Chapter 15, Fluids

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Chapter 15, Fluids

• This is an actual photo of an iceberg, taken by a
  rig manager for Global Marine Drilling in St.
  Johns, Newfoundland. The water was calm and the
  sun was almost directly overhead so that the diver

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Physics 207, Lecture 20, Nov. 10

• Goals

• Chapter 15
• Understand pressure in liquids and gases
• Use Archimedes principle to understand buoyancy
• Understand the equation of continuity
• Use an ideal-fluid model to study fluid flow.
• Investigate the elastic deformation of solids
  and liquids

• Assignment
• HW9, Due Wednesday, Nov. 19th
• Wednesday Read all of Chapter 16

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Fluids (Ch. 15)

• At ordinary temperature, matter exists in one of
  three states
• Solid - has a shape and forms a surface
• Liquid - has no shape but forms a surface
• Gas - has no shape and forms no surface
• What do we mean by fluids?
• Fluids are substances that flow. substances
  that take the shape of the container
• Atoms and molecules are free to move.
• No long range correlation between positions.

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Fluids

• An intrinsic parameter of a fluid
• Density

units kg/m3 10-3 g/cm3
r(water) 1.000 x 103 kg/m3 1.000
g/cm3 r(ice) 0.917 x 103 kg/m3
0.917 g/cm3 r(air) 1.29 kg/m3
1.29 x 10-3 g/cm3 r(Hg) 13.6
x103 kg/m3 13.6 g/cm3
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Fluids

• Another parameter Pressure

• Any force exerted by a fluid is perpendicular to
  a surface of contact, and is proportional to the
  area of that surface.
• Force (a vector) in a fluid can be expressed in
  terms of pressure (a scalar) as

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What is the SI unit of pressure?

• Pascal
• Atmosphere
• Bernoulli
• Young
• p.s.i.

Units 1 N/m2 1 Pa
(Pascal) 1 bar 105 Pa 1 mbar 102 Pa 1
torr 133.3 Pa
1 atm 1.013 x105 Pa 1013 mbar
760 Torr 14.7 lb/ in2 (PSI)
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Pressure vs. DepthIncompressible Fluids
(liquids)

• When the pressure is much less than the bulk
  modulus of the fluid, we treat the density as
  constant independent of pressure
• incompressible fluid
• For an incompressible fluid, the density is the
  same everywhere, but the pressure is NOT!
• p(y) p0 - y g r p0 d g r
• Gauge pressure (subtract p0,
• usually 1 atm)

F2 F1 m g F1 rVg F2 /A
F1/A rVg/A p2 p1 - rg y
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Pressure vs. Depth

• For a uniform fluid in an open container
  pressure same at a given depth independent of the
  container

• Fluid level is the same everywhere in a connected
  container, assuming no surface forces

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Pressure Measurements Barometer

• Invented by Torricelli
• A long closed tube is filled with mercury and
  inverted in a dish of mercury
• The closed end is nearly a vacuum
• Measures atmospheric pressure as
• One 1 atm 0.760 m (of Hg)

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Exercise Pressure

• What happens with two fluids??
• Consider a U tube containing liquids of density
  r1 and r2 as shown
• Compare the densities of the liquids

dI
r2
r1
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Exercise Pressure

• What happens with two fluids??
• Consider a U tube containing liquids of density
  r1 and r2 as shown
• At the red arrow the pressure must be the same on
  either side. r1 x r2 (d1 y)
• Compare the densities of the liquids

dI
r2
y
r1
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Archimedes Principle

• Suppose we weigh an object in air (1) and in
  water (2).
• How do these weights compare?

• Buoyant force is equal to the weight of the fluid
  displaced

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The Golden Crown

• In the first century BC the Roman architect
  Vitruvius related a story of how Archimedes
  uncovered a fraud in the manufacture of a golden
  crown commissioned by Hiero II, the king of
  Syracuse. The crown (corona in Vitruviuss Latin)
  would have been in the form of a wreath, such as
  one of the three pictured from grave sites in
  Macedonia and the Dardanelles. Hiero would have
  placed such a wreath on the statue of a god or
  goddess. Suspecting that the goldsmith might have
  replaced some of the gold given to him by an
  equal weight of silver, Hiero asked Archimedes to
  determine whether the wreath was pure gold. And
  because the wreath was a holy object dedicated to
  the gods, he could not disturb the wreath in any
  way. (In modern terms, he was to perform
  nondestructive testing). Archimedes solution to
  the problem, as described by Vitruvius, is neatly
  summarized in the following excerpt from an
  advertisement
• The solution which occurred when he stepped into
  his bath and caused it to overflow was to put a
  weight of gold equal to the crown, and known to
  be pure, into a bowl which was filled with water
  to the brim. Then the gold would be removed and
  the kings crown put in, in its place. An alloy
  of lighter silver would increase the bulk of the
  crown and cause the bowl to overflow.
• From http//www.math.nyu.edu/crorres/Archimedes/C
  rown/CrownIntro.html

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Archimedes Principle

• Suppose we weigh an object in air (1) and in
  water (2).
• How do these weights compare?

• Why?
• Since the pressure at the bottom of the object
  is greater than that at the top of the object,
  the water exerts a net upward force, the buoyant
  force, on the object.

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Sink or Float?

• The buoyant force is equal to the weight of the
  liquid that is displaced.
• If the buoyant force is larger than the weight of
  the object, it will float otherwise it will sink.

• We can calculate how much of a floating object
  will be submerged in the liquid

• Object is in equilibrium

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Bar Trick
What happens to the water level when the ice
melts?
B. It stays the same
A. It rises
C. It drops
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Exercise
V1 V2 V3 V4 V5 m1 lt m2 lt m3 lt m4 lt
m5 What is the final position of each block?
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Exercise
V1 V2 V3 V4 V5 m1 lt m2 lt m3 lt m4 lt
m5 What is the final position of each block?
But this
Not this
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Exercise Buoyancy

• A small lead weight is fastened to a large
  styrofoam block and the combination floats on
  water with the water level with the top of the
  styrofoam block as shown.
• If you turn the styrofoam Pb upside-down,
• What happens?

(A) It sinks
(C)
(B)
(D)
Active Figure
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ExerciseBuoyancy

• A small lead weight is fastened to a large
  styrofoam block and the combination floats on
  water with the water level with the top of the
  styrofoam block as shown.
• If you turn the styrofoam Pb upside-down,
• What happens (assuming density of Pb gt water)?

(A) It sinks
(C)
(B)
(D)
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Exercise More Buoyancy

• Two identical cups are filled to the same level
  with water. One of the two cups has plastic
  balls floating in it.
• Which cup weighs more?

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Exercise More Buoyancy

• Two identical cups are filled to the same level
  with water. One of the two cups has plastic
  balls floating in it.
• Which cup weighs more?

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Exercise Even More Buoyancy

• A plastic ball floats in a cup of water with half
  of its volume submerged. Next some oil (roil lt
  rball lt rwater) is slowly added to the container
  until it just covers the ball.
• Relative to the water level, the ball will
• Hint 1 What is the buoyant force of the part in
  the oil as compared to the air?

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Exercise Even More Buoyancy

• A plastic ball floats in a cup of water with half
  of its volume submerged. Next some oil (roil lt
  rball lt rwater) is slowly added to the container
  until it just covers the ball.
• Relative to the water level, the ball will
• Hint 1 What is the buoyant force of the part in
  the oil as compared to the air?

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Pascals Principle

• So far we have discovered (using Newtons Laws)
• Pressure depends on depth Dp r g Dy
• Pascals Principle addresses how a change in
  pressure is transmitted through a fluid.

Any change in the pressure applied to an enclosed
fluid is transmitted to every portion of the
fluid and to the walls of the containing vessel.
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Pascals Principle in action

• Consider the system shown
• A downward force F1 is applied to the piston of
  area A1.
• This force is transmitted through the liquid to
  create an upward force F2.
• Pascals Principle says that increased pressure
  from F1 (F1/A1) is transmitted throughout the
  liquid.

• F2 gt F1 with conservation of energy

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Exercise Hydraulics A force amplifierAkathe
lever

• Consider the systems shown on right.
• In each case, a block of mass M is placed on
  the piston of the large cylinder, resulting in
  a difference di in the liquid levels.
• If A2 2 A1, compare dA and dB
• V10 V1 V2
• dA A1 dB A2
• dA A1 dB 2A1
• dA dB 2

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Home Example Hydraulics A force
amplifierAkathe lever

• Consider the system shown on right.
• Blocks of mass M are placed on the piston of
  both cylinders.
• The small large cylinders displace distances
  d1 and d2
• Compare the forces on disks A1 A2 ignoring
  the mass and weight of the fluid in this process

M
M
d1
A1
A2
W2 - M g d2 - F2 d2
V1 V2 A1 d1 A2 d2 d1 / d2 A2 / A1

W1 M g d1 F1 d1
W1 W2 0 F1 d1 - F2 d2 F2 d2 F1 d1
F2 F1 d1/ d2 F1 A2 / A1
implying P1 P2 F1 /A1 F2 /A2
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Fluids in Motion

• To describe fluid motion, we need something that
  describes flow
• Velocity v

• There are different kinds of fluid flow of
  varying complexity
• non-steady / steady
• compressible / incompressible
• rotational / irrotational
• viscous / ideal

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Types of Fluid Flow

• Laminar flow
• Each particle of the fluid follows a smooth
  path
• The paths of the different particles never
  cross each other
• The path taken by the particles is called a
  streamline
• Turbulent flow
• An irregular flow characterized by small
  whirlpool like regions
• Turbulent flow occurs when the particles go
  above some critical speed

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Types of Fluid Flow

• Laminar flow
• Each particle of the fluid follows a smooth
  path
• The paths of the different particles never
  cross each other
• The path taken by the particles is called a
  streamline
• Turbulent flow
• An irregular flow characterized by small
  whirlpool like regions
• Turbulent flow occurs when the particles go
  above some critical speed

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Onset of Turbulent Flow
The SeaWifS satellite image of a von Karman
vortex around Guadalupe Island, August 20, 1999
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Ideal Fluids

• Fluid dynamics is very complicated in general
  (turbulence, vortices, etc.)
• Consider the simplest case first the Ideal Fluid
• No viscosity - no flow resistance (no internal
  friction)
• Incompressible - density constant in space and
  time

• Simplest situation consider ideal fluid moving
  with steady flow - velocity at each point in the
  flow is constant in time
• In this case, fluid moves on streamlines

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Ideal Fluids

• Streamlines do not meet or cross
• Velocity vector is tangent to streamline
• Volume of fluid follows a tube of flow bounded by
  streamlines
• Streamline density is proportional to velocity

• Flow obeys continuity equation
• Volume flow rate Q Av is constant along
  flow tube.
• Follows from mass conservation if flow is
  incompressible.

A1v1 A2v2
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Exercise Continuity

• A housing contractor saves some money by reducing
  the size of a pipe from 1 diameter to 1/2
  diameter at some point in your house.

v1
v1/2

• Assuming the water moving in the pipe is an
  ideal fluid, relative to its speed in the 1
  diameter pipe, how fast is the water going in the
  1/2 pipe?

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Exercise Continuity

• For equal volumes in equal times then ½ the
  diameter implies ¼ the area so the water has to
  flow four times as fast.
• But if the water is moving four times as fast
  then it has 16 times as much kinetic energy.
• Something must be doing work on the water (the
  pressure drops at the neck and we recast the work
  as
• P DV (F/A) (ADx) F Dx )

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Lecture 20, Nov. 10

• Question to ponder Does heavy water (D20) ice
  sink or float?
• Assignment
• HW9, Due Wednesday, Nov. 19th
• Wednesday Read all of Chapter 16
• Next slides are possible for Wednesday

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Conservation of Energy for Ideal Fluid

• Recall the standard work-energy relation W DK
  Kf Ki
• Apply the principle to a section of flowing
  fluid with volume DV and mass Dm r DV (here W
  is work done on fluid)
• Net work by pressure difference over Dx (Dx1
  v1 Dt)
• W F1 Dx1 F2 Dx2
• (F1/A1) (A1Dx1) (F2/A2) (A2 Dx2)
• P1 DV1 P2 DV2
• and DV1 DV2 DV (incompressible)
• W (P1 P2 ) DV

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Conservation of Energy for Ideal Fluid

• W (P1 P2 ) DV and
• W ½ Dm v22 ½ Dm v12
• ½ (rDV) v22 ½ (rDV) v12
• (P1 P2 ) ½ r v22 ½ r v12
• P1 ½ r v12 P2 ½ r v22 const.

Bernoulli Equation ? P1 ½ r v12 r g y1
constant
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Conservation of Energy for Ideal Fluid

• This leads to
• P1 ½ r v12 P2 ½ r v22 const.
• and with height variations

Bernoullis Equation ? P1 ½ r v12 r g y1
constant
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Bernoullis Principle

• A housing contractor saves some money by reducing
  the size of a pipe from 1 diameter to 1/2
  diameter at some point in your house.

v1
v1/2
2) What is the pressure in the 1/2 pipe relative
to the 1 pipe?
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Cavitation
Venturi result
In the vicinity of high velocity fluids, the
pressure can gets so low that the fluid vaporizes.
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Applications of Fluid Dynamics

• Streamline flow around a moving airplane wing
• Lift is the upward force on the wing from the air
• Drag is the resistance
• The lift depends on the speed of the airplane,
  the area of the wing, its curvature, and the
  angle between the wing and the horizontal

higher velocity lower pressure
lower velocity higher pressure
Note density of flow lines reflects velocity,
not density. We are assuming an incompressible
fluid.
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Some definitions

• Elastic properties of solids

• Youngs modulus measures the resistance of a
  solid to a change in its length.
• Bulk modulus measures the resistance of
  solids or liquids to changes in their volume.

elasticity in length
volume elasticity
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EXAMPLE 15.11 An irrigation system
QUESTION
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EXAMPLE 15.11 An irrigation system
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EXAMPLE 15.11 An irrigation system
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EXAMPLE 15.11 An irrigation system
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EXAMPLE 15.11 An irrigation system
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Elasticity
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Elasticity
F/A is proportional to ?L/L. We can write the
proportionality as

• The proportionality constant Y is called Youngs
     modulus.
• The quantity F/A is called the tensile stress.
• The quantity ?L/L, the fractional increase in
  length, is    called strain.With these
  definitions, we can write

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EXAMPLE 15.13 Stretching a wire
QUESTIONS
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EXAMPLE 15.13 Stretching a wire
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EXAMPLE 15.13 Stretching a wire
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Volume Stress and the Bulk Modulus
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Volume Stress and the Bulk Modulus

• A volume stress applied to an object compresses
  its volume slightly.
• The volume strain is defined as ?V/V, and is
  negative when the volume decreases.
• Volume stress is the same as the pressure.

where B is called the bulk modulus. The negative
sign in the equation ensures that the pressure is
a positive number.
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The figure shows volume flow rates (in cm3/s) for
all but one tube. What is the volume flow rate
through the unmarked tube? Is the flow direction
in or out?

• 1 cm3/s, in
• 1 cm3/s, out
• 10 cm3/s, in
• 10 cm3/s, out
• It depends on the relative size of the tubes.

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Rank in order, from highest to lowest, the liquid
heights h1 to h4 in tubes 1 to 4. The air flow is
from left to right.

• h1 gt h2 h3 h4
• h2 gt h4 gt h3 gt h1
• h2 h3 h4 gt h1
• h3 gt h4 gt h2 gt h1
• h1 gt h3 gt h4 gt h2