FLUID MECHANICS
Aim of Course:



to offers basic knowledge in fluid
mechanics
to obtain an understanding for the
behaviour of fluids
to solve some simple problems of the
type encountered in Engineering
practice
Aim of fluid mechanics lectures: It is the aim
of these lectures to help students in this
process of gaining an understanding of, and
an appreciation for, fluid motion—what can
be done with it, what it might do to you,
how to analyze and predict it.
Objective of course
At the end of the course, participants are
expected to be able to:
 Define and use basic fluid properties
 Define and use basic concepts in fluid
mechanics


Perform simple calculations in hydrostatics
and kinematics
Make simple designs in hydraulics
METHODS TO BE USED



Lectures
Workshops (tutorials)
Laboratory works
Assessment methods
 Class assignments,
 Home assignments
 Laboratory reports
 Examination
Lectures and class assignments
Attendance to lectures is compulsory for
all students.
Class works (Tests) will be unannounced.
Students who take all class test also get the
full marks for attendance
Tutorials and Laboratory Works
Tutorials : 2hrs per week outside our usual
schedule.
Laboratory works:
1. Pressure gauges
2. Plane surfaces immersed in fluids
3. Floating bodies
Reports on each laboratory work will be written
by the group and defended at my office
Literature
1.
2.
3.
4.
5.
Fluid Mechanics (including Hydraulic Machines) – Dr. A.
K. Jain, Khanna Publishers, Delhi, 2003
Fluid Mechanics (6th edition) – frank M. White;
McGraw-Hill 2008
Introduction to Engineering Fluid Mechanics.- J. A. Fox
1985
Fluid Mechanics:- J. F. Douglas; J. M. Gasiorek; J. A.
Swaffield
Hydraulics,Fluid Mechanics and Fluid Machines – S.
Ramamrutham
Literature cont.
6. Essentials of Engineering Hydraulics – J. M. K. Dake,
1992
7. Hydrology and Hydraulic Systems – Ram S. Gupta
Mechanics of Fluids – Bernard Massey, revised by john
Ward- Smith
WHAT IS A FLUID?
Molecules of solids are so closely packed
together that the attractive forces between
the molecules are so large that a solid tends
to retain it’s shape unless compelled by
some external forces to change it. Fluids
are composed of molecules with relatively
larger distances between molecules and
therefore the attractive forces between
molecules are smaller than in solids.
WHAT IS A FLUID?
F
F
Shear τ
t1
t0
θ3
θ θ2
θ
Solid
Fluid
Shear τ
t3
t2
Definition of fluid
A fluid may be defined in two perspectives:a) The form in which it occurs naturally :- a
substance that is capable of flowing and has
no definite shape but rather assumes the
shape of the container in which it is placed.
DEFINITION OF FLUID
b) By the deformation characteristics when
acted upon by a shear stress:
A fluid is a substance that deforms
continuously under the action of a
shearing stress no matter how small the
stress. (Examples of fluid: gases (air, lpg),
liquids (water, kerosene, etc)
DISTINCTION BETWEEN
SOLID AND FLUID
There are plastic solids which flow under the
proper circumstances and even metals may
flow under high pressures. On the other hand
there are viscous fluids which do not flow
readily and one may easily confuse them with
solid plastics. The distinction is that any fluid,
no matter how viscous will yield in time to the
slightest shear stress.
SOLID AND FLUID
But a solid, no matter how plastic, requires a
certain limiting value of stress to be exerted
before it will flow. Also when the shape of a solid
is altered (without exceeding the plastic limit) by
external forces, the tangential stresses between
adjacent particles tend to restore the body to its
original shape. With a fluid, these tangential
stresses depend on the velocity of deformation
LIQUID AND GAS
A liquid is composed of relatively closed
packed molecules with strong cohesive forces.
Liquids are relatively incompressible. As a
result, a given mass of fluid will occupy a
definite volume of space if it is not subjected
to extensive external pressures.
GAS
Gas molecules are widely spaced with relatively small
cohesive forces. Therefore if a gas is placed into a
container and all external pressure removed, it will expand
until it fills the entire volume of the container. Gases are
readily compressible. A gas is in equilibrium only when it
is completely enclosed. The volume (or density) of a gas is
greatly affected by changes in pressure or temperature or
both. It is therefore necessary to take account of changes
of pressure and temperature whenever dealing with gases.
FLUID MECHANICS
Fluid mechanics is the science of the mechanics of
liquids and gases and is based on the same
fundamental principles that are employed in solid
mechanics. It studies the behaviour of fluids at
rest and in motion. The study takes into account
the various properties of the fluid and their effects
on the resulting flow patterns in addition to the
forces within the fluid and forces interacting
between the fluid and its boundaries
FLUID MECHANICS
The study also includes the mathematical
application of some fundamental laws :conservation of mass - energy, Newton’s law
of motion ( force - momentum equation ),
laws of thermodynamics, together with other
concepts and equations to explain observed
facts and to predict as yet unobserved facts
and to predict as yet unobserved fluid
behaviour.
FLUID MECHANICS
The study of fluid mechanics subdivides
into:
 fluid statistics

fluid kinematics and

fluid dynamics
Fluid Statics
Fluid statics : is the study of the
behaviour of fluids at rest. Since for a fluid
at rest there can be no shearing forces all
forces considered in fluid statics are normal
forces to the planes on which they act.
Fluid Kinematics
Fluid kinematics: deals with the geometry
(streamlines and velocities ) of motion
without consideration of the forces
causin g the motion. Kinematics is
concerned with a description of how
fluid bodies move.
Fluid dynamics
Fluid dynamics: is concerned with the
relations between velocities and
accelerations and the forces causing the
motion.
SYSTEM AND CONTROL
VOLUME
In the study of fluid mechanics, we make
use of the basic laws in physics namely:
 The conservation of matter (which is called
the continuity equation)
 Newton’s second law (momentum equation)
 Conservation of energy (1st law of
thermodynamics)
 Second law of thermodynamics and
 there are numerous subsidiary laws
In employing the basic and subsidiary laws, either
one of the following models of application is adopted:


The activities of each and every given mass must
be such as to satisfy the basic laws and the
pertinent subsidiary laws – SYSTEM
The activities of each and every volume in space
must be such that the basic and the pertinent
subsidiary laws are satisfied – CONTROL VOLUME
SYSTEM & CONTROL VOLUME
A system is a predetermined identifiable quantity of
fluid. It could be a particle or a collection of
particles. A system may change shape, position
and thermal conditions but must always contain
the same matter.
A control volume refers to a definite volume
designated in space usually with fixed shape. The
boundary of this volume is known as the control
surface. A control volume mode is useful in the
analysis of situations where flow occurs into and
out of a space
SYSTEM & CONTROL VOLUME
A system
Control volume
Control surface
FORCES ACTING ON FLUIDS
(BODY & SURFACE FORCES)
Those forces on a body whose distributions act on
matter without the requirement of direct contact
are called body forces (e.g. gravity, magnetic,
inertia, etc.
Body forces are given on the basis of the force per
unit mass of the material acted on.
Those forces on a body that arise from direct contact
of this body with other surrounding media are called
surface forces eg. pressure force, frictional force,
surface tension
FLUID PROPERTIES
Property :- is a characteristic of a substance
which is invariant when the substance is in a
particular state. In each state the condition
of the substance is unique and is described
by its properties. The properties of a fluid
system uniquely determine the state of the
system.
EXTENSIVE & INTENSIVE
PROPERTIES
Extensive Properties: those properties of the
substance whose measure depends on the
amount of the substance present (weight,
momentum, volume, energy)
Intensive Properties: those properties whose
measure is independent of the amount of
substance present (temperature, pressure,
viscosity, surface tension, mass density etc.
volume per unit mass v and energy per unit mass
e)
PHYSICAL PROPERTIES OF
FLUIDS
Each fluid property is important in a particular
field of application.
Viscosity plays an important role in the
problems of hydraulic friction.
Mass density is important in uniform flow.
Compressibility is a factor in water hammer.
Vapour pressure is a factor in high velocity flow
Mass density & unit
(specific) weight
Mass density and unit weight are the two
important parameters that tend to indicate
heaviness of a substance
 Mass density is the mass per unit volume
usually denoted by the Greek letter “rho”
ρ=M/V
kg/m3
At standard pressure (760 mmHg) and 4o C
density of water = 1000 kg/mm3
Specific Weight
Specific volume : Is the reciprocal of the density ie.
the volume occupied per unit mass of fluid.
Vs = 1/ρ = V/M ( m3 / kg)
Specific (unit ) weight:  (gamma) - Is the weight per
unit volume of the substance (is and indication of
how much a unit volume of a substance weighs.)
 = W/V = Mg/V =ρg ( kgm/s2)
FLUIDS PROPERTYSPECIFIC GRAVITY
Specific Gravity : Is the ratio of the weight of a
substance to the weight of an equal volume
of water at standard conditions.
FLUIDS PROPERTYVISCOSITY
Viscosity : is the property of a fluid to offer
resistance to shear stress. Fluids offer
resistance to a shearing force. Viscosity is a
property of a fluid that determines the
amount of resistance. Viscosities of liquids
vary inversely with temperature, while
viscosities of gases vary directly with
temperature
FLUIDS PROPERTYVISCOSITY
F
b
τ
θ
Y
c
b’
c’
y
u
a
d
U
FLUIDS PROPERTYVISCOSITY
At any point at a distance y from the
lower plate, the velocity
U(y) = Uo * (y/Y)
Uodt/Y =θ
(du/dy) = (Uo/Y)
(θ/dt)=Uo/Y
Experiments show that, other quantities being
held constant F is directly proportional to the
A (area) and the velocity U and inversely
proportional to the distance between the
plates Y
FLUIDS PROPERTYVISCOSITY
AUo
where  is the proportionality factor.
F
Y
The shear stress is defined as Γ (Tau)
du
F Uo
τ= 
; = 
dy
A Y
The shear stress at any point in the fluid ,
du
= 
dy
Dynamic & kinematic viscosity
The constant of proportionality, μ, in the above
equation is called the dynamic viscosity with
units Ns. /m2
Kinematic Viscosity : (nu) is the ratio of the
dynamic viscosity to the density of the fluid.
 =  /
Ns / m2 kgm-3 = m2 / s
NEWTONIAN &
NON-NEWTONIAN FLUIDS
Not all fluids show exactly the same relation
between stress and the rate of deformation.
Newtonian fluids: are fluids for which shear
stress is directly proportional to the rate
angular deformation or a fluid for which the
viscosity  is a constant for a fixed
temperature and pressure. eg. Air, water,
etc. Petroleum, kerosene, steam.
NEWTONIAN &
NON-NEWTONIAN FLUIDS
Non-Newtonian fluids : are fluids which
have a variable proportionality (viscosity  )
between stress and deformation rate. In
such cases, the proportionality may depend
on the length of time of exposure to stress
as well as the magnitude of the stress eg.
Plastics, paint, blood, ink, etc
COMPRESSIBLE AND
INCOMPRESSIBLE FLUIDS
Compressible fluids are fluids whose specific
volume v or (density, ρ) is a function of
pressure. An incompressible fluid is a fluid
whose density is not changed by external
forces acting on the fluid.
Hydrodynamics is the study of the behaviour of
incompressible fluids whereas gas dynamics
is the study of compressible fluid.
Compressibility of fluid
Compressibility of a fluid is a measure of the
change in volume of the fluid when it is
subjected to outside force. It is defined in
terms of an average bulk modulus of
elasticity K.
p
K 
V
V
SURFACE TENSION
Explain from molecular theory
These forces F tend to pull the surface
molecules tightly to the lower layer and
cause the surface to behave as though it
were a membrane. The magnitude of this
force per unit length is defined as surface
tension  (sigma).
Relative magnitude of
molecular surface pressure
Cohesive and adhesive forces
Cohesive and adhesive forces
1)
2)
If the intermolecular cohesive forces between two
molecules of the fluid is greater than the adhesive
forces between the molecules of the container and
the molecule of the fluid, - a convex meniscus is
obtained.
On the other hand if the adhesive force of
molecule of the container and fluid is greater than
the cohesive force of the fluid molecules, case (b)
- concave meniscus is obtained
CAPILLARITY
Is the rise or fall of a column of fluid (in a
narrow tube called capillary tube) inserted
in the fluid
In the contact area between the fluid and
container, we can have two cases ;
CAPILLARITY RISE
CAPILLARITY RISE OR FALL
The rise or fall in the capillary tube is given by:
4 cos 
h
d
Where h – capillary rise
σ – surface tension force per unit length
d – diameter; γ – weight density of fluid and
HYDROSTATICS
Hydrostatic deals with fluid at rest. Hydrostatics
studies the laws governing the behaviour of
fluid at equilibrium when it is subjected to
external and internal forces and bodies at
equilibrium when they are immersed in the fluid.
Shear stress in a fluid at rest is always zero.
Therefore in fluid at rest, the only stress we shall
be dealing with is normal stresses.
WHAT IS HYDROSTATIC
PRESSURE?
HYDROSTATIC PRESSURE
The basic concept of hydrostatics is the
concept of hydrostatic pressure. What is it?
lim it [ p ]
p
.
A0 A
pressure of a given point in a fluid or simply
hydrostatic pressure.
PROPERTIES OF
HYDROSTATIC PRESSURE
1). Hydrostatic pressure is a compressive
stress and always acts along the inside
normal to the element of area.
2). The hydrostatic pressure p at a given
point in a fluid does not depend on the
orientation of the surface i.e. on the incline
of the surface.
DIFFERENTIAL EQUATION
OF A FLUID AT REST
EQUATION OF A FLUID AT
REST CONT.Consider the equilibrium of an elemental
parallelepiped in a fluid.
Since it is in equilibrium, the projection of all
forces on the x, y, z axis should be equal to
zero i.e. Fx=0, Fy=0, Fz=0.
DIFF. EQUATION OF A
FLUID AT REST CONT.Projection of surface forces on the x-axis
Force on side ABCD
dFx=pdydx
Force on side A1B1C1D1
dF1=p1dydz
p
dx
x
p


dF '   p 
dx dy.dz
x


p'  p 
DIFF. EQUATION OF A
FLUID AT REST CONT.Projection of body forces on the x-axis.
The projection of body forces on the x-axis is
the product of the mass of fluid and the
projection of acceleration on the x-axis. i.e.
dRx=dxdydz.
where X is the projection of acceleration of
body forces in the x-axis
DIFF. EQUATION OF A
FLUID AT REST CONT.Applying Newton’s law in the x-axis
Fx=0---sum of surface and body forces in the x-axis
equals zero
p 

 Fx  pdydz   p  x dx dydz  dxdydzX  0
Dividing through by ρdxdydz, we shall obtain
1

1

dp  X
dp.dx  Xdx
DIFF. EQUATION OF A
FLUID AT REST CONT.By analogy, we can write similar equations in the y-axis
and z-axis Fy=0; Fz=0
1

1

1

p.dx  X .dx
p.dy  Y .dy
p.dz  .Z .dz
Adding left hand side and the right hand side;
1  p
p
z 

.dx 
dy 
dz   Xdx  Ydy  Zdz
  x
y
z 
DIFF. EQUATION OF A
FLUID AT REST CONT.Since hydrostatic pressure is a function of
independent coordinates x, y, z, then the first
three functions on the left side of the above
equation being the sum of three partial
differential equals the exact (total) differential.
1
dp  Xdx  Ydy  Zdz

Basic differential equation of hydrostatic
DIFF. EQUATION OF A
FLUID AT REST CONT.Since the left hand side of equation is an exact
(total) differential, then the right hand side
must also be an exact differential of a certain
function say U (x, y, z)
Xdx+Ydy+Zdz= U (x, y, z)
We can write the exact differential dU(x, y, z)
into partial differential
U
U
z
dU ( x, y, z ) 
dx 
dy  dz
x
y
z
DIFF. EQUATION OF A
FLUID AT REST CONT.Therefore;
U
U
U
Xdx  Ydy  Zdz 
dx 
dy 
dz
x
y
z
and we can write
1 p
U
X 
 x
x
1 p
U
Y 
 y
y
1 p
U
Z 
 z
z
DIFF. EQUATION OF A
FLUID AT REST CONT.Since U is a function of only coordinates (x, y, z)
and its partial differential gives the
corresponding projection of body forces per
unit mass (X, Y, Z) on the respective axes,
then the function U is a Potential Function.
Conclusion:
Fluid can be in a state of equilibrium (rest)
when and only when it is acted upon by
potential forces
Integrating the basic differential
equation of hydrostatics
The basic equation is:
1

Integrating;
p=ρU + C
dp  dU
where C is the constant of integration
To find C, we consider a point in a fluid with p and U known.
Assuming at this point when p=p0 when U=U0, then
po=U0+C
and therefore;
p = po +ρ(U-Uo)
General equation of hydrostatics in the integral form
HYDROSTATIC PRESSURE AT A POINT IN A FLUID
WHEN GRAVITY IS THE ONLY BODY FORCE
PRESSURE AT A POINT IN A FLUID
WHEN GRAVITY IS ONLY BODY FORCE
The basic differential equation is:
1
dp  Xdx  Ydy  Zdz

Since force of gravity is the only body force acting, we
shall have the following:
X=0; Y=0; Z=-g
1

and
dp   gdz
dp = -ρg.dz
PRESSURE AT A POINT IN A FLUID
WHEN GRAVITY IS ONLY BODY FORCE
Integrating the above equation, we have
p=-g.z + C
or
p = -γ.z +C
To find C let us consider a point at the surface of fluid. At
that point O, z=0; p=po
po=C
The above equation becomes:
p=-z + po
Now let h be the depth of immersion of the point M. h=-z
Therefore the above equation becomes:
p = po +γh
fundamental equation of hydrostatics
PRESSURE AT A POINT IN A FLUID
WHEN GRAVITY IS ONLY BODY FORCE
P ---- is known as the absolute hydrostatic
pressure at the point M
h --- is the body pressure i.e. pressure due
to the body of column of fluid above M.
Conclusion: the absolute pressure at a point
is the sum of the external surface pressure
and the body pressure (pressure created by
the column of fluid on point).
If the external pressure po is atmospheric, ie
container is opened, then po =pa.
p=pa + h
pa= atmospheric pressure or barometric pressure
p-pa=h-------Gauge or manometric pressure
Manometric (Gauge)
Pressure
Gauge pressure: is the differential (excess) pressure
above atmospheric pressure at a point in a fluid.
In practice we often use the manometric pressure
instead of the absolute pressure. So from now we
shall denote;
PA= absolute pressure
p = γh -- excess or manometric pressure
 pA=po + p
Where pA –absolute pressure; po – external
pressure and p – gauge pressure
PASCAL’S LAW:
HYDRAULIC PRESS
The pressure at a point in a fluid is given by:
p= po + h where po –external pressure
if the external pressure changes from po to
po1=po + po
The pressure at all point in the fluid at rest also
changes by the same value po.
It is therefore evident that liquid possesses the
property of total transmissibility of the external
pressure
Pascal’s law
Pascal’s Law states: pressure (external)
which arises (or which is applied) at the
surface of a liquid at rest is transmitted
throughout the liquid in all direction without
any change.
HYDRAULIC PRESS:
The distinctive characteristic of the hydraulic
press is its ability to produce great forces by
expending fairly small original forces.
The force F1 acts on the piston pump 8 of area
A1 causes it to travel downwards and to exert
pressure on the liquid surface below. This
pressure is
P=F1/A1
Pascal’s law
HYDRAULIC PRESS
From Pascal’s law, this pressure is transmitted
to the piston 5. The result is a useful force
F2 under whose action the material is
pressed.
F2 = p1.A2
Where A2 is the area of piston 5. Therefore
F2 = p1.A2 = (F1/A1).A2 = F1.D2/d2
F2
D2
 2
F1
d
PIEZOMETRIC HEIGHT
PIEZOMETRIC HEIGHT
Considering the point m, we can write the
following relationships;
a) The absolute pressure at the point m with
reference to the closed container
pAm = po + h.
b)The absolute pressure at the point m with
reference to the tube To
pAm = pOT + hA=hA
pAm = pa + hex=hA
PIEZOMETRIC HEIGHT
Piezometric head: is the pressure at a point
in a fluid measured as a column of fluid.
pA
hA 

hA –absolute piezometric head
hex—piezometric head (excess, gauge
pressure, differential, manometric head)
POTENTIAL ENERGY OF
FLUID AT REST
Liquid at rest or in motion possesses a certain amount
of energy i.e. possesses the ability to do a certain
amount of work. Liquid at rest possesses only
potential energy relative to a certain level (datum).
This potential energy is made of two energies:
1.
Energy by virtue of position, a fluid of weight G has
(P.E)Z = z x G relative to O----O
2.
Energy by virtue of pressure at that point, a fluid of
weight G has
(P.E.)p = hex x G
POTENTIAL ENERGY OF
FLUID AT REST
Total work that can be done by the liquid of
weight G located at n is:
P.E. = z x G + z x G = = (P.E.)z + (P.E.)p
P.E. is called the potential energy of the liquid of weight G
located at the point n
SPECIFIC POTENTIAL ENERGY
S.P.E: is defined as the potential energy per
unit weight of the fluid.
S.P.E. = P.E./G = {(z x G) + (hex x G )}/G = z + hex = H
Specific potential Energy is the sum of
i) specific potential energy by virtue of position
(z).
ii). Specific potential energy due to pressure
hex = p/γ
POTENTIAL HEAD
In fluid mechanics (or hydraulics) “head” is used to
denote specific potential energy; i.e. a measure of
energy per unit weight of the liquid.
Therefore the potential head, H can be written as
H = z + hex
Z – is called the geometric head
hex - is called the pressure (or piezometric) head
H = z + hex = z + p/γ
Home work: Show that in a fluid at rest, the value of the
potential head is the same at all points within the liquid.
VARIATION OF PRESSURE IN
THE EARTH’S ATMOSPHERE
Gases are highly compressible and are
characterized by changes in density. The
change in density is achieved by both change in
pressure and temperature. In the treatment of
gases, we shall consider the perfect gas. It
must be recognized that there is no such
thing as a perfect gas, however, air and
other real gases that are far removed from
the liquid phase may be so considered.
Equations of state for gases
The absolute pressure p, the specific volume v, and
the absolute temperature are related by the
equation of state. For a perfect gas, the equation of
state per unit weight is
pv = RT or p/ρ =RT or p = ρRT ----(1)
p

