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FLUID

Characteristics of Fluid Flow (1)

- Steady flow (lamina flow, streamline flow)
- The fluid velocity (both magnitude and direction)

at any given point is constant in time - The flow pattern does not change with time
- Non-steady flow (turbulent flow)
- Velocities vary irregularly with time
- e.g. rapids, waterfall

Characteristics of Fluid Flow (2)

- Rotational and irrotational flow
- The element of fluid at each point has a net

angular velocity about that point - Otherwise it is irrotational
- Example whirlpools
- Compressible and incompressible fluid
- Liquids are usually considered as incompressible
- Gas are usually considered as highly compressible

Characteristics of Fluid Flow (3)

- Viscous and non-viscous fluid
- Viscosity in fluid motion is the analog of

friction in the motion of solids - It introduces tangential forces between layers of

fluid in relative motion and results in

dissipation of mechanical energy

Streamline

- A streamline is a curve whose tangent at any

point is along the velocity of the fluid particle

at that point - It is parallel to the velocity of the fluid

particles at every point - No two streamlines can cross one another
- In steady flow the pattern of streamlines in a

flow is stationary with time

Change of speed of flow with cross-sectional area

- If the same mass of fluid is to pass through

every section at any time, the fluid speed must

be higher in the narrower region - Therefore, within a constriction the streamlines

must get closer together

Kinematics (1)

- Mass of fluid flowing past area Aa ?ava?tAa
- Mass of the fluid flowing past area Ab ?bvb?tAb

Kinematics (2)

- In a steady flow, the total mass in the bundle

must be the same - ? ?avaAa ?t ?bvbAb ?t
- i.e. ?avaAa ?bvbAb
- or ?vA constant
- The above equation is called the continuity

equation - For incompressible fluids
- vA constant

Further reading

Static liquid pressure

- The pressure at a point within a liquid acts in

all directions - The pressure depends on the density of the liquid

and the depth below the surface - P ?gh

Further reading

Bernoullis equation

- Bernoullis equation
- This states that for an incompressible,

non-viscous fluid undergoing steady lamina flow,

the pressure plus the kinetic energy per unit

volume plus the potential energy per unit volume

is constant at all points on a streamline - i.e.

Derivation of Bernoullis equation (1)

- The pressure is the same at all points on the

same horizontal level in a fluid at rest - In a flowing fluid, a decrease of pressure

accompanies an increase of velocity

Derivation of Bernoullis equation (2)

- In a small time interval ?t, fluid XY has moved

to a position XY - At X, work done on the fluid XY by the pushing

pressure - force ? distance moved
- force ? velocity ? time
- p1A1 ? v1 ? ?t

figure

Derivation of Bernoullis equation (3)

- At Y, work done by the fluid XY emerging from the

tube against the pressure - p2A2 ? v2 ? ?t
- Net work done on the fluid
- W (p1A1 ? v1 - p2A2 ? v2)?t
- For incompressible fluid, A1v1 A2v2
- ? W (p1 - p2)A1 v1 ?t

figure

Derivation of Bernoullis equation (4)

- Gain of p.e. when XY moves to XY
- p.e. of XY - p.e. of XY
- p.e. of XY p.e. of YY - p.e. of XX - p.e.

of XY - p.e. of YY - p.e. of XX
- (A2 v2 ?t?)gh2 - (A1 v1 ?t?)gh1
- A1 v1 ?t?g(h2 - h1)

figure

Derivation of Bernoullis equation (5)

- Gain of k.e. when XY moves to XY
- k.e. of YY - k.e. of XX

figure

Derivation of Bernoullis equation (6)

- For non-viscous fluid
- net work done on fluid gain of p.e. gain of

k.e. - (p1 - p2)A1 v1 ?t A1 v1 ?t?g(h2 - h1)

figure

Derivation of Bernoullis equation (7)

- or

figure

Derivation of Bernoullis equation (8)

- Assumptions made in deriving the equation
- Negligible viscous force
- The flow is steady
- The fluid is incompressible
- There is no source of energy
- The pressure and velocity are uniform over any

cross-section of the tube

Further reading

Applications of Bernoulli principle (1)

