b.tech civil 4th semester notes

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SAQIB IMRAN 1
Assala mu alykum My Name is saqib imran and I am the
student of b.tech (civil) in sarhad univeristy of
science and technology peshawer.
I have written this notes by different websites and
some by self and prepare it for the student and also
for engineer who work on field to get some knowledge
from it.
I hope you all students may like it.
Remember me in your pray, allah bless me and all of
you friends.
If u have any confusion in this notes contact me on my
gmail id: Saqibimran43@gmail.com
or text me on 0341-7549889.
Saqib imran.

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Fluid Mechanics Notes
Fluid: A fluid is a substance which conforms continuously under the action of
shearing forces. OR A flowing is a substance which is capable of flowing.OR
A fluid is a substance which deforms continuously when subjected to external shearing stress.
Fluid Mechanics is that branch of science which deals with behaviour of the fluids at rest as well as in
motion.
Fluid: Fluids are substance which area capable of flowing and conforming the shapes of container.Fluids
can be in gas or liquid states.
Mechanics: Mechanics is the branch of science that deals with the state of rest or motion of body under
the action of forces.
Fluid Mechanics: Branch of mechanic that deals with the response or behavior of fluid either at rest or
in motion.
Branches of Fluid Mechanics
Fluid Statics: It is the branch of fluid mechanics which deals with the response/behavior of fluid when
they are at rest.
Fluid kinematics: It deals with the response of fluid when they are in motion without considering the
energies and forces in them.
Hydrodynamics: It deals with the behavior of fluids when they are in motion considering energies and
forces in them.
Hydraulics: It is the most important and practical/experimental branch of fluid mechanics which deals
with the behavior of water and other fluid either at rest or in motion.
Significance of Fluid Mechanics: Fluid is the most abundant available substance e.g., air, gases,
ocean, river and canal etc. It provides basis for other subjects e.g., Public health/environmental
engineering, Hydraulic Engineering, Irrigation Engineering, Coastal engineering, etc
What is fluid: A substance which deforms continuously under the action of shearing forces, however
small they may be. If a fluid is at rest, there can be no shearing forces
All forces in the fluid must be perpendicular to the planes upon which they act.
Solids & Fluids (liquids & gases).

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Matter exist in two principal forms: • Solid, • Fluids.
Fluids are further sub-divided into: • Liquid, • Gas.
For all practical purposes , the liquids and solids can be regarded as incompressible. This means that
pressure and temperature have practically no effect on them. Example, Water, Kerosene, petrol etc. But
Gases are readily compressible fluids. They expand infinitely in the absence of pressure and contract
easily under pressure. Example: air , ammonia etc.
YOU WOKE UP IN THE MORNING AND THE ROOM IS COOL. Coolant circulating Inside it and cool Air
which it gives is Fluid. After that you washed your face at the sink. The water which comes at your tap is
fluid and has come through the piping system which also comes under fluid mechanics. A mixture of fuel
like petrol and air is forced by atmospheric (or greater) pressure into the cylinder through the intake
port. All physical quantities are given by a few fundamental quantities or their combinations. The units
of such fundamental quantities are called base.
Units and dimensions: Units, combinations of them being called derived units. The system in which
length, mass and time are adopted as the basic quantities, and from which the units of other quantities
are derived, is called the absolute system of units.
Absolute system of units
MKS system of units: This is the system of units where the metre (m) is used for the unit of length,
kilogram (kg) for the unit of mass, and second (s) for the unit of time as the base units.
CGS system of units: This is the system of units where the centimetre (cm) is used for length, gram (g)
for mass, and second (s) for time as the base units.
International system of units (SI): SI, the abbreviation of La System International d’Unites, is the system
developed from the MKS system of units. It is a consistent and reasonable system of units which makes
it a rule to adopt only one unit for each of the various quantities used in such fields as science,
education and industry.
There are seven fundamental SI units, namely: metre (m) for length, kilogram (kg) for mass, second (s)
for time, ampere (A) for electric current, kelvin (K) for thermodynamic temperature, mole (mol) for mass
quantity and candela (cd) for intensity of light. Derived units consist of these units.
Dimension: All physical quantities are expressed in combinations of base units. The index
number of the combination of base units expressing a certain physical quantity is called the dimension,
as follows. In the absolute system of units the length, mass and time are respectively expressed by L, M
and T. Put Q as a certain physical quantity and c as a proportional constant, and assume that they are
expressed as follows: Q = cLà
Mß
T
where a, ß and  are respectively called the dimensions of Q for L, M, T.
Physical properties of Fluid
1: Density: The mass of a liquid per unit volume at standard temperature & pressure (STP) is called its
density. It is also termed as mass density or specific mass of the liquid. Thus
Density = ρ = Mass / Volume = M / V
2: Specific Weight: The weight of a liquid per unit volume at standard temperature & pressure (STP) is
called its Specific Weight. It is also termed as weight density of the liquid. Thus
Specific Weight = weight / Volume = W / V
3: Specific Volume: The volume of a liquid occupied by unit mass is called specific volume of the liquid.
Specific Volume = volume of liquid/mass of liquid OR V = 1/density = 1/ρ.

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4: Specific Gravity: The ratio of the specific weight of a liquid to that of the specific weight of the water
at standard temperature & pressure (STP) is called the Specific Gravity of the liquid.
It is also termed as Relative density of the liquid. Thus
Specific Gravity (sp.gr.) = S = sp. Weight of liquid/sp. Weight of water
5: Surface tension: When two liquids of different densities or when a liquid & a gas are in contact,
then the surface of contact will be in tension due to pressure difference due to cohesion which is called
surface tension.
6: Capillary Action: When a tube of small diameter open to the atmosphere is inserted in a liquid, the
liquid rises or falls inside the tube. This behaviour of the liquids is termed as Capillary Action of the
Liquid.
7: Compressibility: The reduction in volume of a liquid on increasing pressure, is called compressibility
of the liquid. The value of compressibility is so small that for all practical purposes it is neglected.
8: Viscosity: The property of a liquid which offers resistance to the movement of one layer of the liquid
over the over adjacent layer of the liquid is called Viscosity. Its unit is called poise &
1 poise = p = dyne – sec/cm2 or p = 1/10 N – sec/m2.
Units of Viscosity: N.s/m2 or kg/m/s
Mollases, tar, glycerine are highly viscous fluids.
Water, air, petrol have very small viscosity and are called thin fluids
Newton’s Law or equation of Viscosity
τ= µ (du/dy)
Where, du/dy = velocity gradient
µ= coefficient of viscosity, absolute viscosity or dynamic viscosity.
Measurement of VISCOSITY
The viscosity of a liquid is measured using a viscometer, and the best viscometers are those which are
able to create and control simple flow fields. The most widely measured viscosity is the shear viscosity,
and here we will concentrate on its measurement, although it should be noted that various extensional
viscosities can also be defined and attempts can be made to measure them, although this is not easy.
Most modern viscometers are computer- or microprocessor-controlled and perform automatic
calculations based on the particular geometry being used. We do not therefore need to go into a great
deal of discussion of calculation procedures, rather we will concentrate on general issues and artifacts
that intrude into measurements. Or
Viscosity is the measure of the internal friction of a fluid. This friction becomes apparent when a layer of
fluid is made to move in relation to another layer. The greater the friction, the greater the amount of
force required to cause this movement, which is called shear. Shearing occurs whenever the fluid is
physically moved or distributed, as in pouring, spreading, spraying, mixing, etc. Highly viscous fluids,
therefore, require more force to move than less viscous materials.

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Isaac Newton defined viscosity by considering the model represented in the figure above. Two parallel
planes of fluid of equal area A are separated by a distance dx and are moving in the same direction at
different velocities V1 and V2. Newton assumed that the force required to maintain this difference in
speed was proportional to the difference in speed through the liquid, or the velocity gradient. To
express this, Newton wrote:
The velocity gradient, dv/dx , is a measure of the change in speed at which the intermediate layers move
with respect to each other. It describes the shearing the liquid experiences and is thus called shear rate.
This will be symbolized as S in subsequent discussions. Its unit of measure is called the reciprocal second
(sec-1).
The term F/A indicates the force per unit area required to produce the shearing action. It is referred to
as shear stress and will be symbolized by F′. Its unit of measurement is dynes per square centimeter
(dynes/cm2).
Using these simplified terms, viscosity may be defined mathematically by this formula:
The fundamental unit of viscosity measurement is the poise. A material requiring a shear stress of one
dyne per square centimeter to produce a shear rate of one reciprocal second has a viscosity of one
poise, or 100 centipoise. You will encounter viscosity measurements expressed in Pascal-seconds (Pa·s)
or milli-Pascal-seconds (mPa·s); these are units of the International System and are sometimes used in
preference to the Metric designations. One Pascal-second is equal to ten poise; one milli-Pascal-second
is equal to one centipoise.
Newton assumed that all materials have, at a given temperature, a viscosity that is independent of the
shear rate. In other words, twice the force would move the fluid twice as fast. As we shall see, Newton
was only partly right.
NEWTONIAN FLUIDS: This type of flow behavior Newton assumed for all fluids is called, not
surprisingly, Newtonian. It is, however, only one of several types of flow behavior you may encounter. A
Newtonian fluid is represented graphically in the figure below. Graph A shows that the relationship
between shear stress (F′) and shear rate (S) is a straight line. Graph B shows that the fluid's viscosity
remains constant as the shear rate is varied. Typical Newtonian fluids include water and thin motor oils.

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What this means in practice is that at a given temperature the viscosity of a Newtonian fluid will remain
constant regardless of which Viscometer model, spindle or speed you use to measure it. Brookfield
Viscosity Standards are Newtonian within the range of shear rates generated by Brookfield equipment;
that's why they are usable with all our Viscometer models. Newtonians are obviously the easiest fluids
to measure - just grab your Viscometer and go to it. They are not, unfortunately, as common as that
much more complex group of fluids, the non-Newtonians, which will be discussed in the next section.
NON-NEWTONIAN FLUIDS: A non-Newtonian fluid is broadly defined as one for which the
relationship F′/S is not a constant. In other words, when the shear rate is varied, the shear stress doesn't
vary in the same proportion (or even necessarily in the same direction). The viscosity of such fluids will
therefore change as the shear rate is varied. Thus, the experimental parameters of Viscometer model,
spindle and speed all have an effect on the measured viscosity of a non-Newtonian fluid. This measured
viscosity is called the apparent viscosity of the fluid and is accurate only when explicit experimental
parameters are furnished and adhered to.
Non-Newtonian flow can be envisioned by thinking of any fluid as a mixture of molecules with different
shapes and sizes. As they pass by each other, as happens during flow, their size, shape, and cohesiveness
will determine how much force is required to move them. At each specific rate of shear, the alignment
may be different and more or less force may be required to maintain motion.
There are several types of non-Newtonian flow behavior, characterized by the way a fluid's viscosity
changes in response to variations in shear rate. The most common types of non-Newtonian fluids you
may encounter include:
Psuedoplastic: This type of fluid will display a decreasing viscosity with an increasing shear rate, as
shown in the figure below. Probably the most common of the non-Newtonian fluids, pseudo-plastics
include paints, emulsions, and dispersions of many types. This type of flow behavior is sometimes called
shear-thinning.

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Dilatant: Increasing viscosity with an increase in shear rate characterizes the dilatant fluid; see the figure
below. Although rarer than pseudoplasticity, dilatancy is frequently observed in fluids containing high
levels of deflocculated solids, such as clay slurries, candy compounds, corn starch in water, and
sand/water mixtures. Dilatancy is also referred to as shear-thickening flow behavior.
Plastic: This type of fluid will behave as a solid under static conditions. A certain amount of force must
be applied to the fluid before any flow is induced; this force is called the yield value. Tomato catsup is a
good example of this type fluid; its yield value will often make it refuse to pour from the bottle until the
bottle is shaken or struck, allowing the catsup to gush freely. Once the yield value is exceeded and flow
begins, plastic fluids may display Newtonian, pseudoplastic, or dilatant flow characteristics. See the
figure below.
So far we have only discussed the effect of shear rate on non-Newtonian fluids. What happens when the
element of time is considered? This question leads us to the examination of two more types of non-
Newtonian flow: thixotropic and rheopectic.

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THIXOTROPY AND RHEOPEXY : Some fluids will display a change in viscosity with time under conditions
of constant shear rate. There are two categories to consider:
Thixotropy: As shown in the figure below, a thixotropic fluid undergoes a decrease in viscosity with
time, while it is subjected to constant shearing.
Rheopexy: This is essentially the opposite of thixotropic behavior, in that the fluid's viscosity increases
with time as it is sheared at a constant rate. See the figure below.
Both thixotropy and rheopexy may occur in combination with any of the previously discussed flow
behaviors, or only at certain shear rates. The time element is extremely variable; under conditions of
constant shear, some fluids will reach their final viscosity value in a few seconds, while others may take
up to several days.
Rheopectic fluids are rarely encountered. Thixotropy, however, is frequently observed in materials such
as greases, heavy printing inks, and paints.
When subjected to varying rates of shear, a thixotropic fluid will react as illustrated in the figure below.
A plot of shear stress versus shear rate was made as the shear rate was increased to a certain value,
then immediately decreased to the starting point. Note that the up and down curves do not coincide.
This hysteresis loop is caused by the decrease in the fluid's viscosity with increasing time of shearing.
Such effects may or may not be reversible; some thixotropic fluids, if allowed to stand undisturbed for a
while, will regain their initial viscosity, while others never will.
The rheological behavior of a fluid can, of course, have a profound effect on viscosity measurement
technique. Later we will discuss some of these effects and ways of dealing with them.
Following Observations can be
made from Newton’s viscosity
Equation:
• Max. shear stress occur when velocity gradient is largest and shear stress disappears where velocity
gradient is zero.
• Velocity Gradient becomes small with distance from the boundary. Consequently the max value of
shear stress occurs at the boundary and it decreases from the boundary.
Consider fluids are full of two parallel walls. A shear stress, τ, is applied
to the upper wall. Fluids are deformed continuously because fluids cannot support shear stresses. The
deformation rate, however, is constant.
Furthermore, if the deformation rate or the so-called rate of strain is proportional to the shear stress,
then the fluid will be classified as a Newtonian
fluid, i.e. τ ∝ dγ / dt , where γ is shear angle or
τ = µ dγ / dt . In addition, dγ / dt = du / dy . Hence, τ = µ du / dy .

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Again, the relationship between shear stress acting on a Newtonian fluid
and rate of strain (or velocity gradient) is linear. If it is not linear, then
1.9 Speed of sound · 9 · the fluid will be called a non-Newtonian fluid. µ is the so-called dynamic
viscosity. Its units are dyne· cms 2 or Poise (cP).
Hydrostatics: that studies the mechanics of fluids at absolute and relative rest. or
The study of pressure exerted by liquids at rest is termed as hydrostatics.
Kinematics: deals with translation, rotation and deformation of fluid without considering the force and
energy causing such a motion.
Hydrodynamics: that prescribes the relation between velocities and acceleration and the forces which
are exerted by or upon the moving fluids. Or
Study of flowing liquids & forces causing their motion is called as hydrodynamics.
Hydraulics: The engineering science of liquid pressure and flow. Hydraulic engineering is the Science of
water in motion and its interactions with the surrounding environment. Water plays a major role in
human perception of the environment because it is an indispensable element.
The term 'Hydraulics' is related to the application of the Fluid Mechanics principles to water engineering
structures, civil and environmental engineering facilities: e.g., canal, river, dam, reservoir, water
treatment plant. Hydraulic engineering is the science of water in motion, and the interactions between
the flowing fluid and the surrounding environment. Hydraulic engineers are concerned with
application of the basic principles of fluid mechanics to open channel flows and real fluid flow
hydrodynamics. Examples of open channels are natural streams and rivers. Man-made channels include
irrigation and navigation canals, drainage ditches, sewer and culvert pipes running partially full, and
spillways.
Fluid Statics
Pressure: The perpendicular force exerted by a fluid per unit area. P = P/A.
Pressure Intensity: The force exerted by the liquid on the unit area of bottom & the sides of the vessel is
called intensity of pressure.
Pressure Head: Pressure in fluids may arise from many sources, for example pumps, gravity,
momentum etc. Since p = ρgh, a height of liquid column can be associated with the
pressure p arising from such sources. This height, h, is known as the pressure head.
The vertical distance (in feet) equal to the pressure (in psi) at a specific point. The pressure head is equal
to the pressure in psi times 2.31 ft/psi.
Absolute pressure: It is the pressure equal to the algebraic sum of the atmospheric and gauge
pressures. Absolute pressure = Gauge pressure + Atmospheric pressure
PA = PG + Patm
The pressure that exists anywhere in the universe is called the absolute pressure, Pabs.
This then is the amount of pressure greater than a pure vacuum. The atmosphere on
earth exerts atmospheric pressure, Patm , on everything in it. Often when measuring
pressures we will calibrate the instrument to read zero in the open air. Any measured
pressure, Pmeas , is then a positive or negative deviation from atmospheric pressure.
We call such deviations a gauge pressure, Pgauge . Sometimes when a gauge pressure
is negative it is termed a vacuum pressure, Pvac. In gauge pressure, a pressure under 1 atmospheric
pressure is expressed as a negative pressure. Or

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Absolute pressure: is defined as the pressure which is measured with reference to
absolute vacuum pressure.
2. Gauge pressure: is defined as the pressure which is measured with the help of a
pressure measuring instrument, in which the atmospheric pressure is taken as
datum. The atmospheric pressure on the scale is marked as zero.
3. Vacuum pressure: is defined as the pressure below the atmospheric pressure.
Note. (i) The atmospheric pressure at sea level at 15°C is 101.3 kN/m2 or 10.13 N/cm2 (ii) The
atmospheric pressure head is 760 mm of mercury or 10.33 m of water.
Measurement of pressure:
Manometers: A manometer (or liquid gauge) is a pressure measurement device which uses the
relationship between pressure and head to give readings. Or
A device which measures the fluid pressure by the height of a liquid column
is called a manometer.
Piezometer: This is the simplest gauge. A small vertical tube is connected to the pipe and its top is
left open to the atmosphere.
What is the relationship between pressure and specific weight?
Pressure varies with height as a function of specific weight.
P = p0 + specific weight * height
Where height is the distance below the reference pressure p0 (usually at a free surface).
What is the relationship between pressure and volume?
For a fixed amount of an ideal gas kept at a fixed temperature, P [pressure] and V [volume] are inversely
proportional (while one increases, the other decreases). As pressure increases and the density increases,
the relationship becomes a bit more complex. Increasing pressure will still decrease the volume but it
becomes less proportional. If you are at a temperature below the critical point, at some point the
pressure will become high enough to cause condensation of a gas to a liquid, or if you are cold enough,
the precipitation of the gas as a solid (the reverse of sublimation). In these cases the relationship
between pressure and volume has a discontinuity as the phase change occurs at constant pressure.
What is the relationship between temperature and pressure?
The relation between temperature and pressure is known as Gay-Lussac's law, one of the gas laws. It
states that the pressure exerted on a container's sides by an ideal gas is proportional to the absolute
temperature of the gas. As an equation this is P=kT In words as the pressure in sealed container goes up,
the temperature goes up, or as temperature goes up pressure goes up. .
What is the relationship between mass and weight?
An object's mass is the quantity of matter that comprises it ...the total protons, neutrons, electrons, lint,
moisture, dirt, wood-chips, and anythingelse of which the object is composed. It belongs to the object,
and doesn'tdepend on where the object is or in what position it is, etc. An object's weight is the
gravitational force between the objectand any other mass. That force depends on both the object's
mass and the other mass,and also on how far apart they are. An object's weight is its mass multiplied by
the acceleration dueto gravity in the place where the object is located at the moment ... so it can
change. For example, your weight would be F W =(your mass inkg)*(9.80m/s 2 ) because 9.80m/s 2 is
the acceleration due to gravity on Earth.
What is the relationship between pressure and temperature?
they increase together well actually they dont increase together they build up holding each other up
while increasing

