PHYS16 – Lecture 30
On a windy day in 1735, a new wig gives Bernoulli an idea.
Fluids: Bernoulli’s Principle
November 12, 2010
Outline for Fluids
• Pressure and Pascal’s Principle
• Buoyant Force and Archimedes’ Principle
• Fluid dynamics
– Ideal Fluids
– Equation of Continuity
– Bernoulli’s Equation
Revisiting Buoyant Force…
Archimedes’ Principle
• Buoyant force = the weight of the water
displaced
 F1  F2  mg  FB
FB  m fluid displaced g   fluidVobject underwater g
http://www.open2.net/open2static/source/file/root/0/30/19/124156/pressure_cube_b.jpg
Sink or Float?
• Floating requires buoyant force to equal
gravity
FB  FG
 fluidVobject underwater g   objectVobject g
http://mcat-review.org/fluids-solids.php
Questions…
1) Does the buoyant force change as you go
deeper underwater? No, assume constant density
2) Does the buoyant force change as you go
higher in the atmosphere? Yes, changing density
3) Is buoyant force on Diet Coke vs. Coke
different? Will Diet Coke or Coke float
higher?
Buoyant force is the same, gravitational force is differet so
Diet Coke floats higher…
Demo…
•
•
•
•
Rock in boat
Sinking boat
Inverting weight + Styrofoam system
Copper ball vs. wood ball
Ideal Fluids
Ideal Fluids
•
•
•
•
Incompressible – density is a constant
Nonviscous – ignore frictional effects
Irrotational – doesn’t rotate
Laminar – no acceleration
Streamlines represent fluid flow
Ideal Fluids
•
•
•
•
Mass is conserved
Energy is conserved
Momentum is conserved
Continuum hypothesis is true – properties
defined at infinitesimal points (density,
pressure, temperature, etc.)
Which fