FLUID MECHNICS

Hydraulics :- Hydraulics is branch of engineering deals with water at rest or in motion.

Fluid Mechanics :- Fluid Mechanics is branch of engineering which deals with the behaviour of fluid under the condition of rest and motion.

Fluid Kinematics :- Branch of fluid mechanics which deals with the study of velocity and acceleration of fluid particles without taking into consideration any force or energy.

Fluid Dynamics :- Branch of science which deals with the study of fluids in motion considering the forces which cause the flow.

DENSITY OR MASS DENSITY

Mass per unit volume is called Density.

Denoted by ρ (rho).

Unit = kg/m3

Density = ρ = m/V=Mass/Volume

WEIGHT DENSITY OR SPECIFIC WEIGHT

Weight per unit volume is called specific weight.

Denoted by w.

Unit = kN/m3 or N/m3 or N/mm3

Specific weight = w = ρg.

Specific weight of sea water is more than pure water.

For water, w = 9.81 kN/m3

SPECIFIC VOLUME

Volume per unit mass of liquid is called Specific Volume.

Denoted by V.

Specific Volume = V = 1/ρ

SPECIFIC GRAVITY

Ratio of specific weight of a liquid to the specific weight of pure water at standard temperature (4℃).

No unit.

Specific gravity of pure water = 1 (Unity).

VISCOSITY

M1 L-1 T-1

Also known as Absolute Viscosity or Dynamic Viscosity.

The property of liquid which offer resistance to the movement of one layer of liquid over another adjacent layer of liquid.

The property of liquid which controls its rate of flow = Viscocity.

Two Reason of viscosity 1. Cohesion & 2. Interaction between particles.

Viscosity of water = 55 x Viscosity of air.

Viscosity of water = 1 Centipoise.

KINEMATIC VISCOSITY

M0 L2 T-1

Ratio of dynamic viscosity to the density of liquid.

SURFACE TENSION

Property of a liquid which enables it to resist tensile stress.

Surface tension is phenomenon due to difference in magnitude between forces due to cohesion and adhesion.

Denoted by σ (sigma).

Force per unit length.

Unit = N/m.

Ball pen work on principle of surface tension.

 More Less Surface Tension Mercury Water Viscosity Water Mercury

COMPRESSIBILITY

Property of liquid by virtue of which liquids undergo a change in volume with the change in pressure.

Compressibility is reciprocal of bulk modulus of elasticity.

Where, Bulk modulus = Compressive Strain/Volumetric Strain

CAPILLARITY

Phenomenon of rise or fall of liquid surface in small vertical tube held in a liquid relative to general level of the liquid.

Height of rise or fall, h = 4 σcosα/w d

Where, α = Angle of contact of liquid surface

D = Diameter of capillary tube

Water rises in tube with an upward concave surface.

Pascal Law :- “Intensity of pressure at any point in fluid at rest is same in all direction.”

PRESSURE OF LIQUID

When liquid is contained in vessel it exerts force on sides and bottom of the vessel. The force per unit area is called intensity of pressure.

Intensity of pressure at any point in liquid at rest is equal to product of weight density of liquid and vertical height from free surface of the liquid.

Intensity of pressure at any point is directly proportional to the depth of liquid from the surface.

Unit = N/m2 = Pascal.

1 Pa = 1 N/m2

1 Bar = 105 Pa.

Atmospheric pressure is also known as Barometric Pressure.

Atmospheric pressure measured by Barometric (A Mercury Manometer)

The atmospheric pressure at sea level is called Standard Atmospheric Pressure.

Standard atmospheric pressure at sea level

= 101.3 kN/m2 or kPa

= 10.3 mm of water

= 760 mm of Hg (Mercury)

The pressure measured with the help of pressure gauge = Gauge Pressure.

All pressure gauge, record the difference between Absolute pressure and Atmospheric pressure.

Absolute Pressure = Atmospheric pressure + Positive gauge pressure

This equation is used for pressure above atmospheric pressure.

For pressure below atmospheric, the gauge pressure is Negative.