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What is the relationship between volume and weight?
The relation between weight and volume -: When the weight of a substance increases, its volume also
increases. Two substances may have the same weight but different volumes. (Example: If you have one
stack of cotton and iron each of the same weight, they will have different volumes. Volume of cotton >
Volume of iron in this case.) Density = Weight/Volume.
What is the relationship between the temperature and pressure of a gas?
Put one more quantity in there and you've got a relationship: the volume of the gas. The product of (
pressure x volume ) is directly proportional to the temperature . Remember that in this relationship, its
the absolute temperature ... the temperature above absolute zero. That makes a difference. On the
absolute scale, the boiling temperature of water is only about 37% higher than the freezing temperature
of water.
What is the relationship between density and specific gravity?
There is a very great relationship between density and specific gravity. Density contributes to the weight
of a substance under specific gravity.
What is the relationships between force and pressure?
Archimedes a Greek mathematician who lived in third century, dicovered how to determine buoyant
force. . Archimedes' principle states that the buoyant force on an object in a fluid is an upward force
equal to the weight of the volume of fluid that the object displaces. . Buoyant force is the upward force
that keeps an object immersed in or floating on a liquid.
What is the relationship between pressure and wind?
pressure = 0.002558 times velocity squared where velocity is miles per hour and pressure is pounds per
square foot for example a wind of 75 mph produces a pressure of 0.002558x75x75 = 14 .39 pounds per
square foot since there are 144 sq in in one sq ft that is 14.39/144 = 0.1 pounds per square inch In
meteorological terms, differences in pressure are what drive wind. Air generally moves toward an area
of low pressure. However, due to the rotation of the earth it gets deflected in large scale weather
patterns. It is deflected to the right in the northern hemisphere and to the left in the southern
hemisphere.
What is the relationship between weight and mass?
Mass is the amount of matter in an object, while weight is the gravitational force applied to an object.
Mass is a function of weight since weight it determined by the amount of force placed on an object of a
certain mass.
Relationship between volume and pressure?
That depends on the substance. In ideal gases, volume is inversely proportional to pressure. That is,
twice the pressure means half the volume. Commonly, real gases are similar to an "ideal gas". Liquids
and solids hardly change their volume if the pressure changes. what is the relationship between the
volume of air and pressure consider some area(some volume) containing some air molecule, if we are
reducing the area of container(ie,volume) keeping the air molecule donot change in
concentration/amount. then we can say that now the presure is larger than first case. 42, the answer is
always 42 For a fixed amount of an ideal gas kept at a fixed temperature, P [pressure] and V [volume]
are inversely proportional (while one increases, the other decreases). As pressure increases and the
density increases, the relationship becomes a bit more complex. Increasing pressure will still decrease
the volume but it becomes less proportional. If you are at a temperature below the critical point, at
some point the pressure will become high enough to cause condensation of a gas to a liquid, or if you
are cold enough, the precipitation of the gas as a solid (the reverse of sublimation). In these cases the

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relationship between pressure and volume has a discontinuity as the phase change occurs at constant
pressure.
What is the Relationship between weight and speed?
Think about it as a toy car on a wooden track. The more the car weighs, the more friction between the
car and track. Therefore, reducing speed(b/c of friction). Hope this helps!
What's the relationship between Specific retention and specific yield?
Both Specific retention and specific yield relate to the ratio of the volume of water (in a permeable unit
of rock and/or sediment) to the total volume of the rock and/or sediment, as it relates to gravity..
Specific retention is the ratio of the volume of water that is RETAINED against the pull of gravity,
...where-as specific yield is the ratio of the volume of water that is EXPELLED (yielded) against the pull of
gravity. Again, ...both as a ratio to the total volume of the rock and/or sediment.
What is the relationship between weight and force?
The weight of an object represents the magnitude of the gravitational force exerted on the object by the
planet, less the effect of immersion in any fluid.
What is the relationship between pressure and depth?
Pressure increases with depth. The formula for pressure is P= Ï•*g*h+Pa where Ï• (the Greek letter Rho)
is the density of the fluid, g is the acceleration of gravity, h is the depth from the fluid surface and Pa is
the pressure at the surface of the fluid. every foot a diver decends you get about 1/2 lb. of pressure. So
at100 foot divide the pressure by 2 and that's approximately thepressure. At sea level the pressure is
14.7 psi. Go down to 33 feetand you have another 14.7 psi. In fresh water it's 34 feet to get
1atmosphere.
Is there a relationship between mass and weight?
Yes there is. Mass is the amount of matter in an object. Weight is the gravitational force exerted on an
object by the larger object on which it rests. Said another way, weight is mass in a gravimetric field. . The
force is given by f = G m 1 m 2 / d 2 where . G is the universal gravitational constant . m 1 is the mass of
one the objects . m 2 is the mass of the other object . d is the distance between the centers of mass of
the two objects. Notice that this formula is symmetric; the force on the larger body by the smaller is
identical to the force exerted on the smaller body by the larger. Back to Newton's third law - action and
reaction are equal and opposite. Notice also that because of the way in which the units were chosen, on
the surface of our earth mass and weight have the same value. A 100Kg mass weighs 100Kg on the
earth's surface . Take it to the moon; it will still have a mass of 100Kg, but weigh only about 15Kg. N.B. In
reference to the above, technically, mass is measured in kilograms, but weight in Newtons. So a 100kg
mass would still have a mass of 100kg on the moon, but its weight on both surfaces should be measured
in Newtons. In everyday use, people use kilograms to describe weight without realising they are actually
talking about the mass of an object.
What is the relationship between pressure and heat?
Heat is the movement of energy in response to a difference in temperature. Heat flows in a direction
from high to low temperature, and has the effect of tending to equalize the temperatures of the objects
in thermal contact. Thus the flow of heat may raise the temperature of one object while lowering the
temperature of the other.
what is the relationship between pressure and temperature?
This in itself is still an ill-posed question (it depends on what is held fixed, e.g., the volume, while the
temperature is changed), but in a general sense the pressure will increase with temperature (although
there are notable exceptions, such as water near freezing).

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Is there a relationship between boiling and pressure?
Yes, there is. Higher pressure increases the boiling point and lower pressure decreases it. That is why a
pressure cooker works and why water boils at lower temperatures in high altitudes.
What is the relationship between weight and density?
Weight is pounds in how fat or skinny you are and density is how your stomach works.
What are the relationships between weight and density?
Given an unchanging volume, if you lower the density you will lower the weight, and the revers is true. if
you lower the weight the the density would lower as well. This applies to any gravitational field if you
are measuring density as a function of weight per volume.
Relationship between liquid pressure and density?
If you were submerged in a liquid more dense than water, the pressure would be correspondingly
greater. The pressure due to a liquid is precisely equal to the product of weight density and depth. liquid
pressure = weight density x depth. also the pressure a liquid exerts against the sides and bottom of a
container depends on the density and the depth of the liquid.
What is the relationship between thrust and pressure?
thrust and pressure are directly proportional 2 each other from d formula pressure =perpendicular force
/area
What is the relationship between force area and pressure?
pressure = force / area Therefore pressure and force are directly proportional, meaning...The greater the
force the greater the pressure and the lower the force the lower the pressure
What is the relationship between ocean depth and pressure?
The pressure (force per cm 2 ) at a particular depth is the weight of water above that square centimetre.
What is the relationship between weight and capacity?
None really. If sent to the International Space Station, objects would have no weight but concave ones
would have some capacity. Those same objects, back on the surface of the earth would have some
weight but the same capacity as before. In stronger gravitational fields, the weight would continue to
increase but there would be no change in the capacity.
Pressure Transducer: A pressure transducer, often called a pressure transmitter, is a transducer
that converts pressure into an analog electrical signal. Although there are various types of pressure
transducers, one of the most common is the strain-gage base transducer.
The conversion of pressure into an electrical signal is achieved by the physical deformation of strain
gages which are bonded into the diaphragm of the pressure transducer and wired into a wheatstone
bridge configuration. Pressure applied to the pressure transducer produces a deflection of the
diaphragm which introduces strain to the gages. The strain will produce an electrical resistance change
proportional to the pressure.
DIFFERENTIAL MANOMETER: A device which is used to measure difference of pressure between the
two fluids which are flowing through the two different pipes or in same pipe at two different points is
known as DIFFERENTIAL MANOMETER.
TYPES OF DIFFERENTIAL MANOMETERS: There are two types of differential manometer as given
below:-
1] U-Tube Differential Manometer
2] Inverted U-Tube Differential Manometer.
U-TUBE DIFFERENTIAL MANOMETER: There are two types of U-Tube Differential Manometer :-

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A] U-Tube Differential Manometer at the same level.
B] U-Tube Differential Manometer at the different level.
A] U-Tube Differential Manometer at the same level: In this type of Manometer, two pipes are in
parallel condition. This type of Manometers are used for measuring the fluid pressure difference
between these two pipes.
B] U-Tube Differential Manometer at the different level: In this case this type of manometer are used
where two pipes are at different place, not in parallel condition. This type of manometers are used for
measuring the fluid pressure between these two pipes.
2] INVERTED U-TUBE DIFFERENTIAL MANOMETER: The inverted U-Tube Differential manometer is
reciprocal of U-Tube Differential manometer at the different level. This type of manometers are used to
measure accuracy of small difference if pressure is increased.
Bourdon gauge: A pressure gauge employing a coiled metallic tube which tends to straighten out when
pressure is exerted within it. Or The Bourdon pressure gauge uses the principle that a flattened tube
tends to straighten or regain its circular form in cross-section when pressurized. This change in cross-
section may be hardly noticeable, involving moderate stresses within the elastic range of easily workable
materials. The strain of the material of the tube is magnified by forming the tube into a C shape or even
a helix, such that the entire tube tends to straighten out or uncoil elastically as it is pressurized. Eugène
Bourdon patented his gauge in France in 1849, and it was widely adopted because of its superior
sensitivity, linearity, and accuracy; Edward Ashcroft purchased Bourdon's American patent rights in 1852
and became a major manufacturer of gauges. Also in 1849, Bernard Schaeffer in Magdeburg, Germany
patented a successful diaphragm (see below) pressure gauge, which, together with the Bourdon gauge,
revolutionized pressure measurement in industry. But in 1875 after Bourdon's patents expired, his
company Schaeffer and Budenberg also manufactured Bourdon tube gauges.
Pressure Measurement Devices
• Bourdon tube: Consists of a hollow metal tube bent like a hook whose end is closed and connected to
a dial indicator needle.
• Pressure transducers: Use various techniques to convert the pressure effect to an electrical effect
such as a change in voltage, resistance, or capacitance. • Pressure transducers are smaller and faster,
and they can be more sensitive, reliable, and precise than their mechanical counterparts.
• Strain-gage pressure transducers: Work by having a diaphragm deflect between two chambers open
to the pressure inputs.
• Piezoelectric transducers: Also called solid state pressure transducers, work on the principle that an
electric potential is generated in a crystalline substance when it is subjected to mechanical pressure.
Forces on immersed bodies
Force on a Submerged Surface: On any surface or body that is submerged in water or any other
liquid, there is a force acting because of the hydrostatic pressure. Learn how to determine the
magnitude of this force. Study of hydrostatic forces on submerged or static surfaces is very important
for the design and engineering processes. Construction of dams, installation of underwater hydraulic
systems, and forces exerted on ships are some of the important and crucial processes that require study
of hydrostatic forces.
Forces on planar surfaces: If the surface is planar, a single resultant point force is found, mechanically
equivalent to the distributed pressure load over the whole surface.

