Lesson 10
Physics 168
Thursday, November 5, 15
Luis Anchordoqui
1
FLUIDS
Luis Anchordoqui
Thursday, November 5, 15
2
Phases of matter
Overview
Three common phases of matter are
➢
solid ☛
➢
liquid ☛ has fixed volume but can be any shape
➢
gas ☛
has definite shape and size
can be any shape and also can be easily compressed
Liquids and gases both flow and are called fluids
C.
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Thursday, November 5, 15
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Density and Specific Gravity
Overview
Density ⇢ of an object is its mass per unit volume:
m
⇢ =
V
SI unit for density is
Water at 4 C has a density of
Specific gravity of a substance
kg/m3
1 g/cm3 = 1.000 kg/m3
ratio of its density to that of water
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Pressure in Fluids
Overview
Pressure is defined as force per unit area ☛
P = F/A
Pressure is a scalar
Units of pressure in SI system are pascals
1 P a = 1 N/m2
Pressure is same in every direction in a fluid at a given depth
if it were not fluid would flow
C.
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Pressure in Fluids (cont’d)
For fluid at rest
Overview
there is no component of force parallel to any solid surface
Fk = 0
Again ☛ if there were fluid would flow
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Pressure in Fluids (cont’d)
Overview
Pressure at depth h below surface of liquid
is due to weight of liquid above it
We can quickly calculate :
P = mg/A = ⇢gh
This relation is valid for any liquid whose density does not change with depth
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Pressure in Fluids (cont’d)
Overview
To a good approximation liquids can be considered incompressible
Gases on other hand are very compressible
and density can vary significantly
Forces on a thin slab of fluids
(shown as a liquid but it could instead be a gas)
We assume fluid is at rest so net force on slab is zero
(P +
P)A
PA
⇢A hg = 0
P = ⇢g h
⇢ ⇡ constant over h
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Thursday, November 5, 15
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Atmospheric Pressure and Gauge Pressure
Overview
At sea level atmospheric pressure is about
1.013 ⇥ 105 N/m2
this is called one atmosphere (atm)
Another unit of pressure is bar:
5
2
1 bar = 1.00 ⇥ 10 N/m
Standard atmospheric pressure is just over 1 bar
This pressure does not crush us
because our cells maintain an internal pressure that balances it
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Thursday, November 5, 15
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Measurement of Pressure manometer
Overview
There are a number of different types of pressure gauges
This one is an open-tube manometer
Pressure in open end is atmospheric pressure
Pressure being measured will cause fluid
to rise until pressures on both sides at same height are equal
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Gauge Pressure
Overview
Most pressure gauges measure pressure above atmospheric pressure
this is called gauge pressure
Absolute pressure is sum of atmospheric pressure and gauge pressure
P = PA + PG
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Measurement of Pressure Barometer
Overview
This is a mercury barometer
developed by Torricelli to measure atmospheric pressure
Height of column of mercury is such that pressure
in tube at surface level is 1 atm
pressure is often quoted in millimeters of mercury
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Measurement of Pressure Barometer (Cont’d)
Overview
Any liquid can serve in a Torricelli-style barometer
but most dense ones are most convenient
This barometer uses water 10.3 m high!
C.
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Thursday, November 5, 15
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13
Buoyancy and Archimedes’ Principle
Overview
Consider an object submerged in a fluid
There is a net force on object
because pressures at top and bottom of it are different
Buoyant force is found to be upward force on same volume of water
F B = F2
F1 = ⇢F g A(h2
= ⇢F g A
h1 )
h
= ⇢F V g
= mF g
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Archimedes principle
Overview
We can derive Archimedes' principle in general
by following simple but elegant argument
Irregularity shaped object D shown in figure
is acted on by force of gravity and buoyant force
To determine buoyant force we next consider a body D' this time made of
fluid itself with shape and size of original object and located at same depth
You might think of this body of fluid as being separated from rest of fluid
by an imaginary membrane
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Archimedes principle (Cont’d)
Overview
Buoyant force on this body of fluid will be exactly same as that on original
object since surrounding fluid which exerts F is in exactly same configuration
Body of fluid D' is in equilibrium
Buoyant force is equal to weight of body of fluid whose volume equals volume
of original submerged object
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Buoyancy and Archimedes’ principle
Net force on object is then difference between buoyant force
and gravitational force
Luis Anchordoqui
Thursday, November 5, 15
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Buoyancy and Archimedes’ principle (cont’d)
For a floating object
Overview
fraction that is submerged
is given by ratio of object’s density to that of fluid
C.
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Fluids in Motion
Flow Rate and Equation of Continuity
Overview
If flow of a fluid is smooth, it is called streamline or laminar flow
Above a certain speed, flow becomes turbulent
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Fluids in Motion
Flow Rate and Equation of Continuity
Overview
We will deal with laminar flow
Mass flow rate is mass that passes a given point per unit time
Flow rates at any two points must be equal, as long as no fluid
is being added or taken away
This gives us equation of continuity
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Equation continuty
Overview
Consider a steady laminar flow of a fluid through an enclosed pipe
Mass flow rate
=
m
t
Volume of fluid passing point1 (that is through area A1 )
in a time
t
is
A1
l1
Distance fluid moves in time
t
Since velocity of fluid passing through point 1 is
v1 =
l1 / t
m1
⇢1 V 1
⇢ 1 A1 l 1
=
=
= ⇢1 A1 v1
t
t
t
C.