Negative gauge pressure is also known as Vacuum or Suction pressure.

Absolute pressure = Atmospheric pressure – Vacuum pressure

MEASUREMENT OF PRESSURE

Pressure of liquid may be measured by Manometers.

Simple Manometer or Piezometer

Used for measuring moderate pressure.

Used for measuring gauge pressure.

Used to measured High pressure, Vacuum pressure and pressure in pipes and channels.

Liquid used in manometer have high surface tension.

Differential Manometer

Used to measure difference of pressure between two points in pipe.

CENTRE OF PRESSURE & TOTAL PRESSURE

Centre of pressure

The point of application of resultant pressure on the surface.

Horizontally Immersed Surface

Total pressure, P = w A ẍ

Vertically Immersed Surface

Total pressure, P = w A ẍ

Depth of centre of pressure, h = IG/A ẍ + ẍ

Inclined Immersed Surface

Total Pressure, P = w A ẍ/sinθ

Depth of centre of pressure, h = IGsin2θ/A ẍ + ẍ

LOCK GATES

Provided in navigation chambers to change the water level in canal or river for navigation.

In lock gates, reaction between two gates, R = P/2sinα

BUOYANCY

Tendency of liquid to uplift an immersed body because of the upward thrust of the liquid is known as Buoyancy.

Depends upon weight of the liquid displaced.

Buoyant force is equal to the weight of the liquid displaced.

The point at which buoyant force act = Centre of Buoyancy.

Force of buoyancy is more than weight of liquid displaced, body will Float.

Force of buoyancy is less than weight of liquid displaced, body will Sink down.

Archimede’s Principle :- When body is immersed wholly or partially in a liquid, it is lifted up by force equal to the weight of liquid displaced by the body.

Static Equilibrium :- If body floating in liquid return back to its original position, when given a small angular displacement then body is said to be in Static Equilibrium.

Unstable Equilibrium :- Body does not return back to its original position.

Neutral Equilibrium :- Body floating in liquid occupies new position and remains at rest in this new position, when given smaller angular displacement.

Metacentre (M) :- Point about which a floating body starts oscillating when given small angular displacement.

Metacentric Height :- Distance between the centre of gravity of the floating body and the metacentre.

 Equilibrium Condition Floating Body Submerged Body Stable M above G B above G Unstable M below G B below G Neutral M, G coincides B, G coincides

Time of oscillation (T) of floating body, T = 2π √(k2/hg)

Where, k = Radius of gyration & h = Metacentric height

Velocity of liquid is maximum near at centre of pipe & minimum near the walls.

DISCHARGE

Liquid flowing per second through pipe or channel.

Rate of flow.

Unit = cumec (m3/s)

1 m3 = 1000 litres

1 litre = 1000 cm3

1 cm3 = 1 millilitres

Uniform Flow :- Liquid particles at all sections of pipe or channel have some velocities.

Non-Uniform Flow :- Liquid particles at difference of pipe or channel have difference velocities.

Streamline Flow :- Each liquid particle has definite path and the path of individual particles do not cross each other.

Turbulent Flow :- Each liquid particle does not have definite path and the path of individual particles cross each other.

Steady Flow :- The quantity of liquid flowing per second is constant.

Unsteady Flow :- The quantity of liquid flowing per second not constant.

COMPRESSIBLE FLOW

Volume of fluid and its density changes during flow.

All gases are considered to have compressible flow.

INCOMPRESSIBLE FLOW

Volume of fluid and its density does not change during flow.

All liquids are considered to have incompressible flow.

Rotational Flow :- Fluid particles also rotate about their own axis while flowing.

Irrotational Flow :- Fluid particles do not rotate about own axes while flowing.

One Dimensional Flow :- Moving particles represented by Straight Line.

Two Dimensional Flow :- Moving particles represented by Curve.

Three Dimensional Flow :- Moving particles represented by Space.

POTENTIAL ENERGY

Potential energy is due to the position above some suitable datum line.

Denoted by Z

KINETIC ENERGY

Kinetic energy is due to the velocity of flowing liquid.