RT
pg
  g 
RT
-----------(2)
Equations of state for gases
Another fundamental equation for a perfect gas:
pvn =p1v1n = p2v2n = const --------(3)
where n may have any value from zero to infinity depending on
the process to which the gas is subjected.
By combining the above equations, the following
useful relationships can be obtained.
(T2/T1) = (v1/v2)n-1 = (p2/p1)(n-1)/n --------(4)
Isothermal Process.
The compression and expansion of a gas may
take place according to various laws of
thermodynamics
If the temperature is kept constant, the
process is called isothermal and the value of
n in eq. (3) is unity; i.e. n = 1.
Isentropic Process.
If a processes is such that there is no heat added
to or withdrawn from the gas (i.e. zero heat
transfer), it is said to be adiabatic process.
An isentropic process is an adiabatic process in
which there is no friction and hence is a
reversible process.
The value of the exponent, n in equation (3) is then
denoted by k which is the ratio of the specific heats
at constant pressure and constant volume. k=
cp/cv = 1.4.
PRESSURE VARIATION IN THE
ATMOSPHERE
The atmosphere may be considered as a static fluid
and as such can be subjected to the basic
differential equation when gravity is the only body
force acting.
dp/dz = -γ
To evaluate the pressure variation in a fluid at rest,
one must integrate the above equation.
For compressible fluids, however, γ must be
expressed algebraically as a function of z and p.
PRESSURE VARIATION IN
THE ATMOSPHERE
Let us illustrate some of the problems dealing with
pressure variation in the atmosphere. Let us compute
the atmospheric pressure at an elevation of H
considering the atmosphere as a static fluid. Assume
standard atmosphere at sea level. Use:
 air at constant density
 constant temperature between sea level and H
 Isentropic conditions
 Air temperature decreasing linearly with elevation
at standard lapse rate of X oC/m
PRESSURE VARIATION IN
THE ATMOSPHERE
Standard atmosphere: po = 760mmHg (101.3kPa; To = 15 oC
or 288oK; γo = 11.99N/m3; ρo =1.2232kg/m3; μo = 1.777 x
10-8 kN/m; zo = 0
Air at constant density
dp
 
dz
dp  dz
Integrating
p  z  C
To.. det er min e..C.., we..use..the..boundary..condition..that..when..z  0,.. p  po
po  C
p  po  z
p H  po  H
Air at constant temperature
between sea level and H
Air under isentropic Conditions
Air under temperature decreasing
linearly with elevation at a lapse
rate of XoC/m
Expression for temperature can be written as:
T=To +Kz
where K = -X
and
To = (273+ 15);
dT = Kdz →→ dz = dT/K
By using one of the fundamental equation of
state:
Air under temperature decreasing linearly
with elevation at a lapse rate of XoC/m
MEASUREMENT OF FLUID
PRESSURE
There are generally two types of pressure measuring
devices:
1. Tube gauges: - are those instruments that work
on the principle that a particular pressure can
support a definite weight of a fluid and this weight
is defined by definite column of fluid.
2. Mechanical gauges: - work on the principle that
the applied pressure will create a deformation in
either a spring or a diaphragm.
Tube Gauges
1. Piezometric Tube
Piezometer is the simplest pressure
measuring tube device and it consists of a
narrow tube so chosen that the effect of
surface tension is negligible. When connected
to the pipe whose pressure is to be measured,
the liquid rises up to a height h, which is an
indicative of the pressure in the pipe p=h
Piezometric tube
Piezometric tube
Pipe
Advantages and Disadvantages
of piezometric tube
Advantages: i) Cheap, easy to install and read
Disadvantages:
i) Requires unusually long tube to measure even
moderate pressures
ii) Cannot measure gas pressures (gases cannot
form free surface)
iii) Cannot measure negative pressures
(atmospheric air will enter the pipe through the
tube).
Manometers:
To overcome the above mentioned limitations of the
piezometer, an improved form of the piezometer
consisting of a bent tube containing one or more
fluids of different specific gravities is used. Such a
tube is called a manometer.
Types of manometers





Simple manometer
Inclined manometer
Micro manometer
Differential manometer
Inverted differential manometer
Manometers:Simple manometer
A simple manometer: consists of a tube bent in
U-shape, one end of which is attached to the
gauge point and the other is opened to the
atmosphere. The fluid used in the bent tube
is called the manometric fluid (usually
mercury) and the fluid whose pressure is to
be measure and therefore exerts pressure on
the manometric fluid is referred to as the
working fluid.
Simple manometer
Simple manometer
measuring gauge pressure
Inclined manometer
Working fluid
Simple manometer
measuring vacuum pressure
Manometric fluid
Simple manometer
By using the principle that the pressure on the horizontal and in the same continuous fluid is
the same, we shall state that:
For diagram A
P1=P2
P1=PA + h11
P2=Pa + h22

PA + h11= Pa + h22
PA-Pa= h22- h11
For diagram B
P2=Pa=P1
P1=PB + h11 + h22

PB + h11 + h22=Pa
PB-Pa= -h11 - h22 = vacuum gauge
INCLINED TUBE MANOMETER
This type is more sensitive than the vertical
tube type. Due to the inclination the distance
moved by the manometric fluid in the narrow
tube will be comparatively more and thus give
a higher reading for a given pressure
Micro manometers
Micro manometers
The pressure on level 1 is P1 and pressure on level 2 is
P2 .
PB=P1 + w (h+X-dh)
PD=P2+γw(dh+X) +m.h
But PB=PD------on the same horizontal and in a
continuous fluid.
P1 + w (h+X-dh)= P2+γw(dh+X) +m.h
ΔP=P1-P2 = γw(dh+X) + m.h - w (h+X-dh)= γwdh +
γwX+ m.h - wh - w X + wdh
ΔP=P1-P2 = m.h - wh + 2wdh
Micro manometers
By equation of volumes,
D2dh/4=d2h/(2x4)  dh=(d/D)2h/2
ΔP=P1-P2 = mh - wh + w (d/D) 2h
ΔP=P1-P2 = mh - w h[1- (d/D) 2]= w h{SG[1-(d/D) 2]
Since d/D is very small, the ratio (d/D) 2 can be
taken as zero
Therefore ΔP=P1-P2 = w h{SG-1}
OTHER TYPES OF MANOMETERS
Differential Manometer :consists of a U-tube
containing the manometric fluid. The two ends of
the tubes are connected to the points, whose
differential pressure is to be measured.
Inverted U-tube Differential Manometer
An inverted U-tube differential manometer is used for
measuring difference of low pressures, where
accuracy is the prime consideration. It consists of
an inverted U-tube containing a light liquid.
MECHANICAL GAUGES
Whenever very high fluid pressures are to
be measured mechanical gauges are
best suited for these purposes. A
mechanical gauge is also used for the
measurement of pressures in boilers or
other pipes, where tube gauges cannot
be conveniently used.
Bourdon’s tube pressure gauge
It can be used to measure both negative (vacuum) and
positive (gauge) pressure. It consists of an elliptical tube
ABC, bent into an arc of a circle. When the gauge tube is
connected to the fluid (whose pressure is to be found) at
C, the fluid under pressure flows into the tube. The
Bourdon tube as a result of the increased pressure tends
to straighten out. With an arrangement of pinion and
sector, the elastic deformation of the Bourdon tube rotates
a pointer, which moves over a calibrated scale to read
directly the pressure of the fluid.
Bourdon’s pressure gauge
Mechanical side with Bourdon
tube
Indicator side with card and
dial
Mechanical Details –
Stationary parts
A: Receiver block. This joins the inlet pipe to the
fixed end of the Bourdon tube (1) and secures the
chassis plate (B). The two holes receive screws
that secure the case.
B: Chassis plate. The face card is attached to this. It
contains bearing holes for the axles.
C: Secondary chassis plate. It supports the outer
ends of the axles.
D: Posts to join and space the two chassis plates
Moving Parts
1. Stationary end of Bourdon tube. This
communicates with the inlet pipe through the
receiver block.
2. Moving end of Bourdon tube. This end is sealed.
3. Pivot and pivot pin.
4. Link joining pivot pin to lever (5) with pins to
allow joint rotation.
5. Lever. This an extension of the sector gear (7).
6. Sector gear axle pin.
Moving Parts
7. Sector gear.
8. Indicator needle axle. This has a spur gear that engages the
sector gear (7) and extends through the face to drive the
indicator needle. Due to the short distance between the
lever arm link boss and the pivot pin and the difference
between the effective radius of the sector gear and that of
the spur gear, any motion of the Bourdon tube is greatly
amplified. A small motion of the tube results in a large
motion of the indicator needle.
9. Hair spring to preload the gear train to eliminate gear lash
and hysteresis.
Diaphragm Pressure Gauge
The principle of work of the diaphragm pressure
gauge is similar to that of the Bourdon tube.
However instead of the tube, this gauges
consists of a corrugated diaphragm. When the
gauge is connected to the fluid whose pressure
is to be measured at C, the pressure in the fluid
causes some deformation of the diaphragm.
With the help of pinion arrangement, the elastic
deformation of the diaphragm rotates the
pointer
Diaphragm Pressure
Gauge
Diaphragm Pressure
Gauge
Dead Weight Pressure Gauge
It is an accurate pressure-measuring instrument and is
generally used for the calibration of other pressure
gauge. A dead weight pressure gauge consists of a
piston and a cylinder of known area and connected to a
fluid by a tube. The pressure on the fluid in the pipe is
calculated by:
p=weight/Area of piston
A pressure gauge to be calibrated is fitted on the other
end of the tube. By changing the weight on the piston
the pressure on the fluid is calculated and marked on
the gauge
Dead Weight Pressure
Gauge
RELATIVE EQUILIBRIUM OF LIQUID
(Liquid under constant acceleration or
constant angular speed)
When fluid masses move without relative motion
between particles, they behave just as much as
solid body and are said to be in relative
equilibrium
Relative equilibrium of a liquid is that
situation in which a liquid being in motion,
stay together as one mass as a solid body
i.e. there is no sliding (displacement) of
some particles over others.
Liquid mass subjected to uniform
linear horizontal acceleration
Consider a tank partially filled and placed on a
tanker truck and given a uniform acceleration
ax in the x-direction. As a result of the
acceleration, within the fluid will emerge an
inertia acceleration in opposition to the
imposed acceleration. The inertia acceleration
has the same magnitude but of opposite
direction.
Liquid mass under uniform linear
horizontal acceleration
Liquid mass under uniform
linear horizontal acceleration
Since this is a a static situation, then we can use the general differential equation of statics, i.e
1
dp  a x dx  a y dy  a z dz ----------------------------( * )

On the accelerating fluid, there are two body forces acting, namely gravity force and inertia
force. From the above equation, we recognise that
ax = -a; ay = 0; az =-g -------------------------------( ** )
Substituting (**) into (*), we shall have
1
dp  adx  gdz

Integrating,
p = ρ(-ax –gz) + c
Liquid mass under uniform
linear horizontal acceleration
The pressure distribution within the accelerating fluid is:
p = ρ(-ax –gz)
The angle the surface of the fluid makes with the
horizontal can be obtained by finding the tangent of
the angle θ.
tan θ = z1/L or tan θ = aL/g.L = a/g
Therefore in a uniform accelerating fluid, the
angle of inclination of the fluid surface to the
horizontal is the ratio of the horizontal body
force acceleration to that of the vertical body
force acceleration
Motion in the vertical plane
with constant acceleration
Z
-
Po
M
g
+
X
Fig 2-8
Motion in the vertical plane
with constant acceleration
The body forces on such a body are the forces of gravity and inertia. The projections of their
acceleration on the axis are;
X=0; Y=0; Z=-g + j -----------------2.40
Where + j – when descending
1

dp  (  g  j )dz
      2.41
-j – when ascending
Integrating
p = + (-g + j) Z + C---------------2.42
When Z=0; p=Po
p =  (-g + j) Z + Po -------------2.43
p =  g(-1 + j/g) Z + Po
p = (-1 + j/g) Z + Po
Motion in the vertical plane
with constant acceleration
Let us represent (-1 + j/g) by k
Then we have
P = -k Z + Po ------------------2.44
Since k is a scalar quantity, we can bring the above expression to the familiar hydrostatic
equation.
Representing -k = 1, we have
p =Po + 1Z ------------------2.45
Though k is a scalar quantity, it can have different values. Let’s look at the different values of k.
1. when j<g, k<1 and  becomes small, so the liquid experiences a certain amount of
weightlessness
2. when j = g, k=0 and  = 0. Liquid experiences a total weightlessness.
EQUILIBRIUM OF A
ROTATING CONTAINER
EQUILIBRIUM OF A
ROTATING CONTAINER
Consider a cylindrical container filled with a
liquid and rotating with a constant angular
velocity ω about the vertical axis. As a result
of the liquid rotating with the same angular
velocity as the container the liquid is
considered to be at rest relative to the
container. Frictional force (both internal, and
external i.e. friction between particles of
liquid walls) is zero.
EQUILIBRIUM OF A
ROTATING CONTAINER
If the coordinate axis shown on the diagram is
considered fixed to the container, then
relative to the rotating vessel, the liquid will
also be at rest. Therefore the basic
differential equation of hydrostatic of Euler is
applicable in the case of a rotating fluid with
the above conditions.
The body forces acting on the fluid are:
EQUILIBRIUM OF A
ROTATING CONTAINER
1.
2.
Gravity dFG = gdM
Centripetal force
or Z = -g
dFCP
v2

dM   2 rdM
r
The centripetal acceleration aCP = v2/r =ω2r
Resolving the accelerations into the axes
X = ω2.x
Y = ω2y
Z = -g
EQUILIBRIUM OF A
ROTATING CONTAINER
Substituting in 2.16 we have
dp = (ω2 x dx + ω2 y dy – gdz) ----------------------2.50
Integrating, we obtain
 2 x2 2 y2

p    

 gz   c
2
 2

2 2 2
p   x  y   gz  c
2
EQUILIBRIUM OF A
ROTATING CONTAINER
To find C we can look at the conditions at the point x=0; y=0, z=0; and p=Po
Therefore
C= Po
Then
p  po 
 2
2
x
2

 y 2  z
Distribution of pressure in the liquid
To find lines of constant pressure (isobars) we put the left land side of the equation to zero.
p=constant.
But since Po is atmospheric, we can put p-po=0
Therefore equation of isobars is given by
ω2 (x2 + y2) - z=0 -----------------------2.53
2
as it can be seen the equation is an equation of a parabola which is rotating (rotating parabola).
At the container x2 + y2 = r 2
ω2 r2 - z=0
2
EQUILIBRIUM OF A
ROTATING CONTAINER
To find C we can look at the conditions at the
point x=0; y=0, z=0; and p=Po
Therefore
C= Po
Then
2
 2
2

p  po 
x  y   z.
2
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
Where are these applied:
1. Irrigation Engineering for water distribution
on the field
2. Dam engineering for all types of gates
3. In River transportation (Locks systems)
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
A
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
Consider in Fig. (above) an open container,
filled with a fluid and an inclined plane OM.
On the inclined plane OM is an arbitrary
plane figure AB with area A.
Our task is two folds:
1. to find the magnitude of the force
2. to find the point (position) of action of this
force.
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
Let us choose an arbitrary point m on the
surface AB immersed in the fluid at a
depth h, and at a distance z from the
axis OZ. At the point m, we choose an
elemental area dA, surrounding the
point m. The hydrostatic force on the
area dA is given by:
dF  pm dA  ( pa  h)dA
But h = z sin θ
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
dF   pa  z. sin  dA
The total force acting on the surface A is
obtained by integrating dF over the whole
surface A.
F    po  z. sin  dA   po dA   . sin   zdA
FA  po A   . sin   zdA
But ..the.. exp ression .. z.dA  ( St ) ox  z C . A
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
The
FA  po . A  z C A. sin 
But ..z C sin   hC
FA  po . A  hC A
Where paA is the force due to atmospheric
pressure, which is transmitted through out
the fluid onto the planes surface AB and
γhCA is the force due to pressure of the
column of fluid on the surface AB
FORCES OF HYDROSTATIC PRESSURE ON
PLANE SURFACES IMMERSED IN FLUIDS
Since in most cases, we shall be interested
only in the gauge pressure, the total force
on a plane surface immersed in a fluid may
be written finally as:
FAB = γhCA
Finding (Centre of Pressure)
To find the centre of pressure, we are going to
use the theory of moments which states that
the moment of the resultant force about a point
(or axis) equals the sum of moments of all the
forces about the same point (or axis). Let ZD be
the centre of pressure and let us write the
equation of moments about the axis Ox.
The moment dM of the elemental force dF about
Ox equals
Finding (Centre of
Pressure)
M (dF ) Ox  dF .z  (hsA) ----------------------(2.9)
The sum of moments of all the individual forces is given by:
 M (dF )
Ox
  z sin  .dA    sin  .z 2 dA -----------(2.10)
The moment of the resultant force about the same axis Ox is given by:
FA.zD = γ.hC .A.zD = γzC.sinθ.A.zD ---------------------(2.11)
Equating equations (2.10) and (2.11) according to the theory of moments, we have
zD 
2
z
 dA
zC .A

I ox
--------------------( St ) ox
Finding (Centre of
Pressure)
Where Iox = ⌠z2dA – 2nd moment of area or moment of inertia of AB about the
axis ox
and (St)ox = zc.A - 1st moment of area or the static moment of AB about the axis
Ox.
It is also known from the theory of moments that the moment of inertia of a body about a given
axis equals the moment of inertia about an axis parallel to the given axis and passing through the
centre of gravity (centroid) plus the product of the area and the square of the distance between
the axes; i.e.
Iox = IC + z2c.A ----------------------------------------(
The point of action (Centre of
Pressure) of the resultant force
The point of action of the resultant force F is
given by:
I C  Z C2 A
IC
Z
 ZC 
ZC A
ZC A
hD  hC 
IC
Sin 2
hC A
or
GRAPHICAL METHOD FOR FINDING
HYDROSTATIC FORCE ON PLANE SURFACES
Ox
O
h
x
b
m
m
C
F
B
γH
H
C
D
D
A
A
PRESSURE DIAGRAM METHOD
Properties of the pressure diagram
 Every ordinate on the pressure diagram gives the
hydrostatic pressure at the point


The area under the pressure diagram gives the
value of the hydrostatic force per unit width of
the gate.
The force F passes through the centre of gravity
of the pressure diagram.
LOCK GATE
Lock gates are hydraulic structures used in
navigation for regulating water levels in
channel for the purposes of creating necessary
levels for navigation.
AB and BC are two lock gates. Each gate is held
in position by two hinges. In the closed
position, the gates meet at B exerting thrust
on one another. Now let us consider the
equilibrium of one of the gate eg. gate AB.
LOCK GATES
t
Let N be the reaction at the common contact
surface of the two gates. Let R be the
resultant reaction of the top and bottom
hinges. The three forces, F, N, and R will all
be in the same horizontal plane (i.e
coplanar). Since F, N and R are coplanar and
they bring about the equilibrium of the gate
AB, then, the three forces must be
concurrent at a point. i.e point D
Angle DBA = angle DAB = θ
Resolving forces along AB
N.cosθ = Rcosθ
N =R
Resolving forces normal toAB
F = N sin θ + R sin θ = 2Rsin θ
Since F = F1 – F2 is known, R can be found
Reaction at the top and
bottom hinges
We know the resultant water pressure F acts
normal to the gate and acts at the middle of
the gate AB. Thus one half of this force is
transmitted to the hinges of the gate and
the other half to the reaction at the
common contact. Let RT and RB be the
reactions of the top and bottom hinges so
that RT + RB = R
Reaction at the hinges
Taking moments about the bottom hinge, we have
F. H  F H 
RT . sin  .H   1 1    2 . 2 
 2. 3   2 3 
Resolving forces in the horizontal direction,
F1 F2 F
RT .sin   RB .sin    
2 2 2
HYDROSTATIC PRESSURE
FORCES ON CURVED SURFACES
Consider a curved surface ABC with length b.
Let Px and Pz be the horizontal and vertical
components of the force due to hydrostatic
pressure acting on the curved surface. To
find these components lets erect the plane
DE. The plane DE will isolate that volume of
liquid ABCED whose equilibrium we wish to
investigate.
The volume ABCED is acted upon by the ff:




the force Ph acting on the vertical side DE
the force RD-reaction of the base EC
RD=[area (C1CED)] b
the reaction R from the curved surface. Rx, Rz is
the horizontal and vertical components
respectively.
force due to liquid’s own weight G
G=[area (ABCED)] b
Now lets resolve all forces acting on the volume
ABCED onto the x- and z- axis.
Rx=0; Ph – Rx = 0
Ph = Rx = Px
Rz=0; G + Rz - RD = 0
 Rz = RD-G
Pz = -Rz = G - RD
Pz= [area ABCED – area C1CED] b
Pz= - [area ABCC1] b
Horizontal component
1. The horizontal component Px of the
force on a curved surface equals the
force of hydrostatic pressure on the
plane vertical figure DE, which is a
projection of the curved surface on the
vertical plane
Vertical Component
2. The vertical component Pz equals the
weight of the imaginary free body of
the fluid ABCC1. This imaginary free
body of the fluid we shall called
"pressure body".
The weight of the pressure body
represent by [area ABCC'] b = Go
Procedure for determining
the horizontal component
1.
2.
3.
Place a vertical plane DE behind the
curved surface.
Project the curved surface onto the
vertical plane to obtain a plane surface.
Determine the horizontal component in a
similar manner as in plane surfaces
immersed in fluid.
Procedure for determining the
vertical component
The cylindrical surface ABC is the surface whose
pressure body is to be found.
1.
First fix the extreme ends A and C of the curved
surface;
2.
Draw vertical lines from these points to the water
surface;
3.
finally note the contour of the pressure body
A'ABCC' ie the body of fluid between the two
vertical lines, the curved surface and the surface of
the fluid.
Procedure for determining the
vertical component cont.The cross-section of the pressure body
(positive or negative) is the area between
the two verticals, the cylindrical surface ABC
and the surface of the fluid (or their
continuation). If the pressure body does not
wet the cylindrical surface, then we have
negative body pressure; however if the
pressure body wets the surface, then the
pressure body is positive
Buoyancy
FLOATING BODIES: ARCHIMEDES PRINCIPLE:
the force, which a fluid exerts on a body
immersed in it equals the weight of the fluid
displaced by the body or when a body is
placed (submerged) in a fluid, it experiences
an upward (upthrust) force which is equal to
the weight of fluid the body displaces.
Buoyancy
d
F
C
G
D
Buoyancy
The body AB with volume V completely
submerged in a fluid. The resultant of all
forces due to pressure acting on the surface
element of the body is determined by the
principle of forces on a curved surface.
R R R R
2
x
2
y
2
z
But Rx=0; Ry=0
Buoyancy
The difference in the pressure force on the strip
is:
dFb=(h2-h1)dA