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This resultant point force acts compressively, normal to the surface, through a point termed the “center
of pressure".
Its magnitude is: F=γzkA, where:
γ is the fluid's specific gravity. For water, it is 9810 N/m3.
zk is the depth in which the center of gravity of the surface, the centroid, is situated.
A is the surface’s area.
The product (γ.zk), is the hydrostatic pressure at the depth of the centroid of the surface. In case the
free surface of the liquid that contains the surface is under atmospheric pressure alone, the above
equation is enough to describe the force. But in case the free surface is under additional pressure, this
pressure will have an additional effect on the acting force. The value of the pressure in the center of
gravity of the surface, is no longer (γ.zk). It is now γ(zk+p/γ), where p is the above-mentioned pressure.
Calculating the magnitude of the force is done as described above. The determination of the point
where this force applies, the “center of pressure," is a little more complicated:
If the surface is inclined at an angle, θ, to the horizontal, the coordinates of the center of pressure, (xcp,
ycp), in a coordinate system in the plane of the surface, with origin at the centroid of the surface, are:
xcp = Ixy/(ykA) and ycp = Ixx/(ykA)
where Ixx is the area moment of inertia, Ixy the product of inertia of the plane surface, both with
respect to the centroid of the surface, and y is positive in the direction below the centroid.
The surface is often symmetrically loaded, so that Ixx = 0, and hence, xcp = 0, or the center of pressure is
located directly below the centroid on the line of symmetry.
If the surface is horizontal, the center of pressure coincides with the centroid. Further, as the surface
becomes more deeply submerged, the center of pressure approaches the centroid, that is, (xcp, ycp)
approaches to (0,0).
Forces on curved surfaces: For general curved surfaces, it may no longer be possible to determine a
single resultant force equivalent to the hydrostatic load; we thus determine separately one or two
horizontal components, and a vertical component. The horizontal component of the force acting on a
curved surface is equal with the force that would be acting on a planar surface. This planar surface is the
projection of the curved surface on the vertical level. For example in figure 2, where one sees reservoir
ABCDEGJPA, the horizontal components with which water pushes surfaces BC and DE, are F2 and
F3respectively. To calculate the magnitude of F2, all we need to do is consider KC, which is the
projection of BC on the vertical plane. By determining the force to KC, we have F2. The same holds for
F3. It is equal to the force on surface MD. The vertical component of the force is equal to the weight of
the volume of liquid that exists between the surface, and the free surface of the liquid. And this is true
whether there is a free surface or not. By this we mean that if the liquid is above the surface, the
postulation is true. In this case, the force is directed downwards. In the opposite case, where, the liquid
is situated below the surface, the same volume counts. The magnitude of the force is still equal to the
weight of the same liquid volume. Only, this volume is now imaginary. The direction of the force in this
case is the inverse: upwards. To express this in another way, if the surface is exposed to the hydrostatic
load from above, like the surface BC in Fig. 2, then the force acts downwards. The magnitude of F1 is
equal to the weight of the volume PBCJP. If the surface is exposed to a hydrostatic load from below, like
the surface DE, then the force acts upwards. And the magnitude of F4 is equal to the weight of the
volume QEDLQ of water. It acts through the center of gravity of that fluid volume. In case a surface is
such that there are both upward and downward vertical components, like surface EDC, the net vertical
force on the surface is the algebraic sum of upward and downward components. For general curved

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surfaces, it may no longer be possible to determine a single resultant force equivalent to the hydrostatic
load; we thus determine separately one or two horizontal components, and a vertical component.
surface is equal with the force that would be acting on a planar surface. This planar surface is the
projection of the curved surface on the vertical level. For example in figure 2, where one sees reservoir
ABCDEGJPA, the horizontal components with which water pushes surfaces BC and DE, are F2 and
F3 respectively. To calculate the magnitude of F2, all we need to do is consider KC, which is the
projection of BC on the vertical plane. By determining the force to KC, we have F2. The same holds for
F3. It is equal to the force on surface MD. The vertical component of the force acting on a curved surface
has a magnitude that is defined as “the weight of the volume of water, or liquid in general, that exists
above the surface, and under the free surface." And this is true, whether there is a free surface, or not.
By this we mean that, if the liquid is above the surface, the postulation is true. In this case, the force is
directed downwards. In the opposite case, where, the liquid is situated below the surface, the same
volume counts. The magnitude of the force is still equal to the weight of the same liquid volume. Only,
this volume is now imaginary. The direction of the force in this case is the inverse: upwards.
To express this in another way, if the surface is exposed to the hydrostatic load from above, like the
surface BC in Fig. 2, then the force acts downwards. The magnitude of F1 is equal to the weight of the
volume PBCJP. If the surface is exposed to a hydrostatic load from below, like the surface DE, then the
force acts upwards. And the magnitude of F4 is equal to the weight of the volume QEDLQ of water. It
acts through the center of gravity of that fluid volume. In case a surface is such that there are both
upward and downward vertical components, like surface EDC, the net vertical force on the surface is the
algebraic sum of upward and downward components. Or
FORCES ON SUBMERGED SURFACES
1: Fluid pressure on a Surface: Pressure is defined as force per unit area. If a pressure p acts on a small
area then the force exerted on that area will be, Since the fluid is at rest the force will act at right-angles
to the surface.
General submerged plane: Consider the plane surface shown in the figure below. The total area is made
up of many elemental areas. The force on each elemental area is always normal to the surface but, in
general, each force is of different magnitude as the pressure usually varies.
We can find the total or resultant force, R, on the plane by summing up all of the forces on the small
elements i.e. This resultant force will act through the centre of pressure, hence we can say
If the surface is a plane the force can be represented by one single resultant force,
acting at right-angles to the plane through the centre of pressure.
Horizontal submerged plane: For a horizontal plane submerged in a liquid (or a plane experiencing
uniform pressure over its surface), the pressure, p, will be equal at all points of the surface. Thus the
resultant force will be given by
Curved submerged surface: If the surface is curved, each elemental force will be a different magnitude
and in different direction but still normal to the surface of that element. The resultant force can be
found by resolving all forces into orthogonal co-ordinate directions to obtain its magnitude and
direction. This will always be less than the sum of the individual forces.
Forces On Plane And Curved Surfaces
Hydrostatic force: Hydrostatic force refers to the total pressure acting on the layer or surface which is in
touch with the liquid or water at rest. If the liquid is at rest then there is no tangential force, and hence
the total pressure will act perpendicular to the surface with contact.

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Center of pressure: The location of total pressure is referred as the center of pressure which is always
below the center of gravity of the surface in contact.
Forces on the horizontal planes: Show the element submerged in the liquid distance (h) from the liquid
surface as in
Figure (1).
Express the forces on the horizontal plane.
Forces on the vertical planes Show the elemental strip of surface area located at x from the free liquid
surface as in Figure (2).
Express the pressure intensity at the elemental surface.
Express the total pressure on the plane.
Consider the number of elemental strips and applying the integration to get total hydrostatic force.

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Therefore, the total pressure is expressed as,
Here, is the moment of total area of contact about free water surface. i.e., the product of the total
area and the distance between free water surface and center of gravity of the contact area. Therefore,
Forces on the curved surface: For forces on the curved surface, there will be two forces required to
determine the resultant hydrostatic force.
Horizontal force on curved surface: The vertical plane shall be considered to determine the horizontal
force, which is the vertical projection of the curved surface generally rectangle. But in case of
hemispherical or spherical, it becomes circular shape.
Express the horizontal component of force.
Vertical force on curved surface: It is the weight of the liquid acting on the curved surface in contact
with the liquid which may be in upward direction due to buoyancy or downward direction due to the
weight of the fluid.
Express the vertical component of force.
Therefore, the resultant force on the curved surface is,
Drag & Lift Forces
Drag Force: The drag force acts in a direction that is opposite of the relative flow velocity. – Affected by
cross-section area (form drag) – Affected by surface smoothness (surface drag). Or
The drag force acting on a body in fluid flow can be calculated
FD = cD 1/2 ρ v2 A
Where, FD = drag force (N), cD = drag coefficient, ρ = density of fluid (kg/m3),
v = flow velocity (m/s), A = body area (m2). or
Drag: Resistive force acting on a body moving through a fluid (air or water). Two types:
Surface drag: depends mainly on smoothness of surface of the object moving through the fluid.
• shaving the body in swimming; wearing racing suits in skiing and speed skating.
Form drag: depends mainly on the cross-sectional area of the body presented to the fluid
• bicyclist in upright v. crouched position
• swimmer: related to buoyancy and how high the body sits in the water.
Lift Force: The lift force acts in a direction that is perpendicular to the relative flow. – The lift force is not
necessarily vertical. Or
The lifting force acting on a body in a fluid flow can be calculated
FL = 1/2 cL ρ v2 A
Where, FL = lifting force (N), cL = lifting coefficient, ρ = density of fluid (kg/m3),
v = flow velocity (m/s), A = body area (m2). Or
Lift Force: Represents a net force that acts perpendicular to the direction of the relative motion of the
fluid;
• Created by different pressures on opposite sides of an object due to fluid flow past the object,

