid will be :
gravity of the liquid will be :
volume of the liquid will be :
V1 < v2
6.5 kN/m3
0.66
1.5 lit/kg

Derive the Bernoulli’s equation with the basic
assumptions
• Euler's equation of motion:
– Statement:
• In an ideal incompressible fluid, when the flow is steady and
continuous, sum of the velocity head, pressure head and
datum head along a stream line is constant.
• Assumptions:
– The fluid is ideal and incompressible.
– Flow is steady and continuous.
– Flow is along streamline and it is 1-D.
– The velocity is uniform across the section and is equal to
the mean velocity.
– Flow is irrotational.

The only forces acting on the fluid are gravity and the pressure
forces. Diagram:

Applies to all points on the streamline and thus provides a useful relationship between p,
the magnitude V of the velocity, and the height z above datum. Eqn. B is known as the
Bernoulli equation and the Bernoulli constant H is also termed the total head
Bernoulli’s equation for real fluid:
Bernoulli‟s equation earlier derived was based on the assumption that fluid is
non viscous andtherefore frictionless. Practically, all fluids are real (and not ideal) and
therefore are viscous andas such always some losses in fluid flow. These losses have,
therefore, to be taken intoconsideration in the application of Bernoulli‟s equation which
gets modified (between sections 1& 2) for real fluids as follows:

Fluid Mechanics Unit-1 (vk-ssm)

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