The sum of all elementalF buoyancy
force on whole
V
V
body AB is:
F    h2  h1 dA    dV  V
0
0
The buoyancy force acts at the centre of gravity of
the displaced liquid AB. The point D is called
centre of buoyancy.
Equation of floating bodies
Therefore the basic equation of floating bodies is:
Rz=0; Fb-G=0 or V-G=0
BUOYANCY: Is the tendency for fluid to exert a supporting
force on a body immersed in it.
Fb < G--- Body sinks and fall to the bed of the fluid where
the reaction of the bed will support to bring the body to
equilibrium
Fb > G--- Body floats partially submerged in fluid
(FLOATING BODIES)
Fb= G --- Body floats totally submerged in the fluid
(SUBMERGED BODIES)
STABILITY OF
SUBMERGED BODIES
STABILITY OF
SUBMERGED BODIES
A body is said to be in a stable
equilibrium , if a slight displacement
generates forces which oppose the
change of position and tend to bring
the body to its original position.
Criterion of stability for
submerged bodies
The criterion of stability for submerged bodies is
the relative positions of D and C. For
submerged body to be stable, i) the weight of
the body G must be equal to the buoyancy
force Fb and ii) the centre of buoyancy D must
always be above the centre of gravity C of the
body. Submarines are submerged bodies,
which use balancing tanks to make Fb equal to
G and trimming tanks to bring the centre of
buoyancy above the centre of gravity.
FLOATING BODIES FB>G
Some basic terms in
floating bodies
O – O – axis of floatation
W-L: - water line –the line of intersection of the free
surface of the fluid with the body.
C- centre of gravity of the body
D – centre of buoyancy of the body when it is upright
D1 – centre of buoyancy of the body when body is
rotated through a small angle θ
M- Metacentre – is the point of intersection of the
axis of floatation and the vertical through D1.
Some basic terms in
floating bodies
MC – metacentric height-the distance between
the metacentre and the centre of gravity.
MD – metacentric radius: - distance between
the meatcentre and the centre of buoyancy
when object is upright.
h – height of floating body
d – draft of floating body
Floating Body
Floating Body
The figures shown above represent floating bodies.
Fig a represents a body in equilibrium. The net
force on the body is zero so it means the buoyancy
force Fb equals in magnitude to the weight of the
body. There is no moment on the body so it means
the weight acting vertically downwards through the
centre of gravity C must be in line with the
buoyancy force acting vertically upwards through
the centre of buoyancy D
Floating Body
Fig (*) (a) shows the situation after the body has
undergone a small angular displacement (angle of
heel θ). It is assumed that the position of the
centre of gravity C remains unchanged relative to
the body. The centre of buoyancy D, however,
does not remain fixed relative to the body. During
the movement, the volume immersed on the right
side increases while that on the left side decreases;
so the centre of buoyancy moves to the new
position D1. The line of action of the buoyancy force
will intersect the axis of floatation at the point M.
Floating Body
Floating Body
On the other hand in Fig (*)(b), the point M is
below the point C and the couple thus
formed is an overturning couple and the
original equilibrium would be unsafe. The
distance MC is known as the metacentric
height and for stability of the body, it must
be positive (i.e.M above C). The magnitude
of MC serves as a measure of stability of
floating bodies.
Condition for stability of
floating bodies
The distance MC is known as the metacentric
height and for stability of the body, it must
be positive (i.e.M above C). The greater
the magnitude of MC, the greater is the
stability of the body. The magnitude of MC
serves as a measure of stability of floating
bodies.
Floating Body
It is important that all floating
bodies do not capsize in water.
It is therefore essential that
we are able to determine its
stability before it is put in
water.
Experimental Determination
of metacentric Height
Determination of metacentric
height
The experiment consists of moving a weight P
across the deck through a certain distance x
and observing the corresponding angle of
heel or roll θ The shifting of the weight P
through a distance x produces a moment Px
which causes the vessel to tilt through an
angle θ. This moment Px is balanced by the
righting moment G x CM θ.
Determination of metacentric
Height
Px = G x MC tan θ
P.x
P.x cot 
MC 

G. tan 
G
It must be noted that the vessel, before the
weight was moved, was in an upright
(vertical) position and G is the total weight of
the vessel (including the weight P)
The metacentric radius DM = I/V0
The metacentric radius
The metacentric radius DM = I/V0
Where I – second moment of area of the
plane of floatation about centroidal axis;
V0 – immersed volume
Periodic Time of Oscillation
The displacement of a stable vessel through
an angle θ from its equilibrium position
produces a righting moment (or torque).
T = G x MC x θ
This torque will produce an angular
acceleration d2θ/dt2 when the force bringing
about the displacement is removed
Time of Oscillation
If I is the mass moment of inertia of the
vessel about its axis of rotation, then
d 2 T G.CM .
CM . .g
 

2
I
dt
k2
G 2
 k
g
Where k – radius of gyration from its axis of
rotation. The negative sign indicates that the
acceleration is in the opposite direction to
displacement
Time of Oscillation
The above equation corresponds to a simple
harmonic motion with the period given by:
t  2


Displaceme nt

 2 
Accerelation
 CM . . g 2
k



  2


 k2 


 CM .g 
From above, it can be inferred that although a large
metacentric height ensures improved stability it
produces a short periodic time of oscillation,
which results in discomfort and excessive stress
on the structure of the vessel
The Hydrometer
The hydrometer is an instrument for
measuring the specific gravity of liquids.
It is based on the principle of buoyancy .
The hydrometer consists of a bulb weighted
at the bottom to make it float upright in
liquid and a stem of smaller diameter and
usually graduated.
The Hydrometer
+Δh
-Δh
Vo
V=Vo+aΔh
V=Vo-aΔh
The Hydrometer
Let the hydrometer read 1.0 when floating in
distilled water of specific gravity 1. The
corresponding weight of water displaced will be
Voγw; where Vo is the volume of distilled water
displaced. In another liquid of higher (or lower)
density, the hydrometer will pop up (or down)
by an amount Δh. If the stem of the hydrometer
is of cross-sectional area a, the reduction (or
increase) in volume of fluid displaced will be
a.Δh
The Hydrometer
Since the weight of the hydrometer is equal to
the weight of the volume of fluid displaced in
each case
G = γwVo = γf(Vo-a.Δh)
h 
 f Vo   wVo
a f
Vo
S.G f  1

a f
KINEMATICS
Kinematics: the study of the geometry of
motion, without considering the forces
causing the motion.
KINEMATICS OF FLUID FLOW
In the 1 8th century, mathematicians sought to
specify fluid motion by mathematical relations.
It must be noted that these relations could be
developed only after certain simplifying
assumptions, notable of which was the concept
of “ideal fluid”, which assumed the fluid as
not having viscosity and not
compressible. The ideal fluid exhibited no
surface tension and could not vaporize if
it was a liquid.
KINEMATICS OF FLUID FLOW
As a result of such assumptions, the relations
obtained for describing the flow of an ideal
fluid may be used to indicate the behaviour
of a real fluid only in certain regions of flow;
e.g. in the regions far removed from
boundaries. The results so obtained may be
only an approximation to the truth, although
in certain cases the theoretical results are
surprisingly close to the actual results.
KINEMATICS OF FLUID FLOW
Irrespective of the way anyone may look at the
relations, they give valuable insight into the
actual behaviour of a real fluid. Therefore in
the forthcoming presentation, we shall only
give an introduction of mathematical
kinematics and its application to a few simple
examples of fluid flow. Attention will be
limited to a steady two-dimensional plane.
TYPES OF FLUID MOTION
(FLOW)
Fluid flow may be classified in a number of ways.
i)
Steady and unsteady flows
ii) Uniform and non-uniform flows
iii) One, two and three dimensional flows
iv) Uniform and non-uniform flows
v) Laminar and turbulent flows
vi) Rotational flow and irrotational flows
vii) Critical, subcritical and supercritical flows
STEADY AND UNSTEADY
FLOWS
STEADY AND UNSTEADY
FLOWS
STEADY AND UNSTEADY
FLOWS
Lets consider a stream contained within the lines a1b1
and a2b2. The point 1 is fixed and we assume that
fluid particles M passes through point 1 at different
times in different particles paths.
Example M' passes through the point at time t', M''-t'',
M'''-t''' etc.
The particle M’ arriving at the point 1 at a time t' has
a velocity U'.
The particle M'' at t''-U''.
The particle M''' at t'''-U'''
UNSTEADY FLOW
U'  U''  U'''
Therefore we have the velocity to be a function
not only of coordinate x, y, z but also of time t.
U = f (x, y, z, t)
U
 x, y , z   0
t
If the fluid velocity at a point is time dependent,
then the motion is called unsteady flow.
STEADY FLOW
When a fluid velocity field does not vary with
time, the flow is called steady flow.
i.e. particles M', M'', M''' arriving at point 1 at
different times have the same velocity i.e.
U'=U''=U'''
U = f (x, y, z)
U
 x, y , z   0
t
UNIFORM AND NONUNIFORM FLOWS
Uniform flow is one in which the free crosssectional area A along the direction of flow
remains constant and the velocities at identical
points in space also remains constant.
V=constant in the direction of flow.
Non-uniform flow is a flow in which:
i) Either the free cross-sectional area changes A
constant or
ii) Velocities at identical points in space do not
remains constant.
One, two and three
dimensional
The velocity of a fluid in the most general case is
dependent upon its position. If any point in
space be defined in terms of space coordinates
(x, y, z) then at any given instant the velocity at
the point is given by V = f(x, y, z). The flow in
such a case is called a three-dimensional flow.
Sometimes, the flow conditions may be such that
the velocity at any point depends only on two
space coordinates say (x, y) at a given instant,
One, two and three
dimensional
i.e., in this case at the given instant,
V = (x, y). In this case the flow conditions are
potential in planes normal to the z-axis. This
type of flow is called two-dimensional.
Example is the flow between two vertical
walls.
One-dimensional flow
One –dimensional flow is that in which all flow
parameters may be expressed as a function of
time and one space coordinate only. The single
space coordinate is usually the distance measured
along the centre line of the conduit in which the
fluid is flowing. For instance, the flow in a pipe is
frequently considered one-dimensional: variations
of pressure, velocity and other properties of fluid
occur along the length of the pipe but any
variation over the cross-section is assumed to be
negligible.
Two & one dimensional
flows
Laminar and turbulent flows
Laminar flow is a type of flow in which the fluid
particles move in layers. There is no transportation
of fluid particles from one layer to another. The
fluid particles in any layer move along well-defined
paths.
Turbulent flow is the most common type of flow
that occurs in nature. The flow shows eddy currents
and the velocity of flow changes in direction and
magnitude from point to point. There is a general
mixing up of the fluid particles in motion. There
are numerous collusion
FLUID PARTICLE & PATH
OF A FLUID PARTICLE
If the volume of fluid under study is so small
that we may neglect changes in its shape
and other physical quantities such as
velocity, pressure, density, temperature etc,
it is called a fluid particle (fluid element)
The curve described by a moving fluid
element is called the path of a fluid
particle (pathline)
STREAMLINES
The flow of a fluid may be described by tracing the
paths of its entire component particles but this is
very complicated. In practice a simpler method is
used. The fluid velocity field is considered given if at
every instant the velocity vector of fluid particles is
known for every point of the fluid in flow. For a
known fluid velocity field, i.e. the distribution of
velocities in the flow and its time dependence, we
can fully determine the motion of the fluid. The
velocity direction of flow is characterised by
streamlines.
STREAMLINES
STREAMLINES
A Streamline: is an imaginary curve whose
tangent line direction at each point coincides
with the velocity vector of the fluid particle
that passes through that point at any given
instant of time. Streamline is an imaginary
curve in the fluid across which, at a given
instant, there is, no flow. Thus the velocity
of every particle of the fluid along the
streamline is tangential to it at that moment.
STREAMTUBE AND FILAMENT
If a series of streamlines are drawn through every
point on the perimeter of a small area dA of the
stream cross-section, they will form a stream tube.
Imaginary lines drawn through every points of a
small closed contour C with an elemental area of
dA cut off from a fluid produces a pipe-like surface
which is called a stream tube.
 Fluid flowing through a stream tube is called the
filament.
STREAMTUBE AND FILAMENT
Properties of (streamtube)
filament when flow is steady
1). Since streamlines in a steady flow do not change with
time, then filament also does not change its form with
time (i.e. constant form).
2). Since the cross-sectional area of a filament is
elemental, the magnitude of the velocity, U, the
pressure, P, and all fluid properties for all point in a
given cross section of the stream tube are considered
equal. Though U and P are not necessary the same
along the flow.
3). Fluid enclosed in the filament can get out of the tube.
Similarly no particle can enter the stream tube.
The area of a filament normal to streamline
direction is called the filament cross-section dA.
Velocity U, and elementary flow rate dQ are two
fundamental quantities that are used in dealing
with fluid in motion. They give an exact (not
average) differential description of the flow.
Elementary flow rate, dQ
Elementary flow rate
Elementary Flow rate: is the volume of fluid
passing through a given filament cross section
in a unit time (i.e. one second).
The equation of elementary flow rate can be
found by considering fig. 3-5
During the time period dt, all fluid particles from
the section n-n might have moved a distance
dS and might have come to section n'-n’.
ds = Udt
Elementary flow rate
Therefore the volume of fluid passing through
the section n-n during the time dt will be
dV = dA.Udt
Therefore, in a unit time, the volume of fluid
passing through the section n-n will be
dQ = dV/dt = UdA
dQ is the elementary fluid flow rate.
FLOW RATE AND MEAN FLOW
VELOCITY OF A STREAM
A stream consist of numerous filaments. Since flow
velocity of each filament is different from the other
it means the flow velocities at different points in a
Q
given cross section of a stream are different. Since
the velocities are different at different points in a
given cross section, the value of the flow rate of a
stream will be given by the summation of all the
elementary flow over the cross-section
A
Q 
 U .dt
A 0
FLOW RATE AND MEAN FLOW
VELOCITY OF A STREAM
Concept of mean flow
velocity
To simplify fluid flow calculations, the concept of mean
flow velocity is introduced. Fluid flowing through a
channel bounded by walls has different velocities at
different points of the cross section. The fluid particles
immediately adjacent to the wall of the tube (channel,
duct etc) adhere to the sides and come to rest. Their
velocities are zero. Filaments in immediate vicinity of the
adhesive particles are dragged because of internal
friction and their velocities are decreased. The farther
the filaments are from the sides of the wall, the greater
their velocity with the maximum at the centre of the
tube
Mean velocity
The mean flow velocity is defined as:
A
v 
Q

A
 U .dA
A 0
A
Velocity profile
Let us represent the area of this diagram of A
and lets suppose the stream has a
rectangular cross section with width b
The flow rate is given by:
Q = A.b
CONDITION FOR
CONTINUITY OF FLOW
CONDITION FOR
CONTINUITY OF FLOW
Consider the sections 1-1 and 2-2 of a filament
in a steady flow. We can write that
dQ1 = U1d1
dQ2 = U2d2
It can be seen that
1). dQ1 not greater than dQ2 (because of
incompressibility of fluid)
2). dQ1 not less than dQ2 (because we never
observe a break in the flow)
CONTINUITY EQUATION
Therefore we can write
dQ1 = dQ2 or U1dA1= U2dA2
or dQ = UdA = const
This equation is equally true for a stream i.e.
For any two sections in a stream,
Q1 = Q2 or 1A2 = 2A2
Q=v.A = const. -------- Continuity equation
for a stream
VELOCITY
The velocity of flow for most engineering
problems is of great importance. For flows
past structural or machine parts, knowledge
of the velocity makes it possible to calculate
pressures and forces acting on the structure.
In other cases of engineering as design of
canals and bridge pier, velocity is of interest
from the point of view of its scouring action.
Therefore it is importance to know how to
determine the velocity of flow.
TWO VIEW POINTS ON
FINDING VELOCITY
As particles move in space, their characteristics,
such as velocity, density, etc may change
with space and time. The flow characteristics
are measured with respect to some coordinate system, fixed or moving.
There exist two approaches for finding the
velocity of flow, namely:
i) The Lagrangian Approach
ii) The Eulerian Approach
The Lagrangian Approach
(“Follow that particle”)
When we choose a co-ordinate system attached
to the particles whiles they move. In this
approach, we follow the movement of
individual particles. This means that the
coordinates x, y, z are not fixed but must
vary continuously in such a way as always to
locate the particle. For any particular particle,
x(t), y(t) and z(t) becomes specific time
function which are different for corresponding
time function of other particles
The lagrangian Approach
If the position vector is known, the velocity
could be obtained by differentiating the
position vector with respect to time. For
example if the position vector is expressed in
terms of its components in the x, y, z as:
F(t) = xi + yj + zk
When the equation is differentiated with
respect to time, we obtain the velocity of the
particle as:
The Lagrangian Approach
dx
dy
dz
u 
.i 
.j 
i
dt
dt
dt
u  u xi  u y j  u z k
u 
u
2
x
u u
2
y
2
z
The Lagrangian Approach
The difficulty in using this method is that
the motion of one particle is inadequate to
describe an entire flow field. It implies that
the motion of all fluid particles must be
Considered simultaneously which is rather
difficult if not an impossible task.
Eulerian Approach
(“Watch that Space”)
In this approach, choose a co-ordinate system
fixed in space and study the motion of fluid
particles passing through these points.
We fix points in the fluid flow and monitor the
velocity field with time. Hence by this technique,
we express at a fixed positions in space the
velocities of a continuous “string” of fluid
particles moving by this position.In this case the
velocity depends on the point in space and time
Eulerian Approach
ux = f1 (x, y, z, t)
uy = f2 (x, y, z, t)
uz = f3 (x, y, z, t)
u = √(u2x + u2y + u2z)
Since it is almost impossible to keep track of
the position of all the particles in a flow field,
the Eulerian approach is favoured over the
Lagrangian approach.
Velocity as function of
position along a streamline
At times it is useful to express velocity as a
function of position along a streamline and
time as
u = f(s, t)
ACCELERATION
The acceleration of a fluid particle is obtained by
differentiating the velocity with respect to time
uX=f1(x, y, z, t)
uy=f1(x, y, z, t)
uz=f1(x, y, z, t)
When we differentiate the component ux with
respect to time t, we shall obtain the
component of acceleration in the x-direction
Acceleration
du x
 x, y , z 
ax 
dt
du u dx u dy u dz u dt
u
u
u u
ax 
 .  .  .  .  u.  v.  w. 
dt x dt dy dt z dt t dt
x
y
z t
ay 
dv v dx v dy v dz v dt
v
v
v v
 .  .  .  .  u.  v.  w. 
dt x dt dy dt z dt t dt
x
y
z t
dw w dx w dy w dz w dt
w
w
w w
az 
 .  .  .  .  u.  v.  w. 
dt x dt dy dt z dt t dt
x
y
z t
ACCELERATION
a  a a a
2
x
2
y
2
z
Angle of inclination of the components of acceleration
is given by:
ay
ax
a
cos   ;..... cos  
;........ cos   z
a
a
a
The first three terms on the right are those terms of
changes of velocity with respect to position and are
called convective accelerations because they are
associated with velocity changes as a particle moves
from one position to another in the flow field.
Tangential and Normal
Acceleration
The last term on the right are called local
accelerations and are the results of velocity
changes with respect to time at a given point
and is characteristic of the unsteady nature
of flow.
If the velocity is expressed as a function of
position along the streamline(s) and time (t)
as u = u(s,t), then
Tangential Acceleration
u ds u dt
u u
at  .  .  u s

s dt t dt
s t
For steady flow,
u 1 du
as  u s

s 2 ds
2
NORMAL ACCELERATION
From mechanics, we know that the normal
acceleration is given by:
2
u
2
aN 
 R
R
The Continuity Equation
The continuity equation is an expression of the
conservation of mass law and it states that for a
steady flow of fluid in the three-dimensional fluid
element (parallelepiped) of size dx, dy, dz,, the
amount entering the element must be equal to
the amount leaving and for unsteady flow, the
difference between the amount entering and
amount leaving must be stored in the
parallelepiped and this is only possible if density
changes occur in the element.
The Continuity Equation
The Continuity Equation
Let us find the mass of fluid entering the side
ABCD and leaving the side A1B1C1D1 of the
element within a certain interval of time dt
Mass entering the side ABCD
δMe = ρu.dt.dy.dz
And the mass leaving the side A1B1C1D1
δMl = ρ’u’ dt.dy.dz
Note that ρ’ =ρ + (δρ/δx).dx
u’ = u + (δu/δx).dx
and
The Continuity Equation
Net mass of fluidbeing retained in the element
= Mass entering and mass leaving within the
time dt is given by:
δMe - δMl = ρu.dt.dy.dz - ρ’u’ dt.dy.dz =
- δ(ρu)/δx).dx.dy.dz.dt
( u )
M x  
dx.dy.dz.dt
x
The Continuity Equation
 ( u )
Similarly ......M y  
dx.dy.dz.dt
y
 ( u )
....................M z  
dx.ddy.dz.dt
z
Therefore the total net gain of mass within the
time dt within the element is given by:
 ( u ) ( v) ( w) 
M  


dx.dy.dz.dt
y
z 
 x
The Continuity Equation
This gain in mass within the element is only
possible if within the period dt there were
changes in density within the element.
If the density of the fluid within the element
at the time t = 0 was ρ and the density at
the end of the period ie time dy was ρ’ then
the mass of the fluid at the beginning of the
period was
The Continuity Equation
∂Mt=o = ρ.dx.dy.dz
and the mass at the end of the period, dt was
∂Mt=dt = ρ’.dx.dy.dz
But ρ’ = ρ + (δρ/δt).dt
Therefore net change in mass within the element
in time dt due to density changes in the
element is given by
M  M t dt  M t 0

 dx.dy.dz.dt
t
The Continuity Equation
The change in mass due to difference in
volume entering and leaving must be equal
to the change in mass due to density
changes within the element with the same
time period dt. Therefore
  ( u )  ( v)  ( w) 




dx
.
dy
.
dz
.
dt

.dx.dy.dz.dt

y
z 
t
 x
   ( u )  ( v)  ( w) 



0
t
y
z 
 x
The Continuity Equation
 ( u) ( v) ( w)



0
t
x
y
z
The above equation is the general equation of
continuity in three dimensions and it is
applicable to any type of fluid flow and for any
fluid whether compressible of incompressible.
For incompressible fluid, the density becomes a
constant and the continuity equation takes the
form:
The Continuity Equation
For incompressible fluid, the density becomes a
constant and the equation takes the form:
 u v w

 
0
t x y z
For steady flow of an incompressible fluid, the
equation becomes:
u v w


0
x y z
PHYSICAL INTEPRETATION OF THE
STEADY, INCOMPRESSIBLE FLOW
EQUATION
PHYSICAL INTEPRETATION OF THE
STEADY, INCOMPRESSIBLE FLOW
EQUATION
Let’s suppose that at the time t, a volume of fluid is in
the position OabcO and it has a linear dimensions of
dx and dz. After a time dt (ie t+dt), it moves to
the position O’a’b’c’O’. Now let’s find the change in
length Oa after moving to O’a’ within the time dt.
It is evident that the distance moved by the point O
within the time dt
= Uxdt
and the distance moved by point a
= Ux!dt =[ Ux +(δUx/δx)]dt.
PHYSICAL INTEPRETATION OF THE
STEADY, INCOMPRESSIBLE FLOW
EQUATION
Therefore the change in the length of Oa
within the time interval dt
dl = (δUx/δx).dx.dt
Change in length per unit time (rate of change
in length) is
dl/dt = (δUx/δx).dx
The relative rate of change in length (rate of
strain) per unit time of Oa along the x-axis
(dl/dt)/dx = δUx/δx
PHYSICAL INTEPRETATION OF THE
STEADY, INCOMPRESSIBLE FLOW
EQUATION
Hence δUx/δx is the velocity of relative elongation (or
contraction) or rate of strain of the element Oa. i.e
the rate of relative linear deformation of Oa.
Therefore the equation
u v w


0
x y z
shows that the sum of the velocities of relative
linear deformations (in all the axis) equals zero.
In other words the equation shows that fluid
flows in such a way that a given mass of fluid
always occupies one and the same volume
IRROTATIONAL AND
ROTATIONAL FLOWS
A fluid in its motion can perform either
translational, rotational and/or distortion
(deformation) or a combination of any of
these. Translational motion occurs when
there is only a normal stress on the fluid and
rotation when there is a torque caused by
normal stresses. That means translational
and rotational motion of a fluid does not
require shear stresses.
However, if shear stress exist, the fluid
element will undergo apart from the
translational and rotational motion,
deformation (distortion of its shape), for an
incompressible fluid the volume of fluid
element remaining constant. This results in
shear strain which causes a change in the
angle between two adjacent sides of the
element
Rotation of a fluid about its
instantaneous centroid.
Rotation of a fluid about its
instantaneous centroid.
We shall be looking at the deformation of a
fluid element which at an initial period was
located at oabc with sides oa and oc being
mutually perpendicular. Within the time
interval dt, the element has moved to the
position o’a’b’c’. Rotation and therefore the
deformation of the element will be
characterised by the average deformation of
the sides oa and oc.
Rotation of a fluid about
its instantaneous centroid.
Rotation of the element will be defined by the
angular velocity which the average rate of
deformation of the mutually perpendicular sides
u 
oa and oc
In moving from oabc to o’a’b’c’, the point O has
moved through a vertical distance dz1 and the
point a has moved through the vertical distance
dz2. within the time dt
Note that
u z 

dx .dt
uz 
’
dz1 = uzdt and dz2 = u zdt = 
x 
z
Rotation of a fluid about its
instantaneous centroid.
u z
dz 2  dz1 
dx.dt
x
u z
tan    
.dx.dt
x
The rate of change of α or the rate of rotation
of side oa
u z

dt

x
Rotation of a fluid
Similarly for the side oc to move to o’c’, the
point o moved through a horizontal distance
of dx1 and point c moved through dx2.
dx1 = uxdt
and
u x
dx2  dx1 
dz.dt
z
u x 

dx2 = u’xdt =  u x  dz .dt
z 

u x
tan    
.dz.dt
z
Rotation of a fluid
Rate of change of β or the rate of rotation of the side
 u x
oc.

dt
z
We adopt a sign convention for rotation. Clockwise
rotation is negative and anticlockwise rotation is
positive.
Rotation of the fluid element about its instantaneous
axis ( in this case about the y-axis) is characterised
by its angular velocity which is defined as the
average rate of deformation of side oa and oc which
Rotation of a fluid
Similarly
1
  1  u z u x 


   
2  dt dt  2  x
z 
1  u z u x 
y  


2  x
z 
1  u y u z 

 x  

2  z
y 
1  u x u y 

 z  

2  y
x 
Rotational and irrotational
flows
When all the components of rotation, i.e Ωx
Ωy, Ωz are equal to zero, it means rotation is
absent and the fluid flow is referred to as
irrotational flow. On the other hand if
even one component of the angular velocity
is not zero, it means rotation exist and the
flow is called rotational flow
VORTICITY
Vorticity is a concept used in fluid mechanics
to define rotation and it is defined as two
times the angular velocity.
x
y
z
 u y
u z

 z  y

u x
 u z


z
 x







u y
 u x

 y  x





VORTICITY
Just like the angular velocity, the vorticity is
defined in the three axes and characterises
rotation of a fluid element about its
instantaneous axis. If vorticity is zero, then
flow is irrotational
CIRCULATION AND
VORTICITY
Consider a closed curve in a two-dimensional
flow field shown in the diagram below.
Streamlines cut the curve. If P is a point of
intersection of the curve with a streamline,
and θ is the angle which the streamline
makes with the curve, then the component of
the velocity along the closed curve at the
point is equal to v.cos θ. The circulation Γ
(gamma) is defined as the line integral of
velocity around a closed curve in a flow
CIRCULATION
CIRCULATION
Thus the differential circulation dΓ along a
small length ds is given by:
dΓ = (vcosθ)ds.

Total circulation =   v. cos  .ds
The line integral is taken around the closed
curve in counter clockwise direction.
CIRCULATION
CIRCULATION
Proceeding from the corner A and remembering
that circulation is considered positive in the
anti-clockwise direction, its value around the
rectangular element is:

u 
 v 
d  u.dx   v  dx dy   u  dy dx  vdy 
y 
 x 

 v u 
 v u 
v
u
 dxdy  dxdy    dxdy    dA
y
y
 x y 
 x y 
CIRCULATION AND
VORTICITY
Γ=ξzdA or ξz=Γ/dA
Vorticity may therefore be defined as the
differential circulation per unit area
Though the above has been obtained for a
regular shape, it is true and applicable to any
shape.
Stokes’ theorem. The circulation around a
contour is equal to the sum of the vorticities
within the area of the contour.
STREAM FUNCTION
Stream function ψ(x,y) (psi) is a function, which
mathematically describes streamlines and
therefore the pattern of fluid flow.
The stream function is a scalar quantity and it is
defined by the function ψ (x,y) such that the
partial derivative of this function with respect to
displacement in any chosen direction is defined


as:
d ( x, y ) 
dx 
dy..........such..that..
x
y


 v........and ........
u
x
y
STREAM FUNCTION
STREAM FUNCTION
The sign convention adopted for stream
function is that an observer looking in
the direction of the stream lines see the
stream function increasing from right to
left.
Consider two points P and P’ lying on two
streamlines ψ and ψ+dψ respectively
STREAM FUNCTION
From the definition of a streamline, it is
known that no flow can cross a streamline
and therefore, the quantity of flow between
the two streamlines must remain constant in
accordance to the continuity equation. Since
the two points have stream functions ψ and
ψ+dψ, then the flow across points P and P’
is dψ.
STREAM FUNCTION
On the other hand the flow passing across PP’
per unit length into the page can be
calculated using the continuity equation as:
dQ = u.dy –v.dx
If ψ is the stream function, then dψ is:


d 
dx 
dy  vdx  udy  dQ
x
y
The flow between any two streamlines is the
difference in the stream function values.
Gradient of the streamline.
For the stream function ψ(x,y), the total
differential is given by:


d 
dx 
dy  vdx  udy
x
y
On a given streamline, the stream function is
the same. Therefore dψ= udy – vdx =0
Then (dy/dx)ψ= v/u.
The gradient of the streamline at any point is
given by the ratio of v to u
VELOCITY POTENTIAL
The velocity potential, φ is another
mathematical concept which is commonly
used in fluid mechanics. The velocity
potential is only a mathematical concept and
does not represent any physical quantity
which could be measured and therefore its
zero position may be arbitrary chosen.
Though an imaginary concept, the velocity
potential is quite useful in the analysis of
flow problems.
VELOCITY POTENTIAL
Whereas the stream function applies to both
rotational and irrotational flows, velocity
potential has meaning only for irrotational flow.
For it is only irrotational flow that movement
from one point to another is independent of
the path taken. For this reason, irrotational
flow is termed potential flow. (after velocity
potential)
VELOCITY POTENTIAL
The existence of a velocity potential in a
flow field ensures that the flow must be
irrotational. If we know that flow is
irrotational, then its velocity potential
must exist.
It is for this reason that an irrotational flow is
often called as potential flow. Lines drawn in a
fluid field joining points of equal velocity
potential gives lines of constant φ-values which
is called equipotential lines.
VELOCITY POTENTIALV
It is a scalar quantity and defined by the
function φ (x,y,z) such that the partial
derivative of this function with respect to
displacement in any chosen direction is equal
to the velocity component in that direction:



 u........................
 v.........................
w
x
y
z
VELOCITY POTENTIAL
The total differential of the function φ in a twodimensional flow can be written as:


d 
dx 
dy  udx  vdy
x
y
Since φ is constant along an equipotential line, we can
write;


d 
x
dx 
y
dy  udx  vdy  0
Which gives the gradient of the equipotential lines as
dy
u
 
dx
v
RELATIONSHIP BETWEEN VELOCITY
POTENTIAL & STREAM FUNCTION
Geometrical relationship
u
 dy 

 
v
Gradient of the equipotential lines  dx  
v
 dy 
Gradient of the streamline

 
u v
 dy   dy 
  .    .  1
v u
 dx    dx 
 dx 
This implies that streamlines intersect
equipotential lines at right angles
u
RELATIONSHIP BETWEEN VELOCITY
POTENTIAL & STREAM FUNCTION
Analytical relationship
For the velocity potential, the component of
velocities are given by:


u   ...........and .........v  
x
y
For the stream function, the component of
velocities are given by


v
...............and ...........u 
x
y
RELATIONSHIP BETWEEN VELOCITY
POTENTIAL & STREAM FUNCTION
Therefore
 