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– example: Airplane wing (hydrofoil)
• Bernoulli’s principle: velocity is inversely proportional to pressure.
– Fast relative velocity lower pressure
– Slow relative velocity higher pressure
Buoyancy: Associated with how well a body floats or how hight it sits in the fluid.
• Archimede’s principle: any body in a fluid medium will experience a buoyant force equal to the weight
of the volume of fluid which is displaced.
Example: a boat on a lake. A portion of the boat is submerged and displaces a given volume of water.
The weight of this displaced water equals the magnitude of the buoyant force acting on the boat.
– The boat will float if its weight in air is less than or equal to the weight of an equal volume of water.
• Buoyancy is closely related to the concept of density.
Density = mass/volume
Buoyancy And Floatation: The first dash point under fluid mechanics is flotation, centre of buoyancy.
These two concepts are put together because floatation is caused by a force known as buoyancy. For an
object to float in water it must be less dense (mass per unit of volume) than the water.
When an object is placed in water it causes the water to be displaced (move upwards). This can be seen
when a person gets into a bath and the water rises. If the bath is filled to the very brim, then when the
person gets into the bath the water that is displaced will pour out of the tub. In order for an object to
float, the water they displace must weight more than they do.
In order for an object to float, the water they displace must weight more than they do. This is essentially
because gravity is seeking to push the water that has been displaced, back down, while also pushing the
person down. If the gravitational force on the water is greater than the force on the object, then the
water will create a buoyant force that will push the object upwards against gravity. Once the two forces
become equal the object will float in this position known as the point of equilibrium. That is the part of
the object below the water has displaced the same weight of water, as the object itself, resulting in a
bouyancy force equal to that of the gravity force acting on the object.
The centre of buoyancy is the centre point of the mass below the water and is the point through which
the buoyant force acts. This is much like the centre of gravity – the point through which gravity acts, but
buoyant forces acts in the opposite direction. In order for the object to not rotate in the water this
buoyant force must pass through the center of mass of the object, if they do not line up the object will
rotate until they do, such that one end of the object will sink further while the other end raises (as seen
in the images to the right).
For an object to have less gravitational force than the water it displaces it must be less dense (mass per
unit of volume) than the water. Not all water has the same density though. Salt water is more dense
than fresh water, and the saltier it is the larger the density. This means that it is easier to flow in the
ocean than it is in a pool.
Flotation and centre of buoyancy relate to performance because the higher an object floats in the water,
the less resistance the water will create to its movement. This applies to all water sports, including:
surfing, kayaking, sailing, skiing, dragon boat racing, water polo, synchronised swimming, and swimming.
These forces also relate to scuba diving, where the person is seeking to sink below the water and remain
submerged. In this instance, the person, with their gear, wants to be the same density as the water in
order to allow them to remain submerged easily, but not have to fight too hard to return to the surface.
This is often achieved using weight belts.

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Buoyancy And Floatation: When a body is immersed in fluid, an upward force is exerted by the fluid
on the body. This upward force is equal to the weight of the fluid displaced by the body and is called the
force of buoyancy.
Causes buoyant force: • Buoyant force is the force on an object exerted by the surrounding fluid.
• When an object pushes water, the water pushes back with as much force as it can.
• If the water can push back as hard, the object floats (boat). If not, it sinks (steel).
Forces Acting on Buoyancy: The buoyant force is caused by the difference between the pressure at
the top of the object (gravitational force), which pushes it downward, and the pressure at the bottom
(buoyant force), which pushes it upward.
• Since the pressure at the bottom is always greater than at the top, every object submerged in a fluid
feels an upward buoyant force.
Buoyancy= “the floating force”: Water is “heavier” than the object…so the object floats
• Low density-more likely to float, • Buoyant force is measured in Newtons (N).
How do you Calculate BF? Buoyant Force = Weight of displaced fluid OR BF = Wair – Wwater
Buoyant Force = Weight of object in air - Weight of object in water.
Floatation: Why do things float?
1. Things float if they are less dense than the fluid they are in.
2. Things float if they weigh less than the buoyant force pushing up on them.
3. Things float if they are shaped so their weight is spread out.
Condition of equilibrium of a floating and sub-merged bodies
Positive buoyancy: Buoyant force is greater than weight so the object floats.
Neutral buoyancy: Buoyant force is equal to weight so the object is suspended in the fluid.
Negative buoyancy: Buoyant force is less than weight so the object sinks.
A ship made of iron floats while an iron needle sinks.
• In the case of ship which is hollow from within, the weight of water displaced by the ship is more than
the weight of the ship hence it floats.
• Incase of iron nail which is compact, the weight of water displaced by it is much less than its own
weight, hence it sinks.
A person weighs 250N while swimming in the dead sea. When outside of the water they weigh
600N. What is the buoyant force acting on them? Will they sink or float?
• BF = Wair – Wwater = 600 – 250 = 350N
• The person will float because their weight in the water is less than the buoyant force.
Centre of Buoyancy: The point through which the force of buoyancy is supposed to act is known as
Centre of Buoyancy.
META-CENTRE: It is defined as the point about which a body starts oscillating when the body is tilted by
a small angle.
• It is the point at which the line of action of the force of buoyancy will meet the normal axis of the body
when the body is given small angular displacement.
Meta-centric Height: It is the distance between the meta-centre of floating body and centre of gravity.
• We can find this height by two methods:-
1. Analytical Method GM I/  BG, Here I=M.O.I m4,  = Volume of sub-merged body.
2. Experimental method for Meta-centric Height: GM  W1
d/ W tan

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Here W = Weight of vessel including, G=centre of gravity og vessel, B=centre of buoyancy
w1=movable weight, d=distance between movable weight.
Condition of equilibrium of a floating and sub-merged bodies
Stability of Sub-merged Body:-
a) Stable Equilibrium:-When W = Fb and point B is above G .
b) Unstable Equilibrium:- When W= Fb but B is below G.
c) Neutral Equilibrium:-When W = Fb and B & G are the same point.
Stability of Floating Body
a) Stable Equilibrium:-If the point M is above G.
b) Unstable Equilibrium:-If the point M is Below G.
c) Neutral Equilibrium:-If the point M is at the G.
Fluid Kinematics
Steady Flow: A flow in which the magnitude & direction of velocity do not change from point to point is
termed as steady flow. Or
A flow whose flow state expressed by velocity, pressure, density, etc., at any position, does not change
with time, is called a steady flow. when water runs out while the handle is stationary, leaving the
opening constant, the flow is steady.
Unsteady Flow: A flow whose flow state does change with time is called an unsteady
flow. Whenever water runs out of a tap while the handle is being turned, the
flow is an unsteady flow.
Laminar Flow: If the particles of a liquid flow along straight & parallel paths, the flow is termed as
laminar flow.
Turbulent Flow: The flow in which the fluid particles move in zig zag way is called as Turbulent Flow.
Uniform Flow: The type of flow in which the velocity at any given time does not change with respect to
space is called uniform flow. (V/S) = 0
Where, V = change in velocity & S = Displacement in any direction.
Non-Uniform Flow: The type of flow in which the velocity at any given point changes with respect to
space is called non-uniform flow. (V/S) ≠ 0
Path line: During flow of a liquid, the path followed by a single fluid particle is called as path line. Or
A path line is the path followed by a fluid particle in motion. A path line shows the direction of particular
particle as it moves ahead. In general this is the curve in three densional space. However, if the
conditions are such that the flow is two dimensional the curve becomes two dimensional.
Stream line: The tangent drawn at any point on the imaginary line in the flow liquid is called stream line.
Or The imaginary line with in the flow so that the tangent at any point on it indicates the velocity at that
point.
Flow Net: A set of flow lines containing both the streams lines & potential lines intersecting each other’s
is called as flow net.
Stream tube: A stream tube is a fluid mass bounded by a group of streamlines. The contents of a stream
tube are known as “current filament”.
Streak line: the streak line is a curve which gives an picture of the location of the fluid particles which
have passed through a given point.
Discharge: The quantity of a liquid flowing per second through a pipe is termed as Discharge.
Cumec & Cusec is the unit of discharge. Formulae of Discharge: “Q = A  V ” Where

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Q = discharge, V = velocity of flowing liquid & A = cross-sectional area of flowing liquid.
Flow velocity: In continuum mechanics the macroscopic velocity, also flow velocity in fluid
dynamics or drift velocity in electromagnetism, is a vector field used to mathematically describe the
motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar.
The flow velocity u of a fluid is a vector field: u = u(x,t), which gives the velocity of an element of fluid at
a position x and Time t. The flow speed q is the length of the flow velocity vector.
Q = ||u|| and is a scalar field.
Velocity potential: It is defined as a scalar function of space and time such that its negative derivative
with respect to any direction gives the fluid velocity in that direction. It is denoted by Φ.
U= -∂Φ/∂x,v=-∂Φ/∂y,w=-∂Φ/∂z.
U,v,w are the velocity in x,y,z direction.
System and control volume: A system refers to a fixed, identifiable quantity of mass which is
separated from its surrounding by its boundaries. The boundary surface may vary with time however no
mass crosses the system boundary. In fluid mechanics an infinitesimal lump of fluid is considered as a
system and is referred as a fluid element or a particle. Since a fluid particle has larger dimension than
the limiting volume (refer to section fluid as a continuum). The continuum concept for the flow analysis
is valid control volume is a fixed, identifiable region in space through which fluid flows. The boundary of
the control volume is called control surface. The fluid mass in a control volume may vary with time. The
shape and size of the control volume may be arbitrary. OR
Fluid: Matter that has no definite shape. (That includes BOTH liquids and gases.)
So, we pick a constant mass and follow it as it flows. With liquids the flow is usually assumed to be
"Incompressible" (The volume is constant but the shape can change). Gasses the flow may be
"compressible" (both the shape and volume can change) or "Incompressible" (like liquids, if the pressure
changes are small, we can assume the volume of a gas does not change as it flows). For all cases above:
The amount of mass in the Control Volume is constant.
Compressible flow is more complicated, of course. The density of the fluid = mass/volume
The mass is constant. You must apply the Ideal Gas Law and Thermodynamics (loss or gain of energy to
affect temperature ) to find the new volume at any point in the flow ,then find the new density.
Continuity Equation - Differential Form Compressible flow
Derivation: The point at which the continuity equation has to be derived, is enclosed by an elementary
control volume.
The influx, efflux and the rate of accumulation of mass is calculated across each surface within the
control volume.

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Fig 9.6 A Control Volume Appropriate to a Rectangular Cartesian Coordinate System
Consider a rectangular parallelopiped in the above figure as the control volume in a rectangular
cartesian frame of coordinate axes.
Net efflux of mass along x -axis must be the excess outflow over inflow across faces normal to x -axis.
Let the fluid enter across one of such faces ABCD with a velocity u and a density ρ.The velocity and
density with which the fluid will leave the face EFGH will be and respectively
(neglecting the higher order terms in δx).
Therefore, the rate of mass entering the control volume through face ABCD = ρu dy dz.
The rate of mass leaving the control volume through face EFGH will be
(neglecting the higher order terms in dx)
Similarly influx and efflux take place in all y and z directions also.
Rate of accumulation for a point in a flow field
Using, Rate of influx = Rate of Accumulation + Rate of Efflux

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Transferring everything to right side
(9.2)
This is the Equation of Continuity for a compressible fluid in a rectangular cartesian coordinate
system.
Continuity Equation - Vector Form or Incompressible Flow
The continuity equation can be written in a vector form as
or,
(9.3)
where is the velocity of the point
In case of a steady flow,
Hence Eq. (9.3) becomes
(9.4)
In a rectangular cartesian coordinate system
(9.5)
Equation (9.4) or (9.5) represents the continuity equation for a steady flow.
In case of an incompressible flow,
ρ = constant
Hence,

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Moreover
Therefore, the continuity equation for an incompressible flow becomes
(9.6)
(9.7)
In cylindrical polar coordinates eq.9.7 reduces to
Eq. (9.7) can be written in terms of the strain rate components as
(9.8)
Hydrodynamics
Different Forms of Energy:
(1). Kinetic Energy: Energy due to motion of body. A body of mass, m, when moving with velocity, V,
posses kinetic energy, KE = 1/2mV2. M & V are Mass & Velocity of the body.
(2). Potential Energy: Energy due to elevation of body above an arbitrary datum
ΡE = mgZ, Z is elevation of body from arbitrary datum, m is the mass of body.
(3). Pressure Energy: Energy due to pressure above datum, most usually its pressure above atmospheric
ΡrE = γh
(4). Internal Energy: It is the energy that is associated with the molecular, or internal state of matter; it
may be stored in many forms, including thermal, nuclear, chemical and electrostatic.
Head: Energy per unit weight is called head.
Kinetic head: Kinetic energy per unit weight is called kinetic head.
Kinetic head = KE/Weight = (1/2mV2)/mg = V2/2g weight = mg
Potential head: Potential energy per unit weight is called potential head.
Potential head = ΡE/Weight = (mgZ)/mg = Z
Pressure head: Pressure energy per unit weight is called pressure head.
Pressure head = ΡrE/Weight = ρ/γ.
TOTAL HEAD = Kinetic Head + Potential Head + Pressure Head
V2/2g Z ρ/γ
Total Head = H = Z + ρ/γ + V2/2g.
Bernoulli’s Equation: It states that the sum of kinetic, potential and pressure heads of a fluid particle
is constant along a streamline during steady flow when compressibility and frictional effects are
negligible. i.e. For an ideal fluid, Total head of fluid particle remains constant during a steady-

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incompressible flow. Or total head along a streamline is constant during steady flow when
compressibility and frictional effects are negligible.
Total Head = H = Z + ρ/γ + V2/2g = constt
Z1 + ρ1/γ + V2
1/2g = Z2 + ρ2/γ + V2
2/2g, or, ρ1/ρg + u1
2/2g + z1 = ρ2/ρg + u2
2/2g + z2
H1 = H2.
Applications of the Bernoulli Equation: The Bernoulli equation can be applied to a great many
situations not just the pipe flow we have been considering up to now. In the following sections we will
see some examples of its application to flow measurement from tanks, within pipes as well as in open
channels. Bernoulli’s equation is used any time we want to relate pressures and velocities in situations
where the flow conditions are close enough to what is assumed in deriving Bernoulli’s equation. You
need to be in a flow that is not changing with time and in a regime for which the fluid behaves pretty
much like an incompressible fluid without viscosity. If the flow is dominated by viscous stresses (low
Reynolds numbers), then Bernoulli’s equation cannot be used. We can still use it for parts of the flow
where viscosity isn’t so strong, but inside the boundary layer, for example, we cannot use it.
If the flow is highly unsteady, then it cannot be used. In some cases, we might be able to use it, but we
have to be careful about how we do it.
The incompressible version can only be used if the effects of compressibility are small. That typically
means lower than about Mach 0.3. But even at somewhat higher Mach numbers, you can still use it to
get a rough idea about the flow. Just remember that your results are distorted, so don’t assume they
have a lot of accuracies.
We use Bernoulli’s equation for A LOT of different fluid flow situations.
Energy Line and Hydraulic Grade line
Energy line: It is line joining the total heads along a pipe line.
HGL: It is line joining pressure head along a pipe line.
ρ/γ + Z + V2/2g = H
Pressure head + Elevation head + Velocity head = Total Head
Ρ + ρgz + ρ V2/2 = contt
Static Pressure: ρ
Dynamic pressure: ρ V2/2
Hydrostatic Pressure: ρgZ
Stagnation Pressure: Static pressure + dynamic Pressure

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Ρ + ρ V2/2 = pstag
Measurement of Heads: Piezometer: It measures pressure head (ρ/γ ).
Pitot tube: It measures sum of pressure and velocity heads i.e.,
ρ/γ + V2/2g.
Introduction to density currents: When waters of two different densities meet, the dense water will
slide below the less dense water. The differing densities cause water to move relative to one-another,
forming a density current. This is one of the primary mechanisms by which ocean currents are formed.
It appears that this effect was first explored by Marsigli who visited Constantinople in 1679 and learned
about a well-known undercurrent in a strait (the Bosphorous) that flows between the Black Sea and the
Mediterranean. Fisherman had observed that “the upper current flows from the Mediterranean to the
Black Sea but the deep water of the abyss moves in a direction exactly opposite to that of the upper
current and so flows continuously against the surface current”.
Marsigli reasoned that the effect was due to density differences. He performed a laboratory experiment:
a container was initially divided by a partition. The left side contained dyed water taken from the
undercurrent in the Bosphorous, while the right side contained water having the density of surface
water in the Black Sea. Two holes were placed in the partition to observe the resulting flow. The flow
through the lower hole was in the direction of the undercurrent in the Bosphorous, while the flow
through the upper hole was in the direction of the surface flow.
We repeat Marsigli’s experiment here. Dense water (dyed) flows rightward through a hole in a partition
near the base: light (clear) fluid returns leftward through a hole toward the top. This experiment was
inspired by one devised by Prof Peter Bannon at Penn State.
Free and Forced Vortex Flow
Vortex flow is defined as flow along curved path. It is of two types namely; (1). Free vortex flow and (2)
forced vortex flow. If the fluid particles are moving around a curved path with the help
of some external torque the flow is called forced vortex flow. And if no external force is acquired to
rotate the fluid particle, the flow is called free vortex flow.
Forced Vortex Flow (Rotational Flow): It is defined as that type of flow, in which some external torque
is required to rotate the fluid mass. The fluid mass in this type of flow rotate at constant angular
velocity, ω. The tangential velocity, V, of any fluid particle is given by: V= ω r,
Where, r is radius of fluid particle from the axis of rotation. Examples of forced vortex flow are;
1. A vertical cylinder containing liquid which is rotated about its central axis with a constant angular
velocity ω,
2. Flow of liquid inside impeller of a centrifugal pump,
3. Flow of water through runner.
Free Vortex Flow (Irrotational flow): When no external torque is required to rotate the fluid mass, that
type of flow is called free vortex flow. Thus the liquid in case of free vortex flow is rotating due to the
rotation which is imparted to the fluid previously. Example of free vortex flow are
1. Flow of liquid through a hole provided at the bottom of container,
2. Flow of liquid around a circular bend in pipe,
3. A whirlpool in river,
4. Flow of fluid in a centrifugal pump casing.
Pressure Conduit: A pressure conduit (such as a penstock) is a pipe which runs under pressure and,
therefore, runs full. This type of conduits prove economical than canals or flumes, because they can

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SAQIB IMRAN 28
generally follow shorter routes. Moreover, their biggest advantage is: that the water or any other fluid
flowing through them is not exposed anywhere and hence, there are no chances or very less chances of
its getting polluted. Hence, these pressure conduits are preferably used for city water supplies. Since the
water wasted in percolation, evaporation, etc is also -saved, when water is carded through these
conduits, they are preferably used when water is scarce. The flow of water through conduit pipes is
generally turbulent, and hence, it will be considered so, while dealing with the hydraulics of flow
through such pipes.
Forces Acting on Pressure Conduits: Pressure pipes must be designed to withstand the following forces :
(1) Internal Pressure of Water. The pressure exerted on the walls of the pipe by
the flowing water, in the form of Hoope's tension, is the internal pressure. The circumferential tensile
stress produced is given as :
cr1 = P1d/2t in KN/m2
where P1 = Internal static pressure in kN/m2
d = Diameter of the pipe in metres.
t= Thickness of the pipe shell in metres.
cr1 = Circumferential tensile stress to be counteracted by providing Hoope's reinforcement.
(2) Water Hammer Pressure. When a liquid flowing in a pipe line is abruptly
stopped by the closing of a valve, the velocity of the water column behind, is retarded,
and its momentum is destroyed. This exerts a thrust on the valve and additional pressure
on the pipe shell behind. The more rapid the closure of the valve, the more rapid is the
change in momentum, and hence, greater is the additional pressure developed. The
pressures so developed are known as water-hammer pressures and may be so high as
to cause bursting of the pipe shell (due to increased circumferential tension) if not
accounted for in the designs. · ·
. The maximum pressure developed in pipe lines due to water hammer is given by
the formula:
p2 = 14.762.v / e1 + K .d/t. where V= Velocity of 1water just before the closing
of the valve in m/sec.
d = Diameter of pipe in metres.
t = Thickness of pipe shell in metres.
K= Constant = Modulus of elasticity of pipe material / Bulk modulus of elasticity of water.
The value of K for steel comes out to be 0.01, for cast iron = 0.02, and for cement
concrete= 0.1.
(3) Stress due to External Loads. When large pipes are buried deep under the
ground, the weight of the earth-fill may produce large stresses in the pipe material. The
stress due to the external earth fill load is given by
F = 22.7
ℎ.𝑑𝟐
𝑡
where h = depth of the earth fill above the crown in metres.
d = diameter of pipe in metres.
f= stress produced in kN/m2.

