Mohd Sharique Ahmad
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Fluid and continuum
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Mechanics: The oldest physical science that deals with both
stationary and moving bodies under the influence of forces.
Statics: The branch of mechanics that deals with bodies at rest
Dynamics: The branch that deals with bodies in motion.
Fluid mechanics: The science that deals with the behavior of
fluids at rest (fluid statics) or in motion (fluid dynamics), and
the interaction of fluids with solids or other fluids at the
Fluid dynamics: Fluid mechanics is also referred to as fluid
dynamics by considering fluids at rest as a special case of
motion with zero velocity.
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Hydrodynamics: The study of the motion of fluids that can be
approximated as incompressible (such as liquids, especially water, and
gases at low speeds).
Hydraulics: A subcategory of hydrodynamics, which deals with liquid
flows in pipes and open channels.
Gas dynamics: Deals with the flow of fluids that undergo significant
density changes, such as the flow of gases through nozzles at high
Aerodynamics: Deals with the flow of gases (especially air) over bodies
such as aircraft, rockets, and automobiles at high or low speeds.
Meteorology, oceanography, and hydrology: Deal with naturally
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Fluid: A substance in the liquid or gas phase. A solid can
resist an applied shear stress by deforming.
A fluid deforms continuously under the influence of a shear
stress, no matter how small.
In solids, stress is proportional to strain, but in fluids, stress
is proportional to strain rate.
When a constant shear force is applied, a solid eventually
stops deforming at some fixed strain angle, whereas a fluid
never stops deforming and approaches a constant rate of
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Types of stresses acting in fluid
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Stress: Force per unit area.
Normal stress: The normal component of a force
acting on a surface per unit area.
Shear stress: The tangential component of a force
acting on a surface per unit area.
Pressure: The normal stress in a fluid at rest.
Zero shear stress: A fluid at rest is at a state of
zero shear stress.
When the walls are removed or a liquid container
is tilted, a shear develops as the liquid moves to
re-establish a horizontal free surface.
The normal stress and shear stress at
the surface of a fluid element. For
fluids at rest, the shear stress is zero
and pressure is the only normal stress
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Intermolecular bonds are strongest in solids and weakest in gases.
Solid: The molecules in a solid are arranged in a pattern that is repeated throughout.
Liquid: In liquids molecules can rotate and translate freely.
Gas: In the gas phase, the molecules are far apart from each other, and molecular ordering is nonexistent.
The arrangement of atoms in different phases:
(a) molecules are at relatively fixed positions in a solid,
(b) groups of molecules move about each other in the liquid phase, and
(c) individual molecules move about at random in the gas phase.
Gas and vapor are often used as synonymous words.
Gas: The vapor phase of a substance is customarily called a gas when it
is above the critical temperature.
Vapor: Usually implies that the current phase is not far from a state of
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Macroscopic or classical approach:
Does not require a knowledge of the
behavior of individual molecules
and provides a direct and easy way to
analyze engineering problems.
Microscopic or statistical approach:
Based on the average behavior of
large groups of individual molecules
On a microscopic scale,
pressure is determined
by the interaction of
we can measure the
pressure on a
macroscopic scale with
a pressure gage.
Physical properties of fluids
Liquid can be easily distinguished from a solid or a gas.
A liquid takes the shape of vessel into which it is poured
The properties of water are of much importance because the
subject of hydraulics is mainly concerned with it. Some properties
of water are :
1. Density 2. Specific Gravity 3.Viscosity
4.Vapour pressure 5.Cohesion 6.Adhesion
7.Surface Tension 8.Capillary 9.Compressibility
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When a fluid is contained in a vessel, it exerts force at all points
on the sides and bottom and top of the container. The force per
unit area is called pressure.
Hydrostatic law: it states that rate of pressure in a vertical
direction is equal to weight density of the fluid at that point.
P = ρ g H
Where p is the pressure at any point in a liquid, g is acceleration
due to gravity and H is the height of free surface above that
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A simple manometer is one
which consists of a glass tube
whose one end is connected
to a point where pressure is to
be measured and the other
end remains open to
Types of Manometer
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on plane and curved surfaces
Total pressure : It is defined as the force exerted by static fluid
on a surface (either plane or curved) when the fluid comes
in contact with the surfaces. This force is always right angle
or normal to the surrface
P = w A ħ
P= Weight of the liquid above the immersed surface
A = total area of surface
w = specific weight of liquid
ħ = Distance of center of pressure from free surface of liquid.
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Centre of pressure
The point through which the resultant pressure acts is
known as center of pressure and is always expressed in
terms of depth from the liquid surface.
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Buoyancy is the ability of an object to float. It is related to
the object’s density
Archimedes Principle states that when a body is immersed
in a fluid either wholly or partially, it is buoyed or lifted up by
a force, which is equal to the weight of fluid dispatched by
The point of application of the force of buoyancy on the body
is known as the centre of buoyancy. It is always centre of
gravity of the volume of fluid displaced.
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Types of equilibrium of floating bodies
Types of Equilibrium of free bodies
When a body is given a small angular displacement (i.e. Tilted
slightly by some external force, then it returns back to its
original position dur to internal forces
If the body does not return to its original position from the from
the slightly displaced angular position and heels farther away,
when given a small displacement
If a body, when given a small angular displacement occupies a
new position and remains at rest in this new position.
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Types of fluid flows
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CLASSIFICATION OF FLUID FLOWS
Viscous flows: Flows in which the frictional effects are significant.
Inviscid flow regions: In many flows of practical interest, there are
regions (typically regions not close to solid surfaces) where viscous
forces are negligibly small compared to inertial or pressure forces.
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Viscous versus Inviscid Regions of Flow
The flow of an
originally uniform fluid
stream over a flat plate,
the regions of viscous
flow (next to the plate
on both sides) and
inviscid flow (away
from the plate)
Steady and unsteady
• The term steady implies no change at a
point with time.
• The opposite of steady is unsteady.
• The term uniform implies no change
with location over a specified region.
• The term periodic refers to the kind of
unsteady flow in which the flow
oscillates about a steady mean.
• Many devices such as turbines,
compressors, boilers, condensers, and
heat exchangers operate for long
periods of time under the same
conditions, and they are classified as
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Oscillating wake of a blunt-based airfoil at Mach number
0.6. Photo (a) is an instantaneous image, while photo (b)
is a long-exposure (time-averaged) image.
Laminar and turbulent flows
Laminar flow: The highly ordered fluid
motion characterized by smooth layers
of fluid. The flow of high-viscosity
fluids such as oils at low velocities is
Turbulent flow: The highly disordered
fluid motion that typically occurs at
high velocities and is characterized by
velocity fluctuations. The flow of low-
viscosity fluids such as air at high
velocities is typically turbulent.
Transitional flow: A flow that alternates
between being laminar and turbulent
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Laminar, transitional, and turbulent flows over
a flat plate.
Compressible and incompressible flows
Incompressible flow: If the density of flowing fluid remains nearly constant throughout (e.g.,
Compressible flow: If the density of fluid changes during flow (e.g., high-speed gas flow)
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When analyzing rockets, spacecraft, and other systems that involve high-speed gas
flows, the flow speed is often expressed by Mach number
Ma = 1 Sonic flow
Ma < 1 Subsonic flow
Ma > 1 Supersonic flow
Ma >> 1 Hypersonic flow
One, Two and Three dimensional flows
• A flow field is best characterized by its velocity
• A flow is said to be one-, two-, or three-dimensional if
the flow velocity varies in one, two, or three
• However, the variation of velocity in certain directions
can be small relative to the variation in other
directions and can be ignored.
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Flow over a car antenna is approximately
two-dimensional except near the top
and bottom of the antenna
The development of the velocity profile in a circular pipe. V = V(r, z) and thus the flow is two-dimensional
in the entrance region, and becomes one-dimensional downstream when the velocity profile fully develops
and remains unchanged in the flow direction, V = V(r).
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When the flow is steady, streamlines are often used to represent the
trajectories of the fluid particles.
A streamline is a line drawn in the fluid such that a tangent to the
streamline at any point is parallel to the fluid velocity at that point.
Steady flow is often called streamline flow.
Dimensional Analysis is a mathematical technique which makes
use of the study of the dimensions for solving several
The various physical quantities in fluids can be expressed in terms
of these fundamental dimensions
Mass M Time T
Lenght L Temperature ϴ
The other quantities which are expressed in these quantities are called
as derieved quanities
Force is expressed as MLT-2
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An orifice plate is a device used for
measuring flow rate, for reducing pressure or
for restricting flow (in the latter two cases it is
often called a restriction plate). Either a
volumetric or mass flow rate may be
determined, depending on the calculation
associated with the orifice plate. It uses the
same principle as a Venturi nozzle, namely
Bernoulli's principle which states that there is
a relationship between the pressure of the
fluid and the velocity of the fluid. When the
velocity increases, the pressure decreases and
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An anemometer is a device used for measuring
the speed of wind, and is also a common
weather station instrument. The term is derived
from the Greek word anemos, which means
wind, and is used to describe any wind speed
measurement instrument used in meteorology
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Hot wire anemometers use a very fine wire (on the order of several
micrometres) electrically heated to some temperature above the ambient. Air
flowing past the wire cools the wire. As the electrical resistance of most
metals is dependent upon the temperature of the metal (tungsten is a popular
choice for hot-wires), a relationship can be obtained between the resistance
of the wire and the flow speed
Notches and Weirs
A notch may be defined as an opening provided in the side
of a tank or vessel such that the liquid surface in the tank is
below the top edge of the opening.