Denoted by v2/2g

PRESSURE ENERGY

Pressure energy is due to the pressure of liquid.

Denoted by p/w

Total Energy = E = Potential Energy + Kinetics Energy + Pressure Energy.

BERNOULLI’S EQUATION

For perfect incompressible liquid, floating in a continuous stream the total energy of particle remains the same, while particle moving from one point to another.

The Bernoulli’s equation applied to Verturimeter, orifice meter & pitot tube.

Bernoulli’s equation is obtained by integrating above Euler’s Equation.

Assumptions

Fluid is Non-viscous.

Fluid is Homogeneous.

Fluid is Incompressible.

Velocity is Uniform.

EULER’S EQUATION

dp/ρ + g.dx + v.dv = 0

VENTURIMETER

Used to measure the discharge of liquid flowing through pipe.

Measure in any direction and in any location.

Three part of Venturimeter :- 1. Divergent cone, 2. Converging cone & 3. Throat.

Length of divergent cone is 3 to 4 times longer than that of divergent cone in order to avoid breaking away the stream of liquid & frictional losses.

Velocity & Pressure of liquid flowing through divergent portion decreases.

To avoid tendency of separation at throat,

Diameter of Throat/Diameter of pipe= 1/8 to 1/4

Orifice Meter :- Used for measuring the discharge of liquid flowing through pipe.

PITOT TUBE

Small open tube bent at right angle.

Used to measure the velocity of flow at required point in pipe.

Determined by measuring rise of liquid in tube.

Law of Momentum :- The net force acting on a mass of fluid is equal to the change in momentum of flow per unit time in that direction.

ORIFICE

Small opening in the wall or base of a vessel through which fluid flows.

Orifice is said to be large if available head of liquid is less than 5 times than the height of orifice.

Orifice is said to be small if available head of liquid is more than 5 times than the height of orifice.

MOUTHPIECE

Mouthpiece is an attachment in the form of a small tube or pipe fixed to the orifice.

L = 2d or 3d, then its called Mouthpiece.

Used to increase the amount of discharge.

Coefficient of discharge for external mouthpiece = 0.855

Coefficient of discharge for internal mouthpiece = 0.5

Coefficient of discharge of external mouthpiece is depends upon length of mouthpiece.

Internal mouthpiece is also known as Re-entrant or Borda’s Mouthpiece.

Jet of liquid after contraction does not touch the sides of mouthpiece, then the mouthpiece said to be running free.

In this above case, length of mouthpiece is equal to diameter of the orifice.

Jet of liquid after contraction explains and fills up the whole mouthpiece, then the mouthpiece said to be running full.

In above case, length of mouthpiece is more than 3 times the diameter of orifice.

Coefficient of Contraction (Cc) :- Ratio of area of jet at vena contracta to area of orifice.

COEFFICIENT OF VELOCITY (Cv)

Ratio of the actual velocity of jet to the theoretical velocity.

Cv = √(x2/4yH)

Average value of Cv = 0.97.

COEFFICIENT OF DISCHARGE (Cd)

Ratio of actual discharge through orifice to the theoretical discharge.

Cd = Cv x Cc [ Cd < Cv ]

VENA CONTRACTA

The point at which streamlines first become parallel is called Vena contracta.

Cross section area of jet at vena contracta is less than that of the orifice.

Theoretical velocity of jet at vena contracta given by,

Vth = √(2gH)

This expression is called Torricelli’s Theorem.

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.

Used for measuring the rate of flow of liquid through channel.

For right angled V-notch, θ = 90°.

WEIR

Any regular obstruction in an open channel over which flow takes place.

Used to measuring the rate of flow of water in rivers.

Discharge over Trapezoidal notch or weir = Discharge over Rectangular notch + Discharge over Triangular notch.

Trapezoidal weir is also known as Cippoletti weir.

Broad Crested Weir :- Width of the crest of the weir is more than half the height of water above the weir crest.

Narrow Crested Weir :- Width of the crest of the weir is less than half the height of water above the weir crest.

Sharp Crested Weir :- Thickness of weir is less than half the height of water.