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Note. In the above formula, it is assumed that the earth to the sides does not give
any lateral support and weighs about 18.4 kN/m3.
(4) Temperatures Stresses. When pipes are laid above the ground, they are exposed to the atmosphere
and-are, therefore, subjected to temperature changes. They
expand during day time and contract at night. If this expansion or contraction· is
prevented due to fixation or friction over the supports, longitudinal stresses are produced
in the pipe material. The amount of these stresses may be calculated by the formula :
F = E. à .T, where E= Modulus of elasticity of the pipe material.
à = Co-efficient of expansion of the pipe material.
T = Change in temperature in °C
(5) Stresses due to Flow around Bends and Change in Cross-Section: Whenever the velocity of a flow
(either magnitude or direction) changes, there is a change in the momentum, and therefore, by
Newton's Second Law, a force is exerted, which is proportional to the rate of change of momentum. The
force required to bring this change in momentum comes from the pressure variation· with in the fluid
and from forces transmitted to the fluid from the pipe walls.
(6) Flexural Stresses. Many a times, steel pipes are laid over concrete supports,
built above the ground ; and sometimes the rain water, etc. may wash off the ground
from below the pipes at intervals. Under all such circumstances, bending stresses get
produced in the pipe, since the pipe then act &. "like a beam with loads resulting from the
weight of the pipe, weight of water in the pipe and any other superimposed loads: The
stresses caused by this beam action may be determined by usual methods of analysis
applied to the beams. However, these stresses are generally negligible except for long
spans or where there are huge superimposed loads.·
Torque in Rotating Machines: A core task for a rotating electric machine is to produce the torque
needed to achieve the required rotation speed under load. In linear machines, correspondingly, force
production is the key element. Torque production is based on forces affecting the stator and the rotor.
There are several ways to study force and torque production. This chapter explores the most important
ways from the electrical drive's point of view. Torque production can be examined by analysing the
energy stored in the magnetic circuit of the machine. Ignoring losses, the torque equation correlates
with power. The voltage of a double‐salient pole reluctance machine can be expressed by applying
Faraday's induction law and Ohm's law. The saliency of an electric machine produces torque if rotor
movement results in a reduction in the reluctance of the main flux path. When applying numerical
methods, Maxwell's stress tensor is often used for the calculation of torque.
Bends & Elbows A BEND is the generic term for what is called in piping as an "offset" - a change in
direction of the piping. A bend is usually meant to mean nothing more than that there is a "bend" - a
change in direction of the piping.
Pipe Bend: Long radius pipeline bends are used in fluid transportation line which required pigging. Due
to their long radius and smooth change of direction, pipe bend has very less pressure drop, and smooth
flow of fluid & pig is possible. 3D and 5D Pipe bends are commonly available. Here, D is the pipe size.
Miter Bend: Miter bends are not standard pipe fittings they are fabricated from pipes. Usually, they are
preferred for size 10” & above because large size elbow is expensive. Use of miter bend is restricted to
the low-pressure water line. Miter bend can be fabricated in 2, 3, & 5 pieces.
An elbow is a pipe fitting installed between two lengths of pipe or tubing to allow a change of direction,
usually a 90° or 45° angle , though 22.5° elbows are also made. The ends may be machined for butt

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welding, threaded (usually female), or socketed, etc. When the two ends differ in size, the fitting is
called a reducing elbow or reducer elbow. Elbows are categorized based on various design features as
below: Long Radius (LR) Elbows – radius is 1.5 times the pipe diameter.
Short Radius (SR) Elbows – radius is 1.0 times the pipe diameter.
90 Degree Elbow – where change in direction required is 90°.
45 Degree Elbow – where change in direction required is 45°.
Couplers & Reducers
Couplers: A coupling connects two pipes to each other. If the size of the pipe is not the same , the fitting
may be called a reducing coupling or reducer, or an adapter. By convention, the term "expander" is not
generally used for a coupler that increases pipe size; instead the term "reducer" is used.
Reducers: A reducer allows for a change in pipe size to meet hydraulic flow requirements of the system,
or to adapt to existing piping of a different size. Reducers are usually concentric but eccentric reducers
are used when required to maintain the same top- or bottom-of-pipe level.
Pipe Reducers: A pipe reducer changes the size of the pipe. There are two types of reducer used in
piping Concentric & Eccentric.
Concentric Pipe Reducer or Conical Reducer: In Concentric reducer which is also known as a conical
reducer, the center of both the ends is on the same axis. It maintains the centerline elevation of the
pipeline. When the center lines of the larger pipe and smaller pipe are to be maintained same, then
concentric reducers are used.
Eccentric Reducer: In Eccentric reducer, the center of both the ends is on different axis as shown in the
image. It maintains BOP (bottom of pipe) elevation of the pipeline. When one of the outside surfaces of
the pipeline is to be maintained same, eccentric reducers are required.
Offset = (Larger ID – Smaller ID) / 2
Swage Reducer: The swage is like reducers but small in size and used to connect pipes to smaller
screwed or socket welded pipes. Like reducers, they are also available in concentric & eccentric type.
Swages are available in different end types. Such as both plain ends or one plain and one threaded end.
Tees: A tee is the most common pipe fitting. It is available with all female thread sockets, all solvent
weld sockets, or with opposed solvent weld sockets and a side outlet with female threads. It is used to
either combine or split a fluid flow. It is a type of pipe fitting which is T-shaped having two outlets, at 90°
to the connection to the main line. It is a short piece of pipe with a lateral outlet. A tee is used for
connecting pipes of different diameters or for changing the direction of pipe runs.
Equal, Unequal.
When the size of the branch is same as header pipes, equal tee is used and when the branch size is less
than that of header size, reduced tee will be used. Most common are tees with the same inlet and outlet
sizes.
Pipe Tee: Pipe tee is used for distributing or collecting the fluid from the run pipe. It is a short piece of
pipe with a 90-degree branch at center. There are two types of Tee used in piping, Equal / Straight Tee
and Reducing / Unequal Tee.
Straight Tee: In straight tee, the diameter of the branch is same as the diameter of the Run (Header)
Pipe.
Reducing Tee: In reducing tee, diameter of the branch size is smaller than the diameter of the Run
(Header) Pipe
Barred Tee: A barred tee which is also known as a scrapper tee is used in pipelines that are pigged. The
branch of the tee has a restriction bar welded internally to prevent the pig or scrapper to enter the

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SAQIB IMRAN 31
branch. The bars are welded in the branch in a way that it will allow restriction free passage of the pig
from the run pipe.
Wye Tee / Lateral: It is a type of Tee which has the branch at a 45° angle, or an angle other than 90°.
Wye tee allows one pipe to be joined to another at a 45° angle. This type of tee reduces friction and
turbulence that could hamper the flow. Wye tee is also known as a lateral.
Cross: Cross fittings are also called 4-way fittings. If a branch line passes completely through a tee, the
fitting becomes a cross. A cross has one inlet and three outlets, or vice versa. They often have solvent
welded socket ends or female threaded ends.
Cross fittings can generate a huge amount of stress on pipe as temperature changes, because they are
at the center of four connection points. A tee is more steady than a cross, as a tee behaves like a three-
legged stool, while a cross behaves like a four-legged stool. (Geocentrically, "any 3 non-colinear points
define a plane" thus 3 legs are inherently stable.) Crosses are common in fire sprinkler systems, but not
in plumbing, due to their extra cost as compared to using two tees.
Pipe Caps: The cap covers the end of a pipe. Pipe caps are used at the dead end of the piping system. It
is also used in piping headers for future connections.
Stub Ends: Stub ends are used with lap joint flange. In this type of flange, the stub is butt welded to the
pipe, whereas flange is freely moved over the stub end. It is basically flange part but covered under
ASME B16.9 that is why it is considered as pipe fittings.
Piping Union: Unions are used as an alternative to flanges connection in low-pressure small bore piping
where dismantling of the pipe is required more often. Unions can be threaded end or socket weld ends.
There are three pieces in a union, a nut, a female end, and a male end. When the female and male ends
are joined, the nuts provide the necessary pressure to seal the joint.
Pipe Coupling: There are three types of coupling available;
Full Coupling: Full Coupling is used for connecting small bore pipes. It used to connect pipe to pipe or
pipe to swage or nipple. It can be threaded or socket ends types.
Half Coupling: Half Coupling is used for small bore branching from a vessel or large pipe. It can be
threaded or socket type. It has a socket or thread end on only one side.
Reducing Coupling: Reducing coupling is used to connect two different sizes of pipe. It is like concentric
reducer that maintains a center line of the pipe but small in size.
Pipe Nipple: Nipple is a short stub of pipe which has a male pipe thread at each end or at one end. It
used for connecting two other fittings. Nipples are used for connecting pipe, hoses, and valves. Pipe
nipples are used in low-pressure piping.
Socket weld and Threaded Pipe Fittings: Socket weld and Threaded Pipe Fittings are forged product and
classified based on its pressure-temperature rating. Socket weld & Threaded end fittings are available
from NPS 1/8” to 4”. These fittings are available in four pressure-temperature rating class.
2000 class fittings are available in only in threaded type.
3000 & 6000 class fittings are available in both Threaded and Socket Weld types.
9000 class fittings are available in only socket weld type.
Socket and threaded fittings are used for small bore and low-pressure piping.
Dimensional Analysis & Similitude : Dimensional Analysis is a mathematical technique making
use of study of dimensions. It deals with the dimensions of physical quantities involved in the
phenomenon. In dimensional analysis, one first predicts the physical parameters that will influence the
flow, and then by, grouping these parameters in dimensionless combinations a better understanding of
the flow phenomenon is made possible. It is particularly helpful in experimental work because it

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provides a guide to those things that significantly influence the phenomena; thus it indicates the
direction in which the experimental work should go. This mathematical technique is used in research
work for design and for conducting model tests.
TYPES OF DIMENSIONS: There are two types of dimensions
• Fundamental Dimensions or Fundamental Quantities
• Secondary Dimensions or Derived Quantities
Fundamental Dimensions or Fundamental Quantities: These are basic quantities. For Example;
• Time, T Time, T
• Distance, L Distance, L
• Mass, M Mass, M
Force = Mass * acceleration = MLT-2 .
Secondary Dimensions or Derived Quantities: The are those quantities which posses more than one
fundamental dimensions. For example;
• Velocity is denoted by distance per unit time L/T
• Acceleration is denoted by distance per unit time square L/T2
• Density is denoted by mass per unit volume M/L3
Since velocity, density and acceleration involve more than one fundamental quantities so these are
called derived quantities.
Dimensional Analysis: When the dimensions of all terms of an equation are equal the equation is
dimensionally correct. In this case, whatever unit system is used, that equation holds its physical
meaning. If the dimensions of all terms of an equation are not equal, dimensions must be hidden in
coefficients, so only the designated units can be used. Such an equation would be void of physical
interpretation.
Utilising this principle that the terms of physically meaningful equations have equal dimensions, the
method of obtaining dimensionless groups of which the physical phenomenon is a function is called
dimensional analysis. If a phenomenon is too complicated to derive a formula describing it,
dimensional analysis can be employed to identify groups of variables which would appear in such a
formula. By supplementing this knowledge with experimental data, an analytic relationship between the
groups can be constructed allowing numerical calculations to be conducted.
Modeling and Similitude: A “model” is a representation of a physical system used to predict the
behavior of the system in some desired respect. The physical system for which the predictions are to be
made is called “prototype”.
Usually, a model is smaller than the prototype so as to conduct laboratory studies and it is less
expensive to construct and operate. However, in certain situations, models are larger than the
prototype e.g. study of the motion of blood cells whose sizes are of the order of micrometers.
“Similitude” in a general sense is the indication of a known relationship between a model and prototype
i.e. model tests must yield data that can be scaled to obtain the similar parameters for the prototype.
Theory of models: A given problem can be described in terms of a set of pi terms by using the principles
of dimensional analysis as,
 1 2 3, ,.......... n    
(8)
This equation applies to any system that is governed by same variables. So, if the behavior of a particular
prototype is described by Eq. (8), a similar relationship can be written for a model of this type i.e.

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SAQIB IMRAN 33
 1 2 3, ,..........m m m nm    
(9)
The form of the function remains the same as long as the same phenomenon is involved in both
prototype and the model. Therefore, if the model is designed and operated under following conditions,
2 2
3 3
.
.
m
m
nm n
  
  
  
(10)
then, it follows that
1 1m  
(11)
Eq. (11) is the desired “prediction equation” and indicates that the measured value of 1m
obtained
with the model will be equal to the corresponding 1
for the prototype as long as the other pi terms
are equal. These are called “model design conditions / similarity requirements / modeling laws”.
Flow similarity: In order to achieve similarity between model and prototype behavior, all the
corresponding pi terms must be equated between model and prototype. So, the following conditions
must be met to ensure the similarity of the model and the prototype flows.
1. Geometric similarity: A model and prototype are geometric similar if and only if all body dimensions
in all three coordinates have the same linear-scale ratio. It requires that the model and the prototype be
of the same shape and that all the linear dimensions of the model be related to corresponding
dimensions of the prototype by a constant scale factor. Usually, one or more of these pi terms will
involve ratios of important lengths, which are purely geometrical in nature. The geometric scaling may
also extend to the finest features of the system such as surface roughness or small perterbance that may
influence the flow fields between model and prototype.
2. Kinematic similarity: The motions of two systems are kinematically similar if homogeneous particles
lie at homogeneous points at homogeneous times. In a specific sense, the velocities at corresponding
points are in the same direction and are related in magnitude by a constant scale factor. This also
requires that streamline patterns must be related by a constant scale factor. The flows that are
kinematically similar must be geometric similar because boundaries form the bounding streamlines. The
factors like compressibility or cavitations must be taken care of to maintain the kinematic similarity.
3. Dynamic similarity: When two flows have force distributions such that identical types of forces are
parallel and are related in magnitude by a constant scale factor at all corresponding points, then the
flows are dynamic similar. For a model and prototype, the dynamic similarity exists, when both of them
have same length-scale ratio, time-scale ratio and force-scale (or mass-scale ratio).
 For compressible flows, the model and prototype Reynolds number, Mach number and specific
heat ratio are correspondingly equal.
 For incompressible flows,
With no free surface: model and prototype Reynolds number are equal.
With free surface: Reynolds number, Froude number, Weber number and Cavitation numbers for model
and prototype must match.

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In order to have complete similarity between the model and prototype, all the similarity flow conditions
must be maintained. This will automatically follow if all the important variables are included in the
dimensional analysis and if all the similarity requirements based on the resulting pi terms are satisfied.
Model scales: In a given problem, if there are two length variables 1l
and 2l
, the resulting requirement
based on the pi terms obtained from these variables is,
1 2
1 2
m ml l
l l

(12)
This ratio is defined as the “length scale”. For true models, there will be only one length scale and all
lengths are fixed in accordance with this scale. There are other ‘model scales’ such as velocity scale
m
v
V
V

 
 
  , density scale
m




 
 
  , viscosity scale
m




 
 
  etc. Each of theses scales is defined
for a given problem.
Distorted models: In order to achieve the complete dynamic similarity between geometrically similar
flows, it is necessary to duplicate the independent dimensionless groups so that dependent parameters
can also be duplicated (e.g. duplication of Reynolds number between a model and prototype is ensured
for dynamically similar flows).
In many model studies, dynamic similarity requires the duplication of several dimensionless groups and
it leads to incomplete similarity between model and the prototype. If one or more of the similarity
requirements are not met, e.g. in Eq. 10, if 2 2m  
, then it follows that Eq. 11 will not be satisfied i.e.
1 1m  
. Models for which one or more of the similar requirements are not satisfied, are called
“distorted models”. For example, in the study of open-channel or free surface flows, both Reynolds
number
Vl

 
 
  and Froude number
V
gl
 
  
  are involved. Then, Froude number similarity requires,
m
m m
V V
g l gl

(13)
If the model and prototype are operated in the same gravitational field, then the velocity scale becomes,
m m
l
V l
V l
 
(14)
Reynolds number similarity requires,
. . . .m m m
m
V l V l 
 

(15)
and the velocity scale is,
. .m m
m m
V l
V l
 
 

(16)
Since, the velocity scale must be equal to the square root of the length scale, it follows that

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 
 
 
3
32
2
m mm m
l
l
l
 

  
 
   
  (17)
Eq. (17) requires that both model and prototype to have different kinematics viscosity scale, if at all both
the requirements i.e. Eq. (13) and (15) are to be satisfied. But practically, it is almost impossible to find a
suitable model fluid for small length scale. In such cases, the systems are designed on the basis of
Froude number with different Reynolds number for the model and prototype where Eq. (17) need not
be satisfied. Such analysis will result a “distorted model”. Hence, there are no general rules for handling
distorted models and essentially each problem must be considered on its own merits.
DIMENSIONAL NUMBERS IN FLUID MECHANICS: Forces encountered in flowing fluids include those due
to inertia, viscosity, pressure, gravity, surface tension and compressibility. These forces can be written as
2 2
Inertia force . V V
dV dV
m a V V L
dt ds
  
 
    
 
Viscousforce
du
A A V L
dy
    
    2
Pressureforce p A p L   
3
Gravityforce m g g L 
Surface tension force L
2
Compressibilityforce ; where is the Bulkmodulusv v vE A E L E 
The ratio of any two forces will be dimensionless. Inertia forces are very important in fluid mechanics
problems. So, the ratio of the inertia force to each of the other forces listed above leads to fundamental
dimensionless groups. These are,
1. Reynolds number
 eR
: It is defined as the ratio of inertia force to viscous force.
Mathematically,
e
VL VL
R

 
  (1)
where V is the velocity of the flow, L is the characteristics length, , and   are the density,
dynamic viscosity and kinematic viscosity of the fluid respectively. If eR
is very small, there is an
indication that the viscous forces are dominant compared to inertia forces. Such types of flows
are commonly referred to as “creeping/viscous flows”. Conversely, for large eR
, viscous forces
are small compared to inertial effects and flow problems are characterized as inviscid analysis.
This number is also used to study the transition between the laminar and turbulent flow regimes.

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SAQIB IMRAN 36
2. Euler number
 uE
: In most of the aerodynamic model testing, the pressure data are usually
expressed mathematically as,
21
2
u
p
E
V


(2)
where p is the difference in local pressure and free stream pressure, V is the velocity of the flow, 
is the density of the fluid. The denominator in Eq. (2) is called “dynamic pressure”. uE
is the ratio of
pressure force to inertia force and it is also called as the pressure coefficient pC
. In the study of
cavitations phenomena, similar expressions are used where p is the difference in liquid stream
pressure and liquid-vapour pressure. The dimensional parameter is called “cavitation number”.
3. Froude number
 rF
: It is interpreted as the ratio of inertia force to gravity force.
Mathematically, it is written as,
.
r
V
F
g L
 (3)
where V is the velocity of the flow, L is the characteristics length descriptive of the flow field and g is
the acceleration due to gravity. This number is very much significant for flows with free surface effects
such as in case of open-channel flow. In such types of flows, the characteristics length is the depth of
water. rF
less than unity indicates sub-critical flow and values greater than unity indicate super-critical
flow. It is also used to study the flow of water around ships with resulting wave motion.
4. Weber number
 eW
: The ratio of the inertia force to surface tension force is called Weber
number. Mathematically,
2
e
V L
W



(4)
where V is the velocity of the flow, L is the characteristics length descriptive of the flow field,  is the
density of the fluid and  is the surface tension force. This number is taken as a index of droplet
formation and flow of thin film liquids in which there is an interface between two fluids. For
1eW
,
inertia force is dominant compared to surface tension force (e.g. flow of water in a river).
5. Mach number
 aM
: It is the key parameter that characterizes the compressibility effects in a
fluid flow and is defined as the ratio of inertia force to compressibility force. Mathematically,
a
v
V V V
M
c dp E
d 
  
(5)