- Jets and nozzles
- Bernoullis equation suggests that for fluid flow

where the potential energy change h?g is very

small or zero, as in a horizontal pipe, the

pressure falls when the velocity rises - The velocity increases at a constriction and this

creates a pressure drop. The following devices

make use of this effect in their action

Applications of Bernoulli principle (2)

- Bunsen burner
- The coal gas is made to pass a constriction

before entering the burner - The decrease in cross-sectional area causes a

sudden increase in flow speed - The reduction in pressure causes air to be sucked

in from the air hole - The coal gas is well mixed with air before

leaving the barrel and this enables complete

combustion

Applications of Bernoulli principle (3)

- Carburettor of a car engine
- The air first flows through a filter which

removes dust and particles - It then enters a narrow region where the flow

velocity increases - The reduced pressure sucks the fuel vapour from

the fuel reservoir, and so the proper air-fuel

mixture is produced for the internal combustion

engine

Applications of Bernoulli principle (4)

- Filter pump
- The velocity of the running water increases at

the constriction - The surrounding air is dragged along by the water

jet and this causes a drop in pressure - Air is then sucked in from the vessel to be

evacuated

Spinning ball

- If a tennis ball is cut it spins as it travels

through the air and experiences a sideways force

which causes it to curve in flight - This is due to air being dragged round by the

spinning ball, thereby increasing the air flow on

one side and decreasing it on the other - A pressure difference is thus created

Further reading

figure

Aerofoil

- A device which is shaped so that the relative

motion between it and a fluid produces a force

perpendicular to the flow - Fluid flows faster over the top surface than over

the bottom. It follows that the pressure

underneath is increased and that above reduced. A

resultant upwards force is thus created, normal

to the flow - e.g. aircraft wings, turbine blades, sails of a

yacht

Pitot tube (1)

- a device for measuring flow velocity and in

essence is a manometer with one limb parallel to

the flow and open to the oncoming fluid - The pressure within a flowing fluid is measured

at two points, A and B. At A, the fluid is

flowing freely with velocity va. At B where the

Pitot tube is placed, the flow has been stopped

Pitot tube (2)

- By Bernoullis equation

where P0 atmospheric pressure

Pitot tube (3)

- ?

- Note
- In real cases, v varies across the diameter of

the pipe carrying the fluid (because of the

viscosity) but if the open end of the Pitot tube

is offset from the axis by 0.7 ? radius of the

pipe, then v is the average flow velocity - The total pressure can be considered as the sum

of two components the static and dynamic

pressures

Pitot tube (4)

- A moving fluid exerts its total pressure in the

direction of flow. In directions at right angles

to the flow, the fluid exerts its static pressure

only - figures

Further reading paragraph of Pitot Static

System near the bottom of the page

Venturi meter (1)

- This consists of a horizontal tube with a

constriction. Two vertical tubes serving as

manometers are placed perpendicular to the

direction of flow, one in the normal part and the

other in the constriction - In steady flow the liquid level in the manometer

connected to the wider part of the tube is higher

than that in the narrower part

figure

Venturi meter (2)

- From Bernoullis principle

(h1 h2)

For an incompressible fluid, A1v1 A2v2 ?

Venturi meter (3)

- Hence

? v1 can be deduced

EXAMPLES

Streamline

Q

P

Change of speed in a constriction

Streamlines are closer when the fluid flows faster

Derivation of Bernoullis equation

v2

Y

v1

Y

p2A2

X

v2?t

X

Area A2

p1A1

h2

v1?t

h1

Area A1

Bunsen burner

Carburettor

air

filter

fuel

to engine cylinder

Filter pump

Spinning ball

Aerofoil

Pitot tube (1)

Pitot tube (2)

Pitot is here

Pitot tube fluid velocity measurement (1)

Fast moving air, lower pressure inside chamber

Static pressure holes

Stagnant air, higher pressure inside tube

Flow of air

Static tube

P1 total pressure P2 static pressure P2 P1

½(?v2)

Total tube

Pitot tube fluid velocity measurement (2)

Ventri meter (1)

Venturi meter (2)

Venturi meter (3)

Density of liquid ?

v1

v2

A2

A1