FLUID

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FLUID
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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

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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

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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

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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

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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

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Kinematics (1)

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

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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
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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
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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.

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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

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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
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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
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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
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Derivation of Bernoullis equation (5)

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

figure
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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
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Derivation of Bernoullis equation (7)

• or

figure
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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
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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

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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

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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

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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

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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
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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

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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

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Pitot tube (2)

• By Bernoullis equation

where P0 atmospheric pressure
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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

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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
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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
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Venturi meter (2)

• From Bernoullis principle

(h1 h2)
For an incompressible fluid, A1v1 A2v2 ?
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Venturi meter (3)

• Hence

? v1 can be deduced
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EXAMPLES
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Streamline
Q
P
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Change of speed in a constriction
Streamlines are closer when the fluid flows faster
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Derivation of Bernoullis equation
v2
Y
v1
Y
p2A2
X
v2?t
X
Area A2
p1A1
h2
v1?t
h1
Area A1
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Bunsen burner
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Carburettor
air
filter
fuel
to engine cylinder
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Filter pump
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Spinning ball
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Aerofoil
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Pitot tube (1)
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Pitot tube (2)
Pitot is here
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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
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Pitot tube fluid velocity measurement (2)
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Ventri meter (1)
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Venturi meter (2)
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Venturi meter (3)
Density of liquid ?
v1
v2
A2
A1