Fluid
 A fluid refers to any substance that is capable of flowing due to pressure difference.
 It includes both liquids and gases.
 Examples of fluid flow include: perfume spray from a perfume bottle, flow of water along a river bed, smoke from chimney etc.
 A flowing fluid experiences internal resistance called viscosity.
Types of Fluid Flow
 There are two types of fluid flow: streamline(steady) and turbulent flows
1. Streamline(steady) flow
 It is a flow in which at any given point each and every particle of the fluid travels in the same direction and with same velocity.
 A streamline refers to the path followed by the particle in a streamline flow. It is represented by a line with an arrow head.
 Note: Streamlines do not cross each other but are closer where the fluid is moving faster.
Characteristics of Streamline Flow
 Streamlines are parallel to each other.
 Streamline flow is smooth and steady.
 Some shapes and bodies are designed to be streamlined to enhance their motion in fluids.
 A body is said to be streamlined if it does not affect the distribution of streamlines behind it.
 Examples of streamlined bodies include: cars, jumbo jets,birds that fly, fish etc.
2. Turbulent Flow
 It is a flow in which the speed and direction of the fluid particles passing at any point vary with time.
 Turbulent flow occurs due to:
 Abrupt change of crosssectional area of the tube of flow.
 Speed of the fluid flow changes sharply or suddenly and beyond a critical velocity.
 An obstacle is placed on the path of streamlines and blocks or breaks the streamlines.
 Abrupt change of crosssectional area of the tube of flow.
Characteristics of Turbulent Flow
 The streamlines are not continuous
 Particles do not travel in same direction and have different velocity.
Notes:
 When bodies which are not streamlined (non streamlined) move in fluids,they cause eddies (turbulence) in the fluid. A body is said to be non streamlined if it produces eddies behind it.
 Critical velocity is the speed of flow of fluid beyond which the fluid exhibits turbulent flow.
Volume Flux(Flow Rate)
 This is the volume of a fluid passing through a given section of a tube of flow per unit time.
Volume flux = ^{volume of fluid passing given section}/_{time the fluid takes to pass the section}  SI unit of volume flux is cubic meter per second (m^{3}/s)
 Consider a fluid flowing through a section B of flow tube shown below.
 If the velocity of fluid through region B is v_{B}, the average crosssection area of tube is A_{B} and the distance covered by the fluid in direction of flow for time, t_{B}, is d_{B}, then the volume flux through that region is:
volume flux or flow rate= ^{Volume}/_{time }=^{V}/_{tB}
But volume = cross–sectionarea × length
V= A_{B} × d_{B}
Volumeflux = ^{(dB x AB)}/_{tB}= ^{dB}/_{tB }x A_{B}
But, ^{dB}/_{tB} =Velocity, v_{B}
∴ Volume flux=v_{B }× A_{B}
Volume flux = velocity × cross section area of tube of flow
Mass Flux
 It is the mass of a fluid tha tflows through a given section of tube of flow per unit time.
mass flux = ^{mass}/_{time}
But,mass = density×volume.
That is, m=ρ×V.
∴mass flux =^{ (ρ×V)}/_{t}But, ^{V}/_{t} =volume flux.
mass flux = density of fluid, ρ×volume flux
∴mass flux = density of fluid × velocity of fluid×crosssection area of tube
The Equation of the Continuity
Assumptions made in deriving the equation of the continuity
 The fluid is flowing steadily (i.e.has a streamline flow)
 The fluid is incompressible
 The fluid is nonviscous.
Deriving Equation of Continuity
 Consider the tube of flow below with changing crosssection areas.
 Section 1 has a crosssection area of A_{1 }while section 2 has crosssectionarea of A_{2}.
 Velocity of fluid in section 1 is v_{1} while in section 2 is v_{2}.
 Volume of fluid flowing through section 1 per unit time is equal to volume o ffluid flowing through section 2 per unit time i.e.flow rate/volume flux is a constant.
Volume flux in section 1=volume flux in section 2
A_{1}v_{1}=A_{2}v_{2}i.e.crosssection area × velocity= constant
Av=constant._{}  This is the equation of continuity which is also called flow rate equation.
Examples
 Water flows through a horizontal pipe at a rate of 1.00m^{3}/min. Determine the velocity of the water at a point where the diameter of the pipe is 1.00cm.
Solution
 In figure below, the tube ABC is filled with a liquid. The piston moves from A to B in 1 second.
 What is the volume of the liquid in point AB
Solution
volume=crosssection area × length
volume=1×10^{4} m^{2 }× 8 × 10^{2} m=8× 10^{6}m^{3}
 What is the velocity of the liquid between A and B?
Solution
 What is the velocity of the liquid between BC?
 What is the volume of the liquid in point AB
Exercise
 A garden sprinkler has small holes, each 2.00mm^{2 }in area. If water is supplied at the rate of 3.0x10^{3}m^{3}s^{1 }and the average velocity of the spray is 10ms^{1}, calculate the number of the holes.
 Oil flows through a 6cm internal diameter pipe at an average velocity of 5ms^{1}. Find the flow rate in m^{3}/s and cm/s
 The velocity of glycerin in a 5cm internal diameter pipe is 1.00m/s. Find the velocity in a 3cm internal diameter pipe that connects with it,both pipes flowing full.
Bernoulli’s Effect
 It states that: provided a fluid is nonviscous, incompressible and its flow streamline, an increase in its velocity produces a corresponding decreases in the pressure it exerts while a decrease in its velocity produces a corresponding increase in pressure.
Bernoulli’s Effect in Practice
 Consider the setup below in which pipe A and C have some diameter tubes.
 When air is blown into the tube by a blower, it is observed that water rises to same level in tube D andF. In E the level of water is higher than D and F.
 Velocity of air in pipe A and C are the same due to same crosssectional areas. Moving air causes a reduction of pressure and since resulting air pressure is the same, atmospheric pressure pushes up the water to the same level.
 The speed of moving air in narrower section B is higher and the resulting pressure is much lower than A and C,hence water rises to higher level in E.
 Consider the setup below in which pipe A and C have some diameter tubes.

