Handout_part_1.pdf

FLUID MECHANICS MC/EL/RN/ES/GL/MN/MR/PE/PG/RP/NG 264
INSTRUCTORS: E. ADAZE/ DR. A. A. KWARTENG 1
CHAPTER 1
BASIC CONCEPTS AND FLUID PROPERTIES
1.1 Why Study Fluid Mechanics?
Fluid mechanics is highly relevant to our daily life. We live in the world full of
fluids. Fluid mechanics covers many areas such as meteorology,
oceanography, aerodynamics, biomechanics, hydraulics, mechanical
engineering, civil engineering, naval architecture engineering, etc. It does
not only explain scientific phenomena but also leads industrial applications.
To fully understand the importance of fluid mechanics, consider the
following examples:
➢ We are often challenged with problems involving water. Design of
aqueducts to transport water from one place to another is possible
through the knowledge of fluid mechanics. Removal of waste water
from towns for the purpose of ensuring cleanliness can be achieved
through the study of fluid mechanics. Finally, engineers developed
water treatment technologies to get rid of waterborne diseases to
remove other forms of hazards.
➢ To come up with other means of transport, the Wright brothers applied
the knowledge from engineering fluid mechanics to develop the
world’s first airplane (flying machine). Later (1940s), engineers
designed practical jet engines to make this means of transport more
possible.
➢ By way of ensuring that homes get access to electrical power,
engineers developed technologies including the water turbine, the
wind turbine, the electric generator, the motor and the electric grid
system.
➢ Most of our farming activities demand that we keep supplying water
to keep them alive. For instance, engineers designed the irrigation
system to supply to our vegetables and other plants of interest.
It can be seen from above that with knowledge in fluid mechanics,
engineers can solve problems and come up with innovative ideas that can
lead to development or improvement of technology.

1.2 What then is Fluid Mechanics?
It is the science that deals with the action of forces on fluids either at rest
(fluid statics) or in motion (fluid dynamics) and their effects on boundaries
such as solid surfaces or interfaces with other fluids.
The subject of fluid mechanics can be subdivided into two (2) broad
categories:
➢ Hydrodynamics
➢ Gas dynamics
Hydrodynamics deals with the flow of fluids for which there is virtually no
density change. Examples include flow of liquids and flow of gases at low
speeds. Gas dynamics deals with fluid that undergo significant density
changes. Examples are high-speed flows of a gas through a nozzles and the
movement of a body through the low density air of the upper atmosphere.
Another area of fluid mechanics is aerodynamics which deals with the flow
of air past aircraft or rockets.
1.3 What is a Fluid?
A fluid is defined as a substance that will continuously deform – that is, flow
under the action of a shear stress (no matter how small that stress may be)
causing its constituent particles to continuously change their positions
relative to one another. The rate of deformation (strain) of a fluid is related
to the applied shear stress by a property called viscosity.
A fluid can be either a gas or a liquid. The differences in behaviour of solids,
liquids and gases are due to the differences in their molecular structures.
In solids, the molecules have definite spacing. Their movement is restricted.
As a result, solids have definite volume and shape. In liquids, the spacing
between the molecules is essentially constant but the molecules can move
with respect to each other when a shearing force is applied. Therefore,
liquids have definite volume but no definite shape. Finally, in gases, the
spacing between the molecules is much wider than that of either solids or
liquids. The spacing is also variable (keeps changing). Therefore, gases have
neither definite shape nor definite volume. Table 1.1 gives a summary on the
comparison of solids, liquids and gases.

Table 1.1 Differences between Solids, Liquids and Gases
From Table 1.1, it can be seen that a liquid is difficult to compress because
the molecules will repulsive forces if they are brought close together. It
changes its shape according to the shape of its container with an upper
free surface. However, a gas is easy to compress because there are no
forces (on average) between the molecules and expands to fill its container.
There is thus no free surface.
Consequently, an important characteristic of a fluid from the viewpoint of
fluid mechanics is its compressibility. Another characteristic is its velocity.
Whereas a solid shows its elasticity in tension, compression or shear stress, a
fluid does so only for compression. In other words, a fluid increases its
pressure against compression, trying to retain its original volume. This
characteristic is called compressibility. Furthermore, a fluid shows resistance
whenever two layers slide over each other. The characteristic is called
viscosity.
In general, liquids are called incompressible fluids and gases compressible
fluids. Nevertheless, for liquids, compressibility must be taken into account

whenever they are highly pressurized, and for gases, compressibility may be
disregarded whenever the changes in pressure is small. Although a fluid is
an aggregate of molecules in constant motion, it is treated as a continuous
isotropic substance.
Meanwhile, a non-existent, assumed fluid without either viscosity or
compressibility is called an ideal fluid or perfect fluid. A fluid with
compressibility but without viscosity is occasionally discriminated and called
a perfect fluid, too. Furthermore, a gas subject to Boyle’s-Charles’ law is
called a perfect or ideal gas.
1.4 The Ideal Gas Law (IGL)
Application of the ideal gas law (IGL) is very common in fluid mechanics. For
instance, in the design of products like air bags, shock absorbers,
combustion systems and aircraft. The IGL is the result of combining three
empirical equations previously developed by Boyle, Charles and Avogadro.
These empirical equations are respectively known as Boyles’s law, Charles
law and Avogadro’s law. The equation of state of IGL is of the form:
RT
p 
= (1.1)
where, p is absolute pressure; T is absolute temperature;  is fluid density; R
is gas constant which is related to the universal gas constant Ru as follows:
M
R
R u
= (1.2)
where, M is molar mass (molecular weight). The universal gas constant (Ru)
is usually taken as 8.314 kJ/kmolK.
1.5 Pascal’s Law
For a given closed system, a pressure change produced at one point in the
system will be transmitted throughout the entire system. This is known as
Pascal’s law. See Figure 1.1 for illustration.
Figure 1.1 Concept of Pascal’s Law

Pressures applied to point 1 and point 2 are equal in terms of Pascal’s law.
Therefore, applying force F1 to the smaller area at point 1 will produce a
larger force F2 on the larger area at point 2. Application of this is found in the
design of devices such as hydraulic brakes, hydraulic jacks and hydraulic
lifts.
1.6 Approaches to Study Fluid Mechanics
1.6.1 Analytical Method
Using advanced mathematics, we can solve governing equations of fluid
motions and obtain specific solutions for various flow problems. For example,
pipe flows.
1.6.2 Experimental Fluid Mechanics
This approach utilizes facility to measure considered flow fields or uses
various visualization methods to visualize flow pattern. For example, Laser
Doppler Anemometer (LDA), hot wire, wind-tunnel test.
1.6.3 Computational Fluid Dynamics (CFD)
For most of flow problems, we cannot obtain an analytical solution. Hence,
we can adopt numerical methods to solve governing equations. The results
are the so-called numerical solutions. On the other hand, costs of
experiments become very expensive. Numerical solutions provide an
alternative approach to observe flow fields without building up a real flow
field. For example, finite difference method (FDM), finite volume method
(FVM), finite element method (FEM), etc.
1.7 Fluid as a Continuum
The concept of a continuum is the basis of classical fluid mechanics.
Because a body of fluid is comprised of molecules, properties are due to
average molecular behavior. That is, a fluid usually behaves as if it were
comprised of continuous matter that is infinitely divisible into smaller and
smaller parts. This concept is called the continuum assumption.
When the continuum assumption is valid, engineers can apply limit concepts
from differential calculus. A limit concept typically involves letting a length,
an area, or a volume approach zero. Because of the continuum
assumption, fluid properties can be considered continuous functions of
position with a value at each point in space. Thus, fluid properties such as
density, temperature, velocity and so on, are considered to be continuous

functions of position and time. To gain insight into the validity of the
continuum assumption, consider a hypothetical experiment to find density.
Fig. 1.2a shows a container of gas in which a volume V
 has been identified.

Figure 1.2 Variation of a physical property with respect to the size of a
continuum
(Density is used as an example)
The idea is to obtain the mass of the molecules m
 inside the volume and
then to calculate density by using:
V
m


=
 (1.3)
The calculated density is plotted in Figure 1.2b. When the measuring volume
V
 is very small (approaching zero), the number of molecules in the volume
will vary with time because of the random nature of molecular motion. That
is, the density will vary as shown by the wiggles in the line. As volume
increases, the variations in calculated density will decrease until the
calculated density is independent of the measuring volume. This condition
corresponds to the vertical line at 1
V
 . If the volume is too large, as shown
by 2
V
 , then the value of density may change due to spatial variations. The
definition of density is provided in subsection 1.9.1.
The continuum assumption is valid in treating the behaviour of fluids under
normal conditions. However, it breaks down whenever the mean free path
of the molecules becomes the same order of magnitude as the smallest
significant characteristic dimension of the problem. In a problem such as
rarefied gas flow (for example, as encountered in flights into the upper
reaches of the atmosphere), we must discard the concept of a continuum
in favour of microscopic and statistical points of view.
There exists a non-dimensional number which is utilized to judge whether
fluids are continuous or not. Its definition is:
L
l
Kn = (1.4)
where, l is the free mean path of a fluid molecule and L is the smallest
characteristic length of a flow field. Kn is the so-called Knudsen number.

1.8 Dimensions and Units
Every fluid has certain characteristics by which its physical condition may be
described. These are known as the properties of the fluid. Properties are
expressed in terms of a number of basic dimensions (length, mass or force,
time temperature) which are quantified by basic units. Using units and
dimensions save a lot of time since errors are easily identified and corrected.
A unit is therefore a quantity selected as a standard against which an
amount can be described. That is, units allow quantification.
Two main systems of units exist. These are the SI (Le Système International d’
Unités) d’Unites) system or international system of units and the traditional
(English or British) units. The basic units of mass, length, time and temperature
are the kilogram (kg), meter (m), and kelvin (K) respectively.
A dimension is an entity that is measured using units. Dimension can be
classified as:
➢ Fundamental, primary or basic dimension with their primary or basic
units
➢ Secondary or derived dimension with the secondary or derived units
Shown in Table 1.2 are the seven fundamental or basic dimensions and their
fundamental SI units. Table 1.3 shows some of the dimensions in the SI and
British units
Table 1.2 Primary Dimensions and Units
Table 1.3 Some Dimensions and Units in SI and British Systems

Common prefixes that usually comes with the SI system of units are
presented in Table 1.4.
Table 1.4 Decimal Relationship between Units: SI System
1.8.1 Dimensional Homogeneity
An equation is dimensionally homogenous if each term in the equation has
the same dimensions or units. The method for verifying an equation for
dimensional homogeneity is to obtain the dimensions or units on each term
and then check to see if each term has the same dimensions or units.
1.9 System and Properties of Fluids
As explained earlier in section 1.8, properties of a fluid are those
characteristics that describe the physical condition of the fluid. Fluid
properties can either be extensive or intensive. Extensive properties are
properties related to the total mass of a system. Examples include mass,
weight, etc. Intensive properties on the other hand, are those properties that
are independent of the amount of fluid. Examples include pressure, density,

etc. A system as used in the definition here, may be defined as a given
quantity of matter under study. The mass of a system is constant. In this
section some of the fluid properties (including those involving the flow of
heat) will be discussed.
1.9.1 Mass Density, 
This is defined as the mass per unit volume of a substance. The density of
some fluids is more easily changed than that of others with temperature. It is
mathematically expressed as:
V
m
volume
mass
=
=
 (1.5)
Water is considered to be incompressible because the density change is
very small over a large pressure change. It has an SI unit of kg/m3.
1.9.2 Specific Weight, 
The weight per unit volume of a substance is called the specific weight. It is
given mathematically as:
g

 = (1.6)
g is the acceleration due to gravity. It is often taken as 9.81 m/s2. The SI unit
of  is N/m3.
1.9.3 Specific Gravity, S
The specific gravity of a fluid is the ratio of the specific weight of the fluid to
the specific weight of water at 4oC. It is a dimensionless quantity. It is
expressed as:
C
water
fluid
C
water
fluid
o
o
S
4
@
4
@




=
= (1.7)
1.9.4 Specific Volume, 
It is defined as the volume per unit mass of a substance. In other words, it is
the reciprocal of density. It is an intrinsic property of matter. It is expressed in
equation (1.8).


1
=
=
m
V
(1.8)
It has an SI unit of m3/kg.

1.9.5 Specific Heat, c
This is the amount of thermal energy that must be transferred to a unit mass
of a substance to raise its temperature by one degree. The specific heat
depends on the process accompanying the change in temperature. For
constant specific volume process, it is represented by cv. It is given as cp for
a constant pressure process. However, the ratio cp/cv often represented by
k.
1.9.6 Specific Internal Energy, u
This property is defined as the energy that a substance possesses due to the
state of the molecular activity in the substance. It is actually the internal
energy per unit mass. The SI unit is kJ/kg. In general, it is a function of
temperature and pressure, thus, f(T, p) but for an ideal gas, it is a function of
only temperature (i.e. f(T)).
1.9.7 Specific Enthalpy, h
This refers to the total energy in a system due to pressure and temperature
per unit mass in that system. See equation (1.9). Often encountered in
equations of thermodynamics and compressible flow.


p
u
p
u
h +
=
+
= (1.9)
It has the same unit as specific internal energy, u, thus, kJ/kg. Similar to
specific internal energy, it is generally a function of temperature and
pressure. i.e. f(T, p). It is a function of temperature only for an ideal gas. i.e.
f(T).
1.9.8 Compressibility or Elasticity of Fluids
When the pressure on a given mass of fluid increases (pressurized), the fluid
contracts (i.e. the total volume is reduced), resulting in an increase in
density. The reverse occurs when the pressure decreases. The amount of
volume change is the compressibility of fluids. In fluid mechanics, we use
bulk modulus of elasticity ( v
E ) to describe the compressibility of a fluid. It is
given mathematically as:


d
dp
V
d
dp
V
Ev =
−
= (1.10)
where, V , dp and V
d are the volume and, the incremental changes in
pressure and volume respectively. A high bulk modulus means that fluids are
not easy to compress. Hence, fluids with a high bulk modulus are

incompressible. Units and dimensions of bulk modulus are the same as
pressure (i.e. N/m2 or Pa). For most of liquids, they have very large bulk
moduli. It means liquids are incompressible. For instance, water has
approximately, Ev = 2.2 GN/m2. For changes of pressure of 1 MN/m2, the
volume of water changes by 0.05%. Therefore, water can be considered to
be incompressible. For most of gases, they are regarded as compressible
fluids due to their small bulk moduli.
1.9.9 Speed of Sound
When disturbances are introduced into a fluid, they are propagated at a
finite velocity. The velocity depends on the compressibility of the considered
fluids. It is called the acoustic velocity or the speed of sound, C. It is defined
as:


v
E
d
dp
C =
= (1.11)
For ideal gas,
( ) RT
d
RT
d
C =
=


(1.12)
1.9.10 Constant Density Assumption
In general, liquids have high value of bulk modulus and are commonly
assumed to be incompressible (density is independent of pressure). High
speed flow of gas such as flow around a jet airplane is compressible. The
criteria for constant density gas flow and variable density gas flow is
achieved through the use of Mach number, M. Mach number is the ratio of
the speed of the flowing fluid, V to the speed of sound, C in that fluid.
C
V
M =
)
(
number
Mach (1.13)
The criteria for considering the gas as a constant density is when M < 0.3.
1.9.11 Surface Tension and Capillarity
The molecules at the surface of a liquid have a greater attraction for each
other than they do for molecules below the surface. The surface behaves as
if it were a “skin” or “membrane” stretched over the fluid mass.
Due to the membrane effect, each portion of the surface exerts “tension”
on adjacent portions of the surface or on objects that are in contact with
the liquid surface. The tension acts in the plane of the surface. The

magnitude of the tension per unit length is defined as the surface tension,
.
 For water-air surface, 073
.
0
=
 N/m. Generally, surface tension can be
calculated from equation (1.14).






=
m
N
interface
the
of
Length
interface
an
along
Force
)
(
Tension
Surface  (1.14)
This implies that the surface tension force )
( 
F = L
 where L is the length over
which the surface tension acts. Shown in the Figure 1.3 is a plot of surface
tension of water against temperature for water/air interface.
Figure 1.3 Surface Tension Variation with Temperature for Water/Air
Interface
The concept of surface tension has been used to explain several commonly
observed phenomena. Examples include:
➢ A steel needle will float on water if placed gently on the surface
because the surface tension force supports the needle.
➢ Capillary action in a small-diameter tube.
➢ The excess pressure created inside droplets and bubbles.
Whenever a fine tube is pushed through the free surface of a liquid, the
liquid rises up (capillary rise) or falls (capillary repulsion) in the tube as shown
in Figure 1.4 owing to the relation between the surface tension and the
adhesive force between the liquid and the solid wall. This phenomenon is
called capillarity. Capillary action is a term used to describe the tendency
of a liquid to rise in narrow tubes or to be drawn into small openings. Water
will rise up a glass tube while mercury will move downward a glass tube.

Figure 1.4 Change of Liquid surface due to Capillary
As shown in Figure 1.5, d is the diameter of the tube,  is the contact angle
of the liquid to the wall,  is the density of the liquid, and ∆h is the mean
height of the liquid surface.
Figure 1.5 Capillarity
An equation for the capillary rise can derived owing to the balance
between the adhesive force of liquid stuck to the wall, trying to pull the liquid
up the tube by the surface tension, and the weight of liquid in the tube as
follows:





 
cos
4
4
force
tension
surface
=
ight
we
2
2
d
h
d
F
h
d
g
=









=









(1.15)
Assuming the contact angle is nearly zero. Thus, 1
cos 
 will yield the
following for the capillary rise:
d
h


4
=
 (1.16)
The attractive force between similar material is called cohesion and the
attractive force between dissimilar materials (e.g. water and glass) is called
adhesion. Water will “wet out” on a surface when adhesion is greater than
cohesion. Such a surface is called hydrophilic (water loving) as shown in
Figure 1.6 ( o
90

 ). Water will bead up on a surface when adhesion is less
than cohesion of the water. Such a surface is called hydrophobic (water
hating) as shown in Figure 1.7 ( o
90

 ).

Figure 1.6 Hydrophilic Surface ( o
90

 )
Figure 1.7 Hydrophobic Surface
1.9.12 Vapour Pressure
The pressure at which a liquid will boil is called its vapour pressure. This
pressure is a function of temperature (vapour pressure increases with
temperature). For instance, water boils at 100oC at sea-level atmospheric
pressure (1 atm abs). The vapour pressure of water at 10oC is 0.01 atm. If the
pressure within water at that temperature is reduced to that value, the water
boils. In flowing fluids, it is possible to have very low pressure due to the fluid
motion. The liquid will boil if the pressure is lowered to the vapour pressure.
Such boiling may occur in flow through the irregular, narrowed passages of
a valve or on the suction side of a pump or in a pipe narrowness. When such
boiling does occur in the flowing liquids, vapour bubbles start growing in
local regions of very low pressure and then collapse in regions of high
downstream pressure. This phenomenon (i.e. the formation and subsequent
collapse of vapour bubbles) is known as cavitation. It can lead to structural
damage and, therefore, should be avoided.
1.9.13 Viscosity, 
The ease with which a fluid can flow is an indication of its viscosity. It is that
property of a fluid, which by virtue of cohesion and intereaction between
the fluid molecules offers resistance to shear deformation. Fluids with high
viscosity (e.g. syrup) deform relatively more slowly than low viscosity fluids
(e.g. water).

In a viscous fluid, the shear stress is proportional to the time strain.
Mathematically, it is expressed as:
dy
dV

 = (1.17)
where,  is the shear stress, V is the fluid velocity and y is the distance
measured from the wall, dV/dy is the rate of strain (which is also called the
velocity gradient normal to the wall). The proportionality factor,  , is the
dynamic or absolute viscosity. It the property of the fluid. The viscosity may
be defined as the ratio of the shear stress to the velocity gradient, i.e.








=
dy
dV

 (1.18)
The SI unit of viscosity is N-s/m2 or Pa-s. Other unit is centipois (cp) which is
0.01 pois. Note that: 1 cp = 0.001 Pa-s.
Consider the velocity distribution next to a boundary as shown in Figure 1.8.
Figure 1.8 Velocity Distribution next to a Boundary
The velocity gradient,
dy
dV also known as the rate of strain, becomes smaller
with distance from the boundary. Therefore, the maximum shear stress is at
the boundary. The fluid velocity is zero at the stationary boundary. Viscosity
causes the fluid to adhere to the surface. This leads to the concept of no-
slip condition.
The no-slip can simply be explained as follows: when a fluid is contact with
a solid body, the velocity of the fluid at the point of contact is the same as
the velocity of the solid body at the same point. For instance, when water
flows in a pipe, the fluid velocity at the wall is equal the velocity of the wall,
which is zero.

1.9.14 Kinematic Viscosity, 
Another definition of viscosity is the kinematic viscosity which is 

 = : Its unit
is m2/s.
1.9.15 Variation of Viscosity with Temperature
The viscosity of a gas increases with the temperature of the gas whereas
that of a liquid decreases with increasing temperature as shown in Figure
1.9.
Figure 1.9 Viscosity Variation with Temperature
The variation of gas viscosity with absolute temperature can be obtained
using Sutherland’s equation given as:






+
+









=
S
T
S
T
T
T o
o
o
2
3


(1.19)
where o
 is the viscosity at temperature To and S is Sutherland’s constant
which can be obtained from tables.
The variation of liquid viscosity with absolute temperature can be obtained
using the equation:
T
b
Ce
=
 (1.20)
where, C and b are constants.
1.10 Newtonian and Non-Newtonian Fluids
Fluids are deformed continuously because fluids cannot support shear
stresses. Fluids for which the shear stress is directly proportional to the rate of
strain are called Newtonian fluids. Most common fluids are Newtonian.
Examples are air, water and oil. Examples of non-Newtonian fluids are
toothpaste, paints and printer’s ink. See Figure 1.10 for illustration. Again, the
relationship between shear stress acting on a Newtonian fluid and rate of

strain (or velocity gradient) is linear and passes through the origin. If it is not
linear, then the fluid will be called a non-Newtonian fluid.
Figure 1.10 Newtonian versus Non-Newtonian Fluids
For another illustration, consider a fluid between two parallel plates, which is
subjected to a shear stress ,
 due to the impulsive motion of the upper plate
as shown in Figure 1.11a.
Figure 1.11 Fluid between Two Parallel Plates
The upper plate moves with a velocity of U while the lower plate is stationary
(velocity is zero). There is no relative motion between the fluid and the
boundary (no-slip condition). That is, fluid in contact with the lower plate is
stationary while fluid in contact with the upper plate moves with a velocity
of U. The fluid deforms, that is, undergoes rate of strain  due to shear stress
 as shown in Figure 1.11b. For a Newtonian fluid:
(a) (b)






=
=
=
viscosity
of
t
coefficien
strain
of
rate









(1.21)
In other words,
dy
du

 (1.22)
Thus,
dy
du

 = (1.23)
This implies,
dy
du
=

 (1.24)
Note: This course focuses on the theory and applications of Newtonian fluids
only.

Problems:
1.1 Calculate the density and specific weight of nitrogen at an absolute
pressure of 1 MPa and a temperature of 40oC.
1.2 Two parallel glass plates
separated by 0.5 mm are
placed in water at 20oC as
shown in Figure P1.2. The
plates are clean, and the
width/separation ratio is large
so that end effects are
negligible. How far will the
water rise between the plates? Figure P1.2
1.3 The dynamic viscosity of air at 15oC is 1.78 × 10-5 N.s/m2. Using
Sutherland’s equation, find the viscosity at 200oC.
1.4 Two plates are separated by 1/8-in space. The lower plate is
stationary; the upper plate moves at a velocity of 25 ft/s. Oil (SAE 10W-
30, 150oF), which fills the space between the plates, has the same
velocity as the plates at the surface of contact. The variation of
velocity of the oil is linear. What is the shear stress in the oil?
1.5 A laminar flow occurs between two horizontal parallel plates under a
pressure gradient dp/ds (p decreases in the positive s direction). The
upper plate moves left (negative) at velocity ut. The expression for the
local velocity u is given as:
( ) H
y
u
y
Hy
ds
dp
u t
+
−
−
= 2
2
1

Is the magnitude of the shear stress greater at the moving plate (y =
H) or at the stationary plate (y = 0)?
1.6 For the conditions of Problem 1.5, derive an expression for the y
position of zero shear stress.
1.7 A water bug is suspended on
the surface of a pond by
surface tension (water does
not wet the legs) as shown in
Figure P1.7. The bug has six
legs, and each leg is in
contact with the water over a
length of 3 mm. What is the
maximum mass (in grams) of
the bug if it is to avoid sinking?
Figure P1.7

CHAPTER 2
FLUID STATICS
2.1 Introduction
Fluid statics is the study of fluids in which there is no relative motion between
fluid particles, i.e. no velocity gradient in the fluid. Therefore, no shearing
forces exist. Only normal forces exist. These normal forces in the fluid are
called pressure forces. For an element of fluid at rest, the element will be in
equilibrium; the sum of the components of the forces in any direction and
the sum of moments of the forces on the element taken about any point
must be zero. Before going into details of Fluid Statics, let us familiarize
ourselves with some terminologies.
2.2 Terminologies
2.2.1 Pressure
Pressure results from a normal compressive force acting on an area.
Mathematically, it is defined as:
dA
dF
A
F
p
A
=


=
→
 0
lim (2.1)
where, F is the normal force acting over the area A. Pressure is a scalar
quantity; it has magnitude only and acts equally in all directions. By
considering a small element of fluid in the form of a triangular prism (as
shown in Figure 2.1) which contains a point P, we can establish a relationship
between the three pressures px in the x-direction, py in the y-direction and
pn in the direction normal to the sloping face.
Figure 2.1 Differential element of fluid

Summing forces in the x-direction gives:
( ) ( ) 0
sin
sin =


−

 
 l
y
p
l
y
p x
n (2.2)
Upon simplification,
x
n p
p = (2.3)
Also summing forces in the z-direction:
( ) ( ) 0
sin
cos
2
1
cos
cos =



−


−


− y
l
l
l
y
p
l
y
p z
n 



 (2.4)
Simplifying equation (2.4) gives,
z
n p
p = (2.5)
That is to say,
z
x
n p
p
p =
= (2.5)
Hence, pressure acts equally in all direction on the fluid element.
2.2.2 Pressure Transmission (Pascal’s Law)
In a closed system (i.e. system with fixed mass of fluid), a pressure change
produced at one point in the system will be transmitted equally throughout
the entire system as shown in Figure 2.2.
Figure 2.2 Illustration of Pascal’s law
This principle is known as Pascal’s law and applies to fluid at rest. The Pascal’s
principle is applied in the development of devices like hydraulic brakes,
hydraulic jacks and hydraulic lifts.

2.2.3 Absolute Pressure, Gauge Pressure and Vacuum
A space that is completely evacuated of all gases is called a vacuum. The
pressure in a vacuum is called absolute zero and all pressures referenced
with respect to this zero pressure are termed absolute pressures. The
atmospheric pressure at sea level is 101.3 kPa measured from the absolute
zero. See Figure 2.3. When pressure is measured relative to the local
atmospheric pressure, the pressure reading is called gauge pressure.
Mathematically, these pressures are related as follows:
gauge
atm
absolute p
p
p +
= (2.6)
Figure 2.3 Illustration of absolute, gauge and vacuum pressures
When the absolute pressure is less than atmospheric pressure, the gauge
pressure is negative. Negative gage pressures are also termed vacuum
pressures, e.g. a gauge pressure of -31.0 kPa can be stated as a vacuum
pressure of 31.0 kPa.
The unit of pressure is the Pascal (Pa) in SI system and pounds per square
inch (psi) in the traditional system. Gauge and absolute pressures are usually
identified after the unit, e.g. 50 kPa gauge; 150 kPa absolute; 8 psig; 22 psia.
Example 2.1
A hydraulic jack has the dimensions
shown in Figure 2.4. If one exerts a
force F of 100 N on the handle of the
jack, what load, F2, can the jack

support? Neglect the weight of the
lifer.
Figure 2.4 Hydraulic jack example
2.3 Pressure Variation with Elevation
2.3.1 Basic Differential Equation
For a static fluid, pressure varies only with elevation within the fluid. To prove
this, consider the cylindrical element of fluid shown in Figure 2.5.
Figure 2.5 Proof of pressure variation with elevation
0
=
 l
F
( ) 0
sin =


−


+
−
 
 l
A
A
p
p
A
p (2.7)
Simplifying gives:

 sin
−
=


l
p
(2.8)
Note that
l
z


=

sin (2.9)

When the element approaches zero
dl
dz
l
z
dl
dp
l
p
=


=


and (2.10)
Combining equations (2.8) and (2.9) taking note of equation (2.10), the
following is obtained:
dl
dz
dl
dp

−
= (2.11)
Hence,

−
=
dz
dp
(2.12)
Equation (2.12) is the basic equation for hydrostatic pressure variation with
elevation. The following can be observed for a static fluid from equation
(2.12).
a. A change of pressure occurs only when there is a change of
elevation. Therefore, pressure is constant everywhere in a horizontal
plane.
b. Pressure changes inversely with elevation.
2.3.2 Pressure Variation for a Uniform-Density Fluid
For constant density (incompressible), the specific weight ( )
 is also
constant. This would make it easier to integrate equation (2.12) to obtain:
constant
=
+ z
p  (piezometric pressure) (2.13)
or
constant
=
+ z
p

(piezometric head) (2.14)
Using equation s (2.13) and (2.14), one can relate the pressure and elevation
at two points in a fluid in the following manner:
2
2
1
1
z
p
z
p
+
=
+


(2.15)
Note:

The equation applies to two points in the same fluid; it does not apply across
an interface of two fluids having different specific weights.
Example 2.2
Water occupies the bottom 1.0 m of a cylindrical tank. On top of the water
is 0.5 m of kerosene, which is open to the atmosphere. If the temperature is
20oC, what is the gauge pressure at the bottom of the tank?
2.3.3 Pressure Variation for Compressible Fluids
When the specific weight varies (variable density) significantly throughout
the fluid, it must be expressed in such a form that equation (2.12) can be
integrated. For an ideal gas, the equation of state is given as:
RT
p 
= (2.16)
This implies:
RT
p
=
 (2.16a)
Multiplying both sides of equation (2.16a) by g, we have:
RT
pg
g =
 (2.16b)
Thus,
RT
pg
=
 (2.16c)
Substituting equation (2.16c) into equation (2.12) gives:
RT
pg
dz
dp
−
= (2.17)
Equation (2.17) is the relation of the pressure variation with elevation in terms
of the pressure and temperature. If we have a relation for temperature with
elevation (altitude) or the pressure with elevation, then it can be substituted
then integrated.
Pressure Variation in the Atmosphere
The atmosphere (about 1000 km thick) is divided into five layers but only two
(layers near the earth surface), namely, troposphere (between sea level and
13.7 km) and stratosphere, will be considered here.

To obtain pressure variation in the
atmosphere, a relation for
temperature variation with altitude
as shown in Figure 2.6, is substituted
into equation (2.17), which is then
integrated.
The temperature variation in the
troposphere can be approximated
(linearly) as:
( )
o
o z
z
T
T −
−
=  (2.18)
To is the temperature at the
reference level zo where the pressure
is known and  is the lapse rate
(slope which is constant). Figure 2.6 Temperature variation
with altitude in the atmosphere
Substituting equation (2.18) into equation (2.17) and integrating for the
pressure p (taking note that at the reference level zo, the pressure is known
as po), the atmosphere pressure variation in the troposphere is given by:
R
g
o
o
o
T
z
z
T
p
p


/
0 )
(
(





 −
−
= (2.19)
where po is the pressure at the reference level zo.
In the lower part of the stratosphere (13.7 to 16.8 km), the temperature is
approximately constant. With that in mind, integrating equation (2.17) for p,
the atmospheric pressure variation in the stratosphere is therefore given by:
RT
g
z
z
o
o
e
p
p /
)
( −
−
= (2.20)
where po is the pressure at the reference level zo. The reference level is
usually the outer edge of the troposphere.
Assignment 2.1
Show that the pressure variation in the troposphere is:
R
g
o
o
o
T
z
z
T
p
p


/
0 )
(
(





 −
−
=
and that in the lower stratosphere as:

RT
g
z
z
o
o
e
p
p /
)
( −
−
=
2.4 Pressure Measurements
During most engineering designs and experimentations, pressure
measurements become very critical. One way to measure pressure is to use
manometry. This is a method which utilizes the change in pressure with
elevation to evaluate pressure. Below are some of the scientific instruments
that can be used for pressure measurements.
2.4.1 Barometer: Atmospheric
Pressure
This device (as shown in Figure 2.7)
is used to measure atmospheric
pressure.
The mercury barometer is
analysed by applying the
hydrostatic equation:
h
p
p Hg
v
atm 
+
= (2.21)
Figure 2.7 A mercury barometer
where, pv is the vapour pressure of mercury which is very small: 6
10
4
.
2 −

=
v
p
atm at 20oC. Usually, pv is neglected and hence,
h
p Hg
atm 
 (2.22)
2.4.2 Piezometer (Simple
Manometer)
A vertical tube (transparent) in which
a liquid rises in response to a positive
gauge pressure. See Figure 2.8.
The gauge pressure at center of
the pipe is computed as:
p
h =
+ 
0 (2.23)
Hence,
h
p 
= (2.24)
Figure 2.8 A piezometer attached
to a pipe

Note that the piezometer is not suitable for measuring large pressures. It is
also not useful for pressure measurement in gases.
2.4.3 The U-tube Manometer
It can be used to measure pressure
as shown in the Figure 2.9. To
evaluate the unknown pressure, the
procedure is to calculate the
pressure changes, step by step, from
one level to the next. The
manometer equation for Figure 2.9 is:
4
1 p
l
h
p m =
−

+ 
 (2.25)
But 0
1 =
p gauge since it is open to
the atmosphere. Therefore,
l
h
p m 
 −

=
4 (2.26)
Since m
 is higher than ,
 h
 will be relatively small. The general manometer
equation between two points 1 and 2 is given as:

 −
+
=
up
down
1
2 i
i
i
i h
h
p
p 
 (2.27)
2.4.4 The Differential Manometer
It is used to measure the difference in pressure between two points in a pipe
as shown in Figure 2.10. The procedure for obtaining the pressure difference
is the same as used for U-tube manometer.
(a) (b)
Figure 2.10 Differential manometers connected to pipes
Note that the fluid is flowing in the pipe and not in the tube from 1 to 2. At
the beginning of the connection, the fluid fills the U-tube and deflect the U-

Figure 2.9 U-tube manometer

tube manometer fluid (manometer fluid could be mercury or any other fluid)
and then stops.
Now, applying the usual technique for calculating the pressure difference
between points 1 and 2 in Figure 2.10a, we start from point 1 and go through
till we reach point 2.
( ) 2
1 p
l
h
h
l
p m =
−

−

+
+ 

 (2.28)
Then,
( ) h
p
p m 
+
=
− 

2
1 (2.29)
The pressure difference is equal to the difference between the specific
weight of the two fluids multiplied by the manometric deflection (Δh).
Repeating this technique for Figure 2.10b, calculating the pressure
difference between point 1 and 2, we have:
( ) ( ) 2
1
2
1 p
z
z
y
h
h
y
p A
B
A =
−
+

−

−

+

+ 

 (2.30)
Then,
( ) ( ) ( )
A
B
A
A h
z
p
z
p 


 −

=
+
−
+ 2
2
1
1 (2.31)
Dividing through by A
 gives:








−

=








+
−








+ 1
2
2
1
1
A
B
A
A
h
z
p
z
p




(2.32)
Thus,








−

=
− 1
2
1
A
B
h
h
h


(2.33)
2.4.5 Bourdon-Tube Gauge
A Bourdon-tube gage consists of a tube that is bent into a circular arc.
Pressure is applied through one end of the tube. The other end carries the
pointer as shown in Figure 2.11. When pressure is applied to the gage, the
curved tube tends to straighten and the pointer deflects proportionately to
read the pressure.

Figure 2.11 Bourdon-tube gauge; (a) View of a typical gauge; (b) Internal
mechanism
This gauge is widely used for steam and compressed gases. The pressure
indicated is the difference between that applied by the system to the
external (atmospheric) pressure or gauge pressure.
2.4.6 Pressure Transducer
A device that converts pressure to an electrical signal. It is designed to
produce electrical signals that can be transmitted to oscillographs or digital
devices and/or to control other devices for process operations. A schematic
diagram is shown in Figure 2.12.
Figure 2.12 Schematic diagram of a pressure transducer
2.5 Hydrostatic Pressure Distribution and Forces on Plance Surfaces
A hydrostatic pressure distribution, as shown in Figure 2.13, describes the
distribution of pressure when pressure varies only with elevation, z, according

to 
−
=
dz
dp . When hydrostatic pressure exists, any panel that is not
horizontal is subjected to a hydrostatic pressure distribution. Discussed below
are force distribution for various surfaces.
2.5.1 Horizontal Surfaces
If a plane surface immersed in a fluid is horizontal, as shown in Figure 2.14,
then:
a. Hydrostatic pressure is uniform (constant) over the entire surface.
b. The resultant force acts at the centroid of the plane.
Figure 2.13 Pressure distribution on a horizontal submerged plane surface
The magnitude of the resultant force will the pressure multiplied by the
surface area. The location of this force will be at the centroid of the plane
and its line of action will be normal to the area.
2.5.2 Inclined Surfaces
However, in the case where the plane is not horizontal (i.e. inclined plane),
as presented in Figure 2.14, the hydrostatic pressure, p, is a function of the
vertical distance, z, and not constant as in the case of horizontal planes. It is
linearly distributed over the surface.
So in the case of a vertical plane or in general, in the case of an inclined
plane, the magnitude and location of the resultant force is not straight
forward. In the following section, we will learn how to calculate the
magnitude and location of the resultant force.
Figure 2.14 Pressure distribution on inclined surfaces
F

Magnitude of the Resultant Force
Consider the plane surface AB immersed in a liquid and inclined at angle
to the liquid surface as shown in Figure 2.15.
Figure 2.15 Hydrostatic pressure distribution on an inclined plane
From Figure 2.15, y is the slanted distance from the liquid surface to the
centroid of the area (plane); y is the distance from the liquid surface to the
differential area; and dA is the differential area.
The pressure from the static fluid, p, on the differential element, dA, is given
as:

 sin


= y
p (2.34)
But, the differential force, dF, on the small differential area, dA, is given by:
pdA
dF = (2.35)
Substituting equation (2.34) into equation (2.35) gives:
dA
y
dF 


= 
 sin (2.36)
For the total force on the whole plane, equation (2.36) is integrated over the
entire area as follows:

 


=
=
A
dA
y
pdA
F 
 sin (2.37)

Since and 
sin are constants, equation (2.37) becomes:
 
=
A
dA
y
F 
 sin (2.38)
But the integration  
A
dA
y represents the first moment of area and it is given
as .
A
y Thus,
A
y
dA
y
A
=

 (2.39)
This implies,
A
y
F sin

= (2.40)
Note that y
sin is the vertical distance from the surface of the liquid to the
centroid of the plane. Then equation (2.40) can be written as:
A
p
F = (2.41)
where, p is the pressure at the surface centroid given as:
h
p 
= (2.42)
where, h is the vertical distance from the surface of the liquid to the centroid
of the plane. It can therefore be concluded that the magnitude of the
resultant hydrostatic force on a plane surface is the product of the pressure
at the centroid of the surface and the area of the surface as shown in
equation (2.41).
Vertical Location of the Line of Action of the Resultant Force
Location of the resultant hydrostatic force lies below the centroid of the
plane (due to pressure increase with depth) as shown in Figure 2.15. The
point where the resultant force acts on the surface is called the center of
pressure (CP). To find the line of action of the resultant hydrostatic force, we
take the sum of moment about o-o and equating to zero. Thus,

= ydF
F
ycp (2.43)
That is, moment of the resultant force is equal to the integration of moments
of all differential forces acting on the differential element. But
dA
y
pdA
dF 
 sin


=
= (2.44)
This implies,

 =


= dA
y
dA
y
F
ycp
2
2
sin
sin 


 (2.45)

But  dA
y2
is the second moment of the area (Io) or area moment of inertia
about o which is given as:
A
y
I
Io
2
+
= (2.46)
where, I is the second moment of area about the centroid.
Upon substitution and rearranging, we have:
y
A
y
I
ycp +
= (2.47)
Note that cp
y is located below y by a distance of
A
y
I
. Values of I for various
surfaces are provided in Figure 2.16.
Figure 2.16 Values of I for different surfaces
2.6 Hydrostatic Forces on Curved Surfaces

For a curved surface, the pressure forces, being normal to the local area
element, vary in direction along the surface (see Figure 2.17a) and thus
cannot be added numerically. The resultant hydrostatic force is computed
by considering the free-body diagram of a body of fluid in contact with the
curved surface, as illustrated in Figure 2.17b.
Figure 2.17 (a) Pressure distribution and equivalent force; (b) Free-body
diagram and action-reaction force pair
The steps involved in calculating the horizontal and vertical components of
the hydrostatic force F are as follows:
1. Summation of forces in the horizontal direction gives
AC
x F
F = (2.48)
where FAC is the hydrostatic force on plane surface AC. It acts through the
center of pressure of side AC.
2. Summation of forces in the vertical direction gives
CB
y F
W
F +
= (2.49)
where W is the weight of the fluid (acting through the center of gravity) of
the free-body diagram and FCB is the hydrostatic force (acting through the
centroid) on the surface CB.
The line of action of Fy is obtained by summing moments about any
convenient axis. The hydrostatic force on the curved surface is equal and
opposite to the force F on the free-body diagram (Figure 2.17b).

Example 2.2
Determine the force acting on one side of a concrete form 2.44 m high and
1.22 m wide (8 ft by 4 ft) that is used for pouring a basement wall. The specific
weight of concrete is 23.6 kN/m3 (150 lbf/ft3).
Example 2.3
An elliptical gate covers the end of
a pipe 4 m in diameter as shown in
Figure 2.18. If the gate is hinged at
the top, what normal force F is
required to open the gate when
water is 8 m deep above the top
of the pipe and the pipe is open to
the atmosphere on the other side?
Neglect the weight of the gate. Figure 2.18 Elliptical gate problem
Example 2.4
Find the force of the gate on the
block as shown in Figure 2.19,
where the vertical distance from
the free surface of the water to the
pivot (d) = 10 m. The gate is 4 m ×
4m in dimension.
Figure 2.19 Block problem
Example 2.5
Surface AB is a circular arc with a radius of 2 m and a width of 1 m into the
paper. The distance EB is 4 m. The ﬂuid above surface AB is water, and
atmospheric pressure prevails on the free surface of the water and on the
bottom side of surface AB. Find the magnitude and line of action of the
hydrostatic force acting on surface AB.

Figure 2.20 Circular arc problem

Problems
2.1 As shown in Figure P2.1, a mouse
can use the mechanical
advantage provided by a
hydraulic machine to lift up an
elephant.
a. Derive an algebraic equation
that gives the mechanical
advantage of the hydraulic
machine shown. Assume the
pistons are frictionless and
massless.
b. A mouse can have a mass of
25 g and an elephant a mass
of 7500 kg. Determine a value
of D1 and D2 so that the mouse
can support the elephant.
Figure P2.1
2.2 For the closed tank shown in
Figure P2.2 with Bourdon-tube
gauges tapped into it, what is the
specific gravity of the oil and the
pressure reading on the gauge C.
Figure P2.2
2.3 A tank is fitted with a manometer
on the side, as shown in Figure
P2.3. The liquid in the bottom of
the tank and in the manometer
has a specific gravity (S.G) of 3.0.
The depth of this bottom liquid is
20 cm. A 15 cm layer of water lies
on top of the bottom liquid. Find
the position of the liquid surface in
the manometer.
Figure P2.3
2.4 What is the maximum gauge
pressure in the odd tank shown in
Figure P2.4? Where will the
maximum pressure occur? What is
the hydrostatic force acting on
the top (CD) of the last chamber
on the right-hand side of the
tank? Assume T=10°C.

Figure P2.4
2.5 What is the pressure at the center of pipe B as shown in Figure P2.5?
Figure P2.5
2.6 The ratio of container diameter to tube diameter is 8. When the air in the
container is at atmospheric pressure, the free surface in the tube is at position
1 as shown in Figure P2.6. When the container is pressurized, the liquid in the
tube moves 40 cm up the tube from position 1 to position 2. What is the
container pressure that causes this deflection? The liquid density is 1200 kg/m3.
Figure P2.6
2.7 Find the pressure at the center of pipe A. T=10oC. See Figure P2.7.
Figure P2.7
2.8 Determine (a) the difference in pressure and (b) the difference in piezometric
head between points A and B. The elevations zA and zB are 10 m and 11 m,
respectively, l1 = 1 m, and the manometer deflection l2 is 50 cm. See Figure P2.8.

Figure P2.8
2.9 The gate shown in Figure P2.9 is rectangular and has dimensions height h = 6 m
by width b = 4 m. the hinge is d = 3 m below the water surface. What is the
force at point A?
Figure P2.9
2.10 For the gate shown in Figure P2.10, α = 45o, y1 = 1 m, and y2 = 4 m. Will the
gate fall or stay in position under the action of the hydrostatic and gravity
forces if the gate itself weighs 150 kN and is 1.0 m wide? Assume T = 10oC. Use
calculations to justify your answer.
Figure P2.10
2.11 A dock gate is to be reinforced with three horizontal beams. If the water acts
on one side only, to a depth of 6 m, find the positions of the beams measured
from the water surface so that each will carry an equal load. Give the load
per meter.
2.12 The profile of a masonry dam is an arc of a circle, the arc having a radius of
30 m and subtending an angle of 60 at the center of curvature which lies in
the water surface. Determine (a) the load on the dam in N/m length; and (b)
the position of the line of action to this pressure.

2.13 The arch of a bridge over a stream is in the form of a semi-circle of radius 2 m.
The bridge width is 4 m. Due to a flood, the water level is now 1.25 m above
the crest of the arch. Calculate (a) the upward force on the underside of the
arch; and (b) the horizontal thrust on one half of the arch.
2.14 A circular lamina 110 cm in diameter is immersed in water so that the distance
of its edge measured vertically below the free surface is varies from 50 cm to
140 cm. Find the total force due to the water acting on one side of the lamina,
and the vertical distance of the centre of pressure below the surface.

CHAPTER 3
FLOWING FLUIDS AND PRESSURE VARIATION (DYNAMICS I)
3.1 Basic Concepts and Terminologies
In order to visualize and describe the features associated with flowing fluids,
engineers and scientists use terms like streamline, streakline, and pathline.
These topics are therefore introduced in the subsequent sections.
3.1.1 Flow Pattern
This refers to the construction of lines in the flow field to indicated the
direction of flow. These lines are known as streamlines. The flow pattern
presents the visualization of the flow field.
3.1.2 Streamlines
These are lines drawn through the flow field in such a manner that the
velocity vector of the fluid at each and every point on the line is tangent to
the line at that instant. In other words, a streamline is a line that is tangent to
the local velocity vector everywhere in the flow field.
Consider a flow of water from a slot in the side of a tank, as shown in Figure
3.1(a). The velocity vectors have been sketched at three different locations. In
the case of flow around a body, part of the flow goes to one side and part to
the other as indicated by the flow over an airfoil shown in Figure 3.1(b).
Figure 3.1 Illustrations of streamlines: (a) Flow through an opening in a tank;
(b) Flow over and airfoil section
The streamline that follows the flow division is called the dividing streamline. At
the location where the dividing streamline intersects the body, the velocity is
zero with respect to the body and is known as the stagnation point.
(a) (b)

The following can be observed:
a. the streamlines describe the direction of the flow.
b. the streamlines form the flow pattern.
Since the velocity of the fluid, expressed in the form, ( )
t
s
V , , (where s is the
distance traveled by a fluid particle along a path and t is the time), is tangent
to the streamlines, it implies:
0
=
 s
V d (3.1)
Note the following facts about streamlines:
• Close to a solid boundary, streamlines are parallel to the boundary.
• Since the fluid is moving in the same direction as the streamlines, fluid
can not cross a streamline.
• Streamlines cannot cross each other. If they were to cross, this would
indicate two different velocities at the same point. This is not physically
possible. This implies that any particle of fluid starting on a streamline
will stay on that same streamline throughout the fluid.
3.1.3 Uniform and Non-Uniform Flow
For a uniform flow, the velocity does not change along a fluid path. In other
words, the velocity is constant in magnitude and direction along a
streamline at each instant in time. Thus,
0
=


s
V
(3.2)
In these flows, the streamlines are rectilinear (straight and parallel) as shown
in Figure 3.2.
Figure 3.2 Examples of Uniform Flows