Submerged or Drowned Weir :- Water level on the downstream side of the weir is above the top surface of weir.

Crest or Sill :- Water flowing over top of the weir is called Crest.

Ogee Weir :- Ogee weir is used as spillway of dam.

NAPPE

Sheet of water flowing over notch is called Nappe.

Three types of Nappe :

Free Nappe

If the atmospheric pressure exists beneath the nappe then its called Free Nappe.

Depressed Nappe

Pressure below the nappe is negative.

Discharge of Depressed nappe = 6 to 7% more than Free nappe.

No air is left below the water and nappe adheres or clings to the downstream of the weir.

PIPES

Closed conduit which is used for carrying fluids under pressure.

Pipes in Series

Total head loss is equal to sum of the head loss in each pipe.

Discharge is same in all pipes.

Compound Pipe

Total head loss & discharge is same.

Pipes in Parallel

Rate of discharge is equal to the sum of discharge in each of the parallel pipes.

Loss of head in each pipe is same.

Open Channel :- When the pipe is partially full of liquid, then it behaves like open channel.

SYPHON

Long bent pipe used to connect two reservoirs at difference levels intervened by high ridge.

Work satisfactorily when minimum pressure in pipe is more than vapour pressure.

Highest point of the syphon is called the Summit.

Air vessel is provided at summit to avoid interruption in the flow.

POWER

Power transmitted (in watt) through the pipe = Weight of water flowing N/s x Head of water in m.

Power is maximum when head lost in friction = 1/3 x total supply head.

Maximum efficiency of transmission through pipe = 66.67%.

Chezy Formula, Q = A.C √(m i)

Manning’s Formula, Q = A.M. m(2/3) i(1/2)

MOST ECONOMICAL SECTION OF A CHANNEL

Rectangular Channel

R = d/2

Trapezoidal Channel

R = d/2

Triangular Channel

Sloping side at 45°with vertical.

CONDITION FOR MAXIMUM DISCHARGE

Rectangular Section

b = 2d

Trapezoidal Section

Sloping side = ½ x Width at the top

Circular Section

Depth of water = 0.95 x Diameter of channel

For Velocity, Maximum at channel in circular channel is depth of water = 0.81 x Diameter of channel.

CHANNEL

Depth of water in channel corresponding to minimum specific energy = Critical Depth =( q2/g) 1/3

Depth of water < Critical depth = Torrential Flow

Depth of water > Critical depth = Tranquil Flow

Discharge corresponding to critical depth = Maximum

In Cylindrical Vessel, Surface of liquid takes the shape of Paraboloid.

HYDRAULIC JUMP

Hydraulic jump is used to reducing the energy of flow.

Critical depth meter is used to measure Hydraulic Jump.

VISCOUS FLOW

S.I Unit of viscosity = N-s/m2

C.G.S Unit of viscosity = Poise

1 Poise = 0.1 N-s/m2

S.I Unit of kinematic viscosity = m2/s

C.G.S Unit of kinematic viscosity = Stoke

1 Stoke = 1 cm2/s = 10-4 m2/s

IDEAL FLUID

Fluid which has no viscosity.

Fluid is Non-viscous.

Frictionless and Incompressible.

Real Fluid :- Fluid which has viscosity.

NEWTONIAN FLUID

Fluid whose viscosity does not change with the rate of deformation.

Water is Newtonian Fluid.

Follow the Newton’s law of viscosity.

Shear stress-strain graph = Straight Line.

NON-NEWTONIAN FLUID

Fluid whose viscosity changes with rate of deformation.

Shear stress-strain graph = Curve.

LAMINAR FLOW

Viscosity flow dominating over inertia force.

It takes place at very low velocities.

TURBULENT FLOW

Inertia force is dominating over viscosity.

It takes place at high velocities.

CRITICAL VELOCITY

Velocity at which the flow changes from the laminar flow to the turbulent flow.

Velocity at which laminar flow stops, its lower critical velocity.

Velocity at which turbulent flow starts, its higher critical velocity.

Re < 2000, then Laminar Flow.