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Thursday, November 5, 15
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Equation continuty (cont’d)
Similarly at point
2
⇢ 2 A2 v 2
Overview
flow rate is
Since no fluid flows in or out sides flow rates through A1 and A2 must be equal
m1
=
t
m2
t
⇢1 A1 v1 = ⇢2 A2 v2
If
⇢ =
constant continuity equation becomes
A1 v 1 = A2 v 2
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Thursday, November 5, 15
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In humans, blood flows from heart into aorta, from which it passes into
major arteries
Overview
These branch into small arteries (arterioles), which in turn branch into
myriads of tiny capilars
Blood returns to heart via veins
Radius of aorta is about
1.2 cm, and blood passing
through it has a speed of about
40 cm/s
A typical capilar has a radius of about 4 ⇥ 10 4 cm
and blood flows through it at a speed of about 0.0005 m/s
Estimate number of capillars that are in body
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Thursday, November 5, 15
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Overview
2
2
v2 A2 = v1 A1 ) v2 N ⇡rcap
= v1 ⇡raorta
2
v1 rcap
9
=
7
⇥
10
N=
2
v2 raorta
C.
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Thursday, November 5, 15
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Bernoulli’s equation
Overview
Consider a small parcel of air moving along a streamline into a region
of reduced pressure
dv
F = m
dt
F = PA
Parcel is so small that
P
(P +
P)A =
A P
can be accurately expressed using
differential approximation
P
dP
=
)
l
dx
dP
P =
dx
l
Substituting
dP
dv
A
l = ⇢A l
dx
dt
dv
dP = ⇢
dx = ⇢ v d v
dt
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Thursday, November 5, 15
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Bernoulli’s equation (Cont’d)
Overview
Integrating both sides
Z
P2
dP =
⇢
P1
Z
v2
v dv
v1
We obtain Bernoulli equation
P2
1 2
P1 = ⇢ v 1
2
1 2
⇢ v2
2
or equivalently
1 2
1 2
P2 + ⇢ v2 = P1 + ⇢v1
2
2
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Thursday, November 5, 15
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Bernoulli’s equation (Cont’d)
Overview
A fluid can also change its height
By looking at work done as it moves, we find:
1 2
P + ⇢v + ⇢gy = constant
2
Bernoulli’s equation tells us that
as speed goes up, pressure goes down
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Applications of Bernoulli’s Principle: Airplanes
Overview
Lift on an airplane wing is due to different air speeds
and pressures on two surfaces of wing
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Figure shows tropical storm Katrina as observed by NASA's QuikSCAT
satellite on August 25, at 4:37 am in Florida
At this time storm had 50 miles per hour sustained winds
storm does not appear to yet have reached hurricane strength
Assume that air pressure outside storm is atmospheric pressure
and that speed of wind in this region is negligible to give a rough
estimate air pressure inside storm (1 mile approx 1.6 km)
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Hurricane Katrina near peak strength on August 28, 2005
Storm reached category 5 hurricane, with wind speed of 300 km/h
It was sixth-strongest Atlantic hurricane ever recorded and thirdstrongest hurricane on record that made landfall in United States
Estimate air pressure inside category 5 hurricane and compare results
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Applications of Bernoulli’s Principle: Hurricanes
Overview
1 2
1 2
Pinside + ⇢vinside + ⇢gyinside = Poutside + ⇢voutside + ⇢gyoutside
2
2
measurement taken @ same altitude
yinside = youtside
voutside ⇡ 0
storm
Pinside = 1.013 ⇥ 105 Pa
hurricane
Pinside = 9.7 ⇥ 104 Pa ⇡ 0.96 atm
C.
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Thursday, November 5, 15
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Viscosity
Overview
Real fluids have some internal friction called viscosity
Viscosity can be measured
It is found from relation
v
F = ⌘A
l
coefficient of viscosity
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Coefficients of Viscosity for Various Fluids
Overview
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Poiseuille’s Equation
Overview
Rate of flow in a fluid in a round tube depends on viscosity of fluid
pressure difference and dimensions of tube
Volume flow rate is proportional to pressure difference,
inversely proportional to length of tube L
and proportional to fourth power of radius R of tube
⇡ R4 (P1
P2 )
Q =
8⌘L
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If cholesterol build-up reduces diameter of an artery by 15%
Overview
what will be effect on blood flow?
4
Qfinal
Qinitial
Qfinal
Rfinal
4
=
)
=
=
0.85
= 0.52
4
4
4
Rfinal
Rinitial
Qinitial
Rinitial
Flow rate is 52% of original value
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Thursday, November 5, 15
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Giraffes are a wonder of cardiovascular engineering
Calculate difference in pressure (in atmospheres) that blood vessels of a
Overview
giraffe's head have to accomodate as head is lowered from a full upright
position to ground level for a drink
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Overview
Height of an average giraffe is about 6 m
P = ⇢g
h
P = 0.6 atm
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Turbulence: Reynolds Number
Overview
When flow speed of a fluid becomes sufficiently great
laminar flows breaks down and turbulence flow sets in
Critical speed above which flow through a tube is turbulent
depends on density and viscosity of fluid and on radius of tube
Flow of a fluid can be characterized
by a dimensionless number called Reynolds number
NR
2r ⇢v
=
⌘
Experiments have shown that
flow will remain laminar if Reynolds number is less than about 2000
and turbulent if it is greater than 3000
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Luis
Anchordoqui
Thursday, November 5, 15
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Smoke from a burning cigarrete
At first smoke rises in a regular
stream, but simple streamline
quickly becomes turbulent and
smoke begins to swirl irregularly
Luis Anchordoqui
Thursday, November 5, 15
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