In non-uniform flows, the velocity changes along a fluid path (streamline). In
other words, the velocity is not constant in magnitude, direction or both.
Thus,
0



s
V
(3.3)
Here, the flow has streamline curvature and as such tends to be non-uniform.
See Figure 3.3 for examples.
Figure 3.3 Examples of Non-Uniform Flows: (a) Converging Duct; (b) Vortex
Flow
3.1.4 Steady and Unsteady Flows
Flows can either be considered to be steady or unsteady. For a steady flow,
the velocity remains constant with time at every location in space. In other
words, the velocity at a given point on a fluid path does not change with
time in steady flows. Thus,
0
=


t
V
(3.4)
For an unsteady flow, the velocity keeps changing with time (at least at
some points). Thus,
0



s
V
(3.5)
3.1.5 Pathlines and Streaklines
Pathline refers to a line drawn through the flow field in such a way that it
defines the path that a given (actual) particle of fluid has taken. In other
words, a pathline is the path of a fluid particle as it moves through the flow
field. A pathline is formed by following the actual path of a fluid particle as
shown in Figure 3.4.

Figure 3.4 Fluid Particle Moving Along a Pathline
A streakline on the other hand, is the line generated by a tracer fluid, such
as dye, continuously injected in the flow field at the starting point. In other
words, the locations or traces of injected dye or smoke (not the actual
particle) at a given point in the flow field as it travels downstream is the
definition of a streakline. Streaklines are the most common flow pattern
generated in a physical experiment. See Figure 3.5.
Figure 3.5 Streaklines Produced by Coloured Fluid Introduced Upstream
Since the flow in Figure 3.5 is steady, the streaklines are the same as
streamlines and pathlines.
Note the following differences between pathlines, streaklines and
streamlines:
• In steady flows, all three lines are coincident (the same) if they start
from the same point.
• In unsteady flows, the pathline, streakline and streamline can be three
distinct lines.
• Pathlines and streaklines provide history of the flow field while
streamlines indicate the current flow pattern.
3.1.6 Streamtube
A streamtube is a term used to define a tubular region of fluid which
surrounded by streamlines as presented in Figure 3.6. It is usually adopted in

fluid flow analysis to consider only a part of the total fluid in isolation from the
rest.
Figure 3.6 Streamtube Representations
3.1.7 Laminar and Turbulent Flows
Laminar flow is well-ordered state of flow in which adjacent fluid layers move
smoothly with respect to each other. This type of flow has a very smooth
appearance with no mixing phenomena and eddies (swirling of fluid). A
typical example is the flow of honey or thick syrup from a pitcher. Laminar
flow is characterised by the following:
• Smooth appearance layer of flow
• Velocity distribution (profile) is parabolic (less uniform). See Figure 3.7a
• Velocity is constant with time at any given position (no fluctuation)
Figure 3.7 Velocity Distribution: (a) Laminar Flow; (b) Turbulent Flow
Turbulent flow is an unsteady flow characterised by a mixing action
throughout the flow field and this mixing is caused by eddies of varying sizes
within the flow. Examples of turbulent flow include the flow in the wake of a
ship, the transport of smoke from a smoke stack on a windy day, and so on.
See Figure 3.7b for the velocity profile of a turbulent flow.
(a) (b)

Since the flow is unsteady, the velocity at any point in the pipe fluctuates
with time. The standard way of analysing turbulent flow is to represent the
velocity as a time-averaged value plus a fluctuating quantity, u
u
u 
+
=
(where, u is the time-averaged value and uis the fluctuating quantity). A
turbulent flow is often designated as “steady” if the time-averaged velocity
is unchanging with time. Turbulent flows are characterised by the following:
• Full of irregularities, eddies and vortices (mixing flow). Flow is more
uniform little away from the wall.
• Fluctuating velocity. Velocity field is stochastic that is, the velocity
components are random variables described by their statistical
properties.
In general, laminar pipe flows are associated with low velocities and
turbulent flows with high velocities. Laminar flows can occur in small tubes,
highly viscous flows or flows with low velocities, but turbulent flows are, by
far, the most common. Table 3.1 compares laminar and turbulent flows.
Table 3.1 Comparison Between Laminar and Turbulent Flows
3.1.8 Real Fluid Flow and Ideal Fluid Flow
Fluid flow which involves friction effects (existence of viscosity) is known as
real fluid flow. It is called viscous fluid flow.
Ideal fluid flow on the other hand, is a hypothetical fluid flow. It assumes no
friction (no viscosity). It is sometimes referred to as inviscid flow.
3.1.9 One-Dimensional and Multi-Dimensional Flows
The dimensionality of a flow field is characterised by the number spatial
dimensions needed to describe the velocity field.

For one-dimensional (1D) flow, consider the velocity distribution for an
axisymmetric flow in a circular duct as shown in Figure 3.8.
Figure 3.8 One-Dimensional (1D) Flow
In 1D, the flow is uniform, or fully developed, so the velocity does not change
in the flow direction (z). The velocity depends on only one spatial dimension,
namely, the radius r. Hence the name, one-dimensional (1D) flow.
For a two-dimensional (2D) flow, the velocity depends on two dimensions (in
the case presented here, x and y). Consider the velocity distribution for a
uniform flow in a square duct as shown in Figure 3.9 for more illustration.
Figure 3.9 Two-Dimensional (2D) Flow
For a three-dimensional (3D) flow, the velocity depends of three (3)
dimensions, namely x, y and z. Consider the velocity distribution in a square
duct as shown in Figure 3.10. Note that the duct cross-sectional area is
expanding in the flow direction (z-direction). This implies that the velocity will
depend on the z direction as well as the x and y.
Figure 3.10 Three-Dimensional (3D) Flow
Another good representation of a 3D flow is turbulence, since the velocity
components at any one time depend on the three coordinate directions.

3.2 Lagrangian and Eulerian Viewpoints
There are two ways to express the equations for fluids in motion, namely,
Lagrangian and Eulerian.
3.2.1 Lagrangian Viewpoint
Under this viewpoint, the motion of a specific fluid particle is recorded for all
time (watch individual, single particle all the time, at all locations x, y, and z)
as shown in Figure 3.11.
Figure 3.11 Lagrangian Viewpoint
This is the familiar approach in dynamics. Using the Cartesian coordinate
system, the position vector (R) is expressed as:
R (t) = xi + yj + zk (3.6)
where, i, j and k are the unit vectors in the x, y and z direction respectively.
The velocity of the particle is then obtained by differentiating the position
vector of the particle with respect to time. Thus,
( ) ( ) k
dt
dz
j
dt
dy
i
dt
dx
dt
t
dR
t
V +
+
=
= (3.7)
which yields,
V (t) = ui + vj + wk (3.8)
where, u, v and w are the component velocities in the x, y and z direction
respectively.

Handout_part_1.pdf

Recommended

Recommended

More Related Content

Similar to Handout_part_1.pdf

Similar to Handout_part_1.pdf (20)

Recently uploaded

Recently uploaded (20)

Handout_part_1.pdf