Re > 2800, then Turbulent Flow.

Loss of Head due to friction in pipe in terms of RN = 16RN

Co-efficient of viscosity may be found out by

1. Capillary Tube Method

2. Orifice Type Viscometer

3. Rotating Cylinder Method

4. Falling Sphere Method

Velocity or Celerity of Sound Wave = C = kρ

Stagnation Point :- The point at which velocity of fluid is zero.

Drag :- The component of pressure in direction of flow of the liquid is known as Drag.

Lift :- The component of this pressure right angle to the direction of the flow of liquid is called Lift.

MACH NUMBER

Mach Number = Velocity of fluid Velocity of sound wave

To determine Mach number, Value of bulk modulus of fluid required.

Mach angle = sin−11𝑀

Mach number < 1, Subsonic Flow.

Mach number = 1, Sonic Flow.

Mach number = 1 to 6, Supersonic Flow.

Mach number > 6, Hypersonic Flow.

Inertia Force = Mass x Acceleration

Viscous Force = Shear stress due to viscosity x Cross-section area of flow

Gravity Force = Mass x Acceleration due to Gravity.

Surface Tension Force = Surface tension per unit length x Length of the surface

Pressure Force = Intensity of pressure x Area of flowing liquid

Elastic Force = Elastic stress x Area of flowing liquid

Reynolds’s Number = Inertia Force/Viscous Force

Froude’s Number = Inertia Force/Gravity Force

Weber’s Number = Inertia Force/Surface Tension Force

Euler’s Number = Inertia Force/Pressure Force

Mach’s or Cauchy’s Number = Inertia Force/Elastic Force

NEWTON’S NUMBER

Newton’s Number = Pressure Force/Inertia Force

Reciprocal of Euler’s Number.

Stability of Dam is checked for,

Tension at the base

Overturning of the wall or dam

Sliding of the wall.

Flow through a long pipe at constant rate = Steady Uniform Flow.

Flow through a long pipe at decreasing rate = Unsteady Uniform Flow.

Flow through expanding tube at constant rate = Steady Non-uniform Flow.

Flow through expanding tube at decreasing rate = Unsteady Non-uniform Flow.

Hydraulic Mean Depth for Circular pipe of d diameter = d/4

SURGE TANK

Purpose of surge tank

To control the pressure variations

To eliminate water hammer possibilities

To regulate flow of water to turbine

VENTURIFLUME

Flumed structure constructed across a channel by restricting its width.

Width of flumed structure is less than width of channel.

In venturiflume, the flow takes place at atmospheric pressure.

Used to measure discharge of liquid.

Co-efficient of venturiflume = 0.95 to 1.0.

D-Alembert’s Principle :- “A moving fluid mass may be brought to a static equilibrium position, by applying an imaginary inertia force of the same magnitude as that of the accelerating force but in the opposite direction.”

Density of Mercury = 13.6 x Density of Water.

In phenomenon of Cavitations, the characteristic fluid property involved is Vapour Pressure.

For stable equilibrium of floating body = GM = I/V – BG

Path traced by single particle of smoke issuing from cigarette = Path line.

Steady irrotational flow of an incompressible fluid = Potential Flow.

Flow of water in river is example of Three Dimensional Flow.

Vorticity is given by Two times Rotation.

p/w = Pressure energy per unit weight

v2/2g = Kinetic energy per unit weight

Z = Potential energy per unit weight

The Bernoulli’s Equation deals with the law of conservation of energy.

Relation between friction factor (Cf) & Coefficient of friction (f) is Cf = 4 f.

Moody diagram is used to obtain friction factor.

Value of Gas constant (R) = 287 J/kg°k

Isothermal Process :- When gas is heated or expanded in such a way that the product of its pressure and

volume remains constant, it is called Isothermal Process.

The velocity of sound is largest in steel.

Water is 80 times more compressible than steel.

Fr = 1, then Critical Flow.

Fr < 1, then Subcritical Flow or Streaming.

Fr > 1, then Supercritical Flow or Shooting.

Continuity Equation is based on Mass.