short version : 15. fluid motion. fluid = matter that flows under external forces = liquid &...
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Short Version : 15. Fluid Motion
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
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
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
15.2. Hydrostatic Equilibrium
Hydrostatic equilibrium :
Fnet = 0 everywhere in fluid
Fluid is at rest.
Fext 0 gives rise to pressure differences.
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 )
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
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
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
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
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
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
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.
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
Conservation of Mass: The Continuity Equation
Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines.
flow tubes do not cross streamlines.
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.
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
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
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
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
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
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
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