 When air is blown above the opening of the flask shown the pith ball is observed to rise from the bottom.
 The blown air causes reduction of pressure at the top therefore, there is a net force upwards as the pressure difference pushes the pith ball upwards.
 When air is blown above the opening of the flask shown the pith ball is observed to rise from the bottom.

 When air is blown horizontally between two suspended balloons in the horizontal direction, the balloons are observed to move towards each other.
 Moving air leads to reduced pressure on the inner sides of the balloons. The higher atmospheric pressure acting on the outer surfaces causes the balloons to move closer to each other.

 A light paper held in front of the mouth and air blown horizontally over it is observed to rise. This is because the velocity of air above paper increases leading to reduction in pressure. The higher atmospheric pressure acting from below produces a force that lifts the paper upwards.
 A light paper held in front of the mouth and air blown horizontally over it is observed to rise. This is because the velocity of air above paper increases leading to reduction in pressure. The higher atmospheric pressure acting from below produces a force that lifts the paper upwards.
Bernoulli’s Principle
 It states that: ”provided the fluid is nonviscous incompressible and has a streamline flow, the sum of pressure, kinetic energy per unit volume and potential energy per unit volume is a constant”.
Mathematical Expression for Bernoulli’s Principle
 Consider a fluid of density,ρ, mass,m, flowing through a pipe with a velocity, v and pressure at any given point, P.
Applications of Bernoulli’s Principle
1. The Aerofoil
 It is a structure designed in such way that the fluid moving above it moves with a higher speed than the one moving below.
 The pressure above the aerofoil is therefore lower than the pressure below it.The pressure difference between the top and bottom gives rise to the lift of the aerofoil.
 This is called dynamic lift.
2. Bunsen burner
 When gas is made to flow into the Bunsen burner, its velocity increases as it passes through the nozzle; this decreases the pressure above the nozzle.
 Because of higher atmospheric pressure outside the barrel, air is then drawn in.
 The air and the gas then mix as they rise up and when ignited a flame is produced.
3. Spray Gun
 When the piston is pushed forward, air is made to flow through the barrel and therefore causes low pressure in the barrel.
 High atmospheric pressure on the surface of the liquid compels the liquid to move up the tube.
 The velocity of the liquid is increased as it pushes through the nozzle due to reduced crosssection area. The liquid therefore emerges as a fine spray.
4. The carburetor.
 Air velocity at constriction is higher. This makes the pressure at the constriction drop. The atmospheric pressure being higher pushes the petrol to the constriction.
Revision Exercise
 The figure below shows a pith ball placed in a flask. When a jet of air is blown over the mouth of the flask as shown, the pith ball is observed to rise from the bottom.
Explain this observation.
 State Bernoulli’s principle
 A pipe of radius 6mm is connected to another pipe of radius 9mm. If water flows in the wider pipe at the speed of 2ms^{1}, what is the speed in the narrower pipe?
 The figure below shows a tube of varying crosssection area.v_{1},v_{2},v_{3},v_{4}, represent the speed of water as it flows steadily through the sections of the tube.
Arrange the speeds v_{1},v_{2},v_{3},v_{4} in decreasing order starting with the highest.  The figure below shows a sheet of paper rolled into a tube.
When a fast stream of air is blown into the tube as shown in the diagram, the paper tube collapses. Explain the observation.  The figure below shows a horizontal tube with two vertical tubes X and Y. Waterflows through the horizontal tube from right to left. The water level in tube X is higher than water level in tube Y.
Explain this observation