 

..............and ............. 

y x
x y
The above equations are known as the CauchyRiemann equations and they enable the stream
function to be calculated if the velocity potential
is known and vice versa. For example, if the
velocity potential ф is known, then
But the stream function is dψ=-vdx+udy


u
.....and ...v  
x
y
COMBIMING FLOW PATTERNS
If two or more flow patterns are combined,
the resultant flow pattern is described by a
stream function that at any point is the
algebraic sum of the stream functions of the
constituent flows at that point. By this
principle, any complicated fluid motion may
be considered as a combination of simple
flows.
Rectilinear (straight line) uniform
flows and their combination
The simplest flow patterns are those in which the
streamlines are all straight lines parallel to each
other. In analyzing the flow about solid bodies
immersed in a fluid stream, the approaching fluid
is assumed to be of an infinite extent and
possesses straight parallel streamlines and
uniform velocity distribution.
If the velocity of the rectilinear flow, v is inclined to
the x-axis at an angle α, then the components
are:
ux = v cos α; and uy =v sinα
Rectilinear (straight line) uniform
flows
Stream function equation
The stream function ψ(x,y) is:


d 
.dx 
.dy
x
y
d  u y dx  u x dy
    v sin dx   v cos  .dy  c
  vx. sin   vy cos   c
By choosing the reference streamline ψo =0 to pass through the origin, we can make the constant
go to zero and the stream function becomes:
Ψ = v(-xsinα +y cosα)
Velocity potential
The velocity potential, φ:


d  .dx  .dy
x
y
d  u x dx  u y dy  v cos  .dx  v sin  .dy
    v cos dx   v sin dy  c
  vx cos   vy sin   c
  v( x cos   y sin  )...by.choo sin g.  0 ..at.x  0;. y  0
Uniform, straight line flow in the Ox
direction with uniform velocity U in the xdirection.
Stream function for uniform
velocity U in the x-direction
For a straight line flow in the x-direction,
Ux=U; and uy =0
Let the stream function be ψ(x,y)
dψ = (δψ/δx).dx +(δψ/δy).dy;
dψ = -uy.dx +uxdy = 0 +Udy
Integrating
ψ(x,y) = Uy +c
Using the condition that ψ0 passes through the origin,
c then becomes zero and
ψ(x,y) = Uy
Velocity potential for uniform
velocity U in the x-direction
The velocity potential φ
dφ= (δφ/δx).dx +(δφ/δy).dy =
-ux.dx –uy.dy
dφ = -Udx +0
Φ = -Ux +c or
φ=-Ux
after making φ0 pass through the origin and
c=0.
Uniform straight line in the O(y)
direction
Stream function equation for
uniform straight line in the O(y)
direction
For this flow, ux=0; and uy =V
dψ=(δψ/δx).dx+(δψ/δy).dy =
-uydx +ux dy
dψ = -Vdx +0
Ψ(x,y) = -Vx +c
Ψ(x,y) =-Vx
Velocity potential equation for
uniform straight line in the O(y)
direction
The Velocity Potential φ.
dφ = (δδ/δx).dx +(δφ/δy).dy =
= -ux.dx-uy.dy=
dφ = -Vdy
Integrating
Φ= -Vy +c ↔↔↔φ = -Vy
Combination of streamlines
Combination of streamlines
Combined flow consisting of a uniform flow u = 2ms-1
along the Ox axis and uniform flowv = 4ms-1 along the
y-axis.
When the stream functions of a flow field are not known as a
function of x and y, the graphical approach is an alternative,
which may be used to combine the flow fields. The graphical
method to such problems uses the definition of the stream
function and considers the flow rate between streamlines and
the origin for both the individual and combined flow fields. For
the graphical solution, the stream functions for
the two flow fields are written as ψ1(x,y) Uy = 2y and
ψ2(x,y) = -Vx = -4x.
Combination of streamlines
Values of x and y are assigned and the corresponding
stream function values computed and plotted as shown
in the diagram. At the intersection of any two
streamlines, the stream function values are added
algebraically and the value put at the point of
intersection.
By joining points of the same value of stream functions,
we obtain streamlines of different stream function values.
The same results may be obtained by algebraically
summing the stream functions as:
Ψcomb = ψ1 +ψ2 = 2y -4x
Combination of streamlines
This equation represents a family of straight
lines, each line being assigned a definite
value of ψ; eg ψ=0; ψ=1; ψ=2; etc.
TRANSFORMATION OF POLAR
TO CATESIEAN
TRANSFORMATION OF POLAR
TO CATESIEAN
The velocity V is defined in the polar coordinates
by the distance r from the origin and the angle
θ the radius makes with the reference, which is
usually the horizontal. The velocity V can be
resolved in the polar coordinate as Vθ and Vr ie
the transverse and radial components of the
velocity V. The same velocity can also be
resolved into the x-y components as Vx and Vy
i.e the horizontal and vertical components
respectively.
TRANSFORMATION OF POLAR TO
CATESIEAN
It is clear the forgoing are valid.
x=r Cos θ;
dx/dr = Cos θ; dx/dθ = -r Sin θ
y = r Sin θ; dy/dr = Sin θ; dy/dθ = r Cos θ
Vr = Vx Cos θ + Vy Sin θ - -------Radial component of
velocity
Vθ = -Vx Sin θ + Vy Cos θ - -------Transverse
component of the velocity
Expressing Vr and Vθ in terms of the velocity potential
and the stream function.
Radial velocity in terms of the
velocity potential
d  dx  dy 




. cos  
. sin  
dr
x dr y dr x
y
 v x cos   v y sin   (v x cos   v y sin  )  v r
d
 v r
dr
d
vr  
dr
Transverse velocity in terms of
the velocity potential
d  dx  dy





r sin  
r cos  
d x d y d
x

 v x r sin   v y r cos   r v x sin   v y cos    rv
d
  rv
d
1 d
v  
r d
Radial velocity in terms of the
stream function
d  dx  dy





r sin  
r cos  
d
x d y 
x
y
 v y .r sin   v x .r cos   r v x cos   v y sin    rv r
d
 rv r
d
1 d
vr 
r d
Transverse velocity in
terms of stream function
d  dx  dy 




. cos  
sin  
dr
x dx y dr x
y
 v y . cos   v x . sin    v y cos   v x . sin    v
d
 v
dr
d
v  
dr
SOURCE & SINK
A source is a point in space from which fluid
issues uniformly in all directions at a constant
rate. For two-dimensional flow, the flow
pattern is made up of streamlines uniformly
spaced and directed radially outwards from one
point in the reference plane. Continuity
principles shows that the velocity will diminish
as the streamlines spread and the symmetry
will require that all velocities will be the same
at the same radial distance from the origin.
SOURCE
Across all circles, the same discharge will pass.
Therefore the velocity at any point r in the flow field
may be determined by the continuity equation as:
q=Vr x A = Vr x 2πrx1 considering unit depth.
Vr = q/2πr
Where q is the constant rate of flow per unit depth
issuing out of the source.
q – usually called the “strength” of the source per unit
depth of the source.
SOURCE
SOURCE
This implies that any circle drawn to enclose the
source will be discharging the same flow q. The
velocity at any point r in the flow field may be
determined as:
Vr = q/2πr; Vθ = 0
It is customarily for the stream function ψ =0 to be
made coincident with the x-axis. Since it is
convenient to express the source in the polar
coordinate, the stream function is written as a
function or r and θ i. e ψ(r, θ).
STREAM FUNCTION FOR
SOURCE


 
.dr 
d  v .dr  r.v r d
r

Integrating
 q 
 (r , )    v .dr   rv r d  0   r 
d  C
 2r 
q
 (r , ) 
 C
2
Choo sin g...boundary...conditions..such..that..when..  0 o ,...  0,..then..C  0
q
 (r , ) 
.
2
VELOCITY POTENTIAL FOR
SOURCE
d (r , ) 


.dr  .d
r

dφ(r,θ) = -vr.dr –r.vθ .dθ
Φ(r,θ) = ∫-(q/2πr).dr +∫0 + C
Φ(r,θ) = -(q/2π).lnr +c
choose lnr =0 when φ=0
Φ(r,θ) = -(q/2π).lnr
SINK

Sink: is the exact opposite of a source i.e. a
point in space to which fluid converges
uniformly and from which fluid is
continuously removed. As a result, a sink is
treated as a negative source flow and the
expression for velocities and the functions ψ
and φ are the same as those for a source but
with the signs reversed.
SINK
Consequently the stream function for a
sink is given by:
ψ(r,θ) = - (q/2π).θ
and the velocity potential of a sink given
as:
Φ(r,θ) = (q/2π).lnr
Combination of a source and
a uniform rectilinear flow
A uniform flow in the x- direction with a stream
function ψu = U.y and a source of uniform
strength of stream function ψs = (q/2π).θ
located at the origin O. When these flows are
brought together, the resulting stream function
is obtained by adding the respective stream
functions. i.e
ψcomb = U.y + (q/2π).θ
and since y= r. sin θ
Ψcomb =U.r.sinθ + (q/2π).θ
Graphical superposition of the
two streamlines
Graphical representation is obtained using
Rankine’s method by superposition of the
two streamlines. For example for the ψcomb =
8 is obtained by adding at the points of
intersections ψ1=8 + ψ0 =0; ψ1=7 + ψ2 =1;
ψ1=6 + ψ2 =2; ψ1=5 + ψ2 =3; ψ1=4 + ψ2
=4; ψ1=3 + ψ2 =5; ψ1=2 + ψ2 =7; ψ1=1 +
ψ2 =7; ψ1=0 + ψ2 =3; etc
Graphical superposition of the
two streamlines
All streamlines of the combined flow are obtained
in such manner. It must be observed that the
resulting streamlines are grouped into two
distinct sets. In one set, all streamlines emerge
from the origin (under ψ=8.0 streamline) and in
the other they approach the rectilinear
asymptotically at some distance upstream of
ψ=9. The two sets are separated by the
streamline ψ=8, which passes through the point
S.
Graphical superposition of
the two streamlines
This point is a stagnation point, where the
velocity from the source equals to the
uniform rectilinear velocity, so that the
resultant is zero at S. The distance OS =a
can be obtained by equating the uniform
velocity to that of the source at a radius a,
from the origin. Thus
U= q/2πa
Or a = q/2πU
Graphical superposition of the
two streamlines
Assignment: Plot graphically the combination
of the ff. streamlines. Uniform flow + source
Group 1 & 8: u=1m/s; q=8m3/s/m
Group 2 & 9: u=2
q=16
Group 3 & 10: u=1.5
q=6
Group 4 & 11: u= 2.5
q=12
Group 5 & 12: u=1.25
q=10
Group 6 & 13: u=1.75
q= 7.5
Group 7 & 14: u=2.75
q= 14
Assignment
Try also the combination of a source of
strength 8m3/s/m and a sink of equal
strength.