A notch may be regarded as an orifice with the water surface
below its upper edge. It is generally made of metallic plate.
A Weir may be defined as any regtangular obstruction in an
open stream over which the flow takes place. It is made of
A weir may be used for measuring the rate of flow of water in
rivers or streams.
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Models are designed on the basis of force which is
dominating in the flow situation. The laws in which the
models are designed for dynamic similarity are called model
or similarity Laws. These are
1. Renolds model law
2. Froude model law
3. Euler model law
4. Weber model law
5. Mach model law
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Kinematics and dynamic similarity
It is the similarity of motion. If at the corresponding points in
the model and in the prototype, the velocity or accelerations
ratios are same and velocity or acceleration vector points in the
same direction, the two flows are said to be kinematically
It is the similarity of forces. The flows in the model and in the
prototype are dynamically similar if at all the corresponding
points, identical types of forces are parallel and bear the same
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Laminar and turbulent flows
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Fluid flow mean velocity and pipe
diameter for known flow rate
Velocity of fluid in pipe is not uniform across section area.
Therefore a mean velocity is used and it is calculated by the
continuity equation for the steady flow as
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Pipe diameter can be calculated when
volumetric flow rate and velocity is known as
where is: D - internal pipe diameter;
q - volumetric flow rate;
v - velocity;
A - pipe cross section area
Reynolds number, turbulent and laminar
The nature of flow in pipe, by the work of Osborne Reynolds, is
depending on the pipe diameter, the density and viscosity of the
flowing fluid and the velocity of the flow. Dimensionless Reynolds
number is used, and is combination of these four variables and may be
considered to be ratio of dynamic forces of mass flow to the shear
stress due to viscosity. Reynolds number is
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where : D - internal pipe diameter;
v – velocity
ρ - density;
ν - kinematic viscosity
μ - dynamic viscosity;
Flow in pipes is considered to be laminar if Reynolds number is
less than 2320, and turbulent if the Reynolds number is greater
than 4000. Between these two values is "critical" zone where
the flow can be laminar or turbulent or in the process of change
and is mainly unpredictable.
When calculating Reynolds number for non-circular cross
section equivalent diameter (four time hydraulic radius
d=4xRh) is used and hydraulic radius can be calculated as:
Rh = cross section flow area / wetted perimeter
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Bernoulli equation - fluid flow head
If friction losses are neglected and no energy is added to, or taken from a
piping system, the total head, H, which is the sum of the elevation head, the
pressure head and the velocity head will be constant for any point of fluid
This is the expression of law of head conservation to the flow of fluid in a
conduit or streamline and is known as Bernoulli equation:
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where: Z1,2 - elevation above reference level;
p1,2 - absolute pressure v1,2 – velocity
ρ1,2 - density; g - acceleration of gravity
Pipe flow and friction pressure drop
As in real piping system, losses of energy are existing and energy is being
added to or taken from the fluid (using pumps and turbines) these must be
included in the Bernoulli equation.
For two points of one streamline in a fluid flow, equation may be written as
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where : Z1,2 - elevation above reference level
p1,2 - absolute pressure; hL - head loss due to friction in the pipe;
Hp - pump head; HT - turbine head;
Head energy loss
Flow in pipe is always creating energy loss due to friction.
Energy loss can be measured like static pressure drop in the
direction of fluid flow with two gauges. General equation for
pressure drop, known as Darcy's formula expressed in meters of
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where is: hL - head loss due to friction in the pipe;
f - friction coefficient; L - pipe length;
D - internal pipe diameter
To express this equation like pressure drop in newtons per square meter
(Pascals) substitution of proper units leads to:
where: Δ p - pressure drop due to friction in the pipe;
f - friction coefficient; L - pipe length
D - internal pipe diameter; Q - volumetric flow
George Gabriel Stokes derived an expression, now known as Stokes'
law, for the frictional force – also called drag force – exerted on
spherical objects with very small Reynolds numbers in a viscous fluid
The force of viscosity on a small sphere moving through a viscous fluid
is given by
Fd = 6π η R ν
Fd is the frictional force – known as Stokes' drag – acting on the
interface between the fluid and the particle
η is the dynamic viscosity
R is the radius of the spherical object
v is the flow velocity relative to the object.
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In the study of turbulence in fluids, a common practical strategy
for calculation is to ignore the small-scale vortices (or eddies)
in the motion and to calculate a large-scale motion with an eddy
viscosity that characterizes the transport and dissipation of
energy in the smaller-scale flow (see large eddy simulation).
Values of eddy viscosity used in modeling ocean circulation
may be from 5×104 to 1×106 Pa·s depending upon the
resolution of the numerical grid.
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The word siphon is used to refer to a
wide variety of devices that involve the
flow of liquids through tubes. In a
narrower sense, the word refers
particularly to a tube in an inverted 'U'
shape, which causes a liquid to flow
upward, above the surface of a reservoir,
with no pump, but powered by the fall of
the liquid as it flows down the tube
under the pull of gravity, then
discharging at a level lower than the
surface of the reservoir from which it
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Water hammer (or, more generally, fluid hammer) is a
pressure surge or wave caused when a fluid (usually a liquid
but sometimes also a gas) in motion is forced to stop or
change direction suddenly (momentum change). A water
hammer commonly occurs when a valve closes suddenly at
an end of a pipeline system, and a pressure wave propagates
in the pipe. It is also called hydraulic shock
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Boundary layer over a flat plate
In fluid mechanics, a boundary layer is an important
concept and refers to the layer of fluid in the immediate
vicinity of a bounding surface where the effects of viscosity
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Boundary layer visualization, showing transition
from laminar to turbulent condition
Boundary laer Thickness
It is defined as that distance from boundary in which the
velocity reaches 99 percent of thee velocity of the free stream
(u = 0.99U). It is denoted by the symbol δ.
The commomnly adopted boundary layer thickness are
Displacement Thickness δ*
Momentum Thickness ϴ
Energy thickness δe
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Displacement Thickness δ*
It is the distance measured perpendicular to the boundary
by which the main/free stream is displaced on account of
formation of boundary layer.
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Momentum Thickness ϴ
It is defined as the distance measured perpendicular to the
boundary of the solid body, by which the boundary should be
displaced to compensate for reduction in the momentum of the
flowing fluid on account of boundary layer formation
Energy Thickness δe
It is defined as the distance, measured perpendicular to the
boundary of the solid body, by which the boundary should
compensate for the reduction in K.E. Of the flowing fluid
on account of boundary layer formation. δe
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The laminar boundary is a very smooth flow, while the
turbulent boundary layer contains swirls or "eddies." The
laminar flow creates less skin friction drag than the
turbulent flow, but is less stable. Boundary layer flow over a
wing surface begins as a smooth laminar flow. As the flow
continues back from the leading edge, the laminar
boundary layer increases in thickness.
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Application of momentum equation
Von Karman suggested a method based on the momentum
equation by the use of which the growth of a boundary layer
along a flat plate, the wall shear stress and the drag force
could be determined
τ₀/ρU2 = dϴ/dx
The above equation is von Karman momentum equation for
boundary layer flow, and is used to find out the frictional drag
on smooth flat plate for both laminar and turbulent boundary
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Turbulent boundary layer
At some distance back from the leading edge, the smooth
laminar flow breaks down and transitions to a turbulent
flow. From a drag standpoint, it is advisable to have the
transition from laminar to turbulent flow as far aft on the
wing as possible, or have a large amount of the wing surface
within the laminar portion of the boundary layer. The low
energy laminar flow, however, tends to break down more
suddenly than the turbulent layer
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Separation and its control,
The forces acting on the fluid in the boundary layer are
Inertia Force Viscous force Pressue force
If these forces act over a over a long stretch, the boundary
layer gets separated from the surface and moves into the
main stream. This phenomenon is called separation. The
point of the body of which the boundary layer is on the
verge of separation from the surface is called point of
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Drag and lift
The component of force in the direction of flow (free stream) on a submerged body
is called the drag force FD
FD = CD A (ρU2/ 2)
Lift Force :
The component of force at right angles to the direction of flow is called the lift
FL = CL A (ρU2/ 2)
WhereCD = Co-efficient of drag (dimensionless)
CL = Co- efficient of lift (dimensionless)
ρ = Density of fluid
U = Relative velocity of fluid w.r.t the body
A = Area.
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An Aerofoil, Magnus effect
An aerofoil is a streamlined body which
may be either symmetrical or
The generation of lift by spinning
cylinder in a fluid stream is called
The effect has been successfully
employed in the propulsion of ships
The magnus effect may also be used
with advantage in games like table
tennis, golf, cricket etc.
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Introduction to compressible flow
A compressible flow is that flow in which the density of the
fluid changes during flow.
Compressibility affects the drag co-efficients of bodies by
formation of shock wave, discharge co-efficients of
measuring devices such as orificemeters, venturimeters and
pitot tubes, stagnation pressure and flows in converging-
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