short version : 15. fluid motion
DESCRIPTION
Short Version : 15. Fluid Motion. Fluid = matter that flows under external forces = liquid & gas. 15.1. Density & Pressure. thousands of molecules. Avogadro’s number N A = 6.022 10 23 / mol . 1 mole = amount of substance containing N A basic elements. - PowerPoint PPT PresentationTRANSCRIPT
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Short Version : 15. Fluid Motion
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Fluid = matter that flows under external forces
= liquid & gas.
solid liquid gas
inter-mol forces strongest medium weakest
volume fixed fixed variable
shape fixed variable variable
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15.1. Density & Pressure
Avogadro’s number NA = 6.022 1023 / mol . 1 mole = amount of substance containing NA basic elements. ( with NA = number of atoms in 12 g of 12C ).
Fluid: average position of molecules not fixed.
Macroscopic viewpoint: deformable continuum.
Density = mass / vol, [ ] = kg / m3 .
31 /water g cm 1 /g cc 31000 /kg m1 /kg liter 1 1000liter cc1000 ml
310air water
Incompressible = density unchanged under pressure
Liquid is nearly incompressible (molecules in contact).
Gas is compressible.
dVfluid pointdV 0
thousands of molecules
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Pressure
Pressure = normal force per unit area
Fp
A 2/p N m
pascal Pa
1 101,300atmosphere atm Pa 101.3 kPa
Pressure is a scalar.
The pressure at a point in a fluid is the magnitude
of the radial force per unit area acting on a fluid
point at that position.
14.7 pounds per square inchpsi
A
F n
A n
F
Fluid point
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15.2. Hydrostatic Equilibrium
Hydrostatic equilibrium :
Fnet = 0 everywhere in fluid
Fluid is at rest.
Fext 0 gives rise to pressure differences.
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netF F x F x x
P x P x x A
PV
x
netF d Pf
V d x
P f
x
Let f be the force density within the fluid :
Force experienced by the fluid element:
( f is the force per unit volume experienced by a small fluid element due to pressure differences )
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Hydrostatic Equilibrium with Gravity
Fluid element: area A, thickness dh, mass dm.
Net pressure force on fluid element:
pdF p d p A p A A d p
Gravitational force on fluid element:
gdF g d m g A d h
Hydrostatic Equilibrium : 0p gdF dF
d p g d h d pg
d h
Liquid (~incompressible):
0p p g h p g h
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Measuring Pressure
Barometer = device for measuring atmospheric pressure
0p p g h
0 0p vacuum inside tube:
313.6 /Hg g cm 3 313.6 10 /kg m
Hgp g h 2 3 39.8 / 13.6 10 /m s kg m h
133.28 /kPa m h
For p = 1 atm = 101.3 kPa :
101.3
133.28 /
kPah
kPa m 0.760 m 760 mm
Cf. h = 10 m for a water barometer
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Manometer
Manometer = U-shaped tube filled with liquid to measure pressure differences.
fluid atm Hgp p g h
Gauge pressure = excess pressure above atmospheric.
Used in tires, sport equipments, etc.
E.g., tire gauge pressure = 30 psi tire pressure = 44.7 psi
Pascal’s law:An external pressure applied to a fluid in a closed vessel is uniformly transmitted throughout the fluid.
equal p
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Example 15.2. Hydraulic Lift
In a hydraulic lift, a large piston supports a car.
The total mass of car & piston is 3200 kg.
What force must be applied to the smaller piston to support the car?
11
2
m g AF
A
11
1
Fp
A
2p
2
1
2
dm g
d
2
2 153200 9.8 /
120
cmkg m s
cm
490 N
Pascal’s law2
m g
A
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15.3. Archimedes’ Principle & Buoyancy
Archimedes’ Principle:
The buoyancy force on an object is equal to the
weight of the fluid it displaces.
Buoyancy force:
Upward force felt by an object in a fluid
Neutral buoyancy :
average density of object is the same as that of fluid.
fluid element in equilibrium
Fb unchanged after replacement
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Example 15.4. Tip of the Iceberg
Average density of a typical iceberg is 0.86 that of seawater.
What fraction of an iceberg’s volume is submerged?
0g bF F
b subF m g
g iceF m g
water subV g
0.86
ice iceV g
sub ice
ice water
V
V
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Center of Buoyancy
Buoyancy force acts at the center of buoyancy (CB),
which coincides with the CM of the displaced water.
CM must be lower than CB to be stable.
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15.4. Fluid Dynamics
Moving fluid is described by its flow velocity v( r, t ).
Streamlines = Lines with tangents everywhere parallel to v( r, t ).
Spacing of streamlines is inversely proportional to the flow speed.
Steady flow: , t v r v r
Small particles (e.g., dyes) in
fluid move along streamlines.
e.g., calm river.
Example of unsteady flow: blood in arteries ( pumped by heart ).
Fluid dynamics: Newton’s law + diffusing viscosity Navier-Stokes equations
slow fast
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Conservation of Mass: The Continuity Equation
Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines.
flow tubes do not cross streamlines.
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Steady flow
Conservation of mass:
1 1 1 1m A v t Mass entering tube:
2 2 2 2m A v t Mass leaving tube:
1 1 1 2 2 2A v A v
A v const v A
Equation of continuity for steady flow :
Mass flow rate = [ v A ] = kg / s
Volume flow rate = A v constLiquid:
[ v A ] = m3 / s v A
Liquid : flows faster in constricted area.
Gas with v < vs ound: flows faster in constricted area.
Gas with v > vsound : flows slower in constricted area.
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Conservation of Energy: Bernoulli’s Equation
Same fluid element enters & leaves tube:
2 22 1
1
2K m v v
Work done by pressure upon its entering tube:
1 1 1 1W p A x
Work done by pressure upon its leaving tube: 2 2 2 2W p A x
Work done by gravity during the trip: 2 1gW m g y y
W-E theorem: 1 2 gW W W K 2 21 1 2 2 2 1 2 1
1
2p V p V m g y y m v v
1 1p V
2 2p V
Incompressible fluid: 1 2V V V m
V
21
2p v g y const
Bernoulli’s Equation
Viscosity & other works neglected
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Example 15.6. Draining a Tank
A large open tank is filled to height h with liquid of density .
Find the speed of liquid emerging from a small hole at the base of the tank.
atmp p
21
2 holev g h
y h
At top surface :
0v
21
2p v g y const
At hole:
atmp p 0y holev v
2holev g h
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Example 15.7. Venturi Flowmeter
Find the flow speed in the unconstricted pipe of a Venturi flowmeter.
1 1 2 2v A v A
2 21 1 2 2
1 1
2 2p v p v
Bernoulli’s eq.
Continuity eq.
12 1
2
Av v
A
2
211 2 1
2
11
2
Av p p p
A
1 2
1
2
2
1
pv
A
A
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Bernoulli Effect
A ping-pong ball supported by downward-flowing air.
High-velocity flow is inside the narrow part of the funnel.
Bernoulli Effect: p v
Example: Prairie dog’s hole
Dirt mound forces wind to accelerate over hole
low pressure above hole
natural ventilation
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Flight & Lift
Aerodynamic lift
Top view on a curved ball : spin
Blade pushes down on air
Air pushes up (3rd law) Faster flow, lower P : uplift.
Top view on a straight ball : no spin
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Application: Wind Energy
A chunk of air, of speed v & density ,
passing thru a turbine of area A in time t,
has kinetic energy
21
2K v v A t
31
2v A t
available power per unit area = 31
2A vP
Better analysis 38
27vP
3381.2 / 10 /
27kg m m sP 2350 /W m
For 10 / 36 /v m s km h
Present tech gives 80% of this.
0.6 A P
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15.6. Viscosity & Turbulence
Smooth flow becomes turbulent.
Viscosity: friction due to momentum transfer between
adjacent fluid layers or between fluid & wall.
B.C.: v = 0 at wall
• drag on moving object.
• provide 3rd law force on propellers.
• stabilize flow.
flow with no viscosity
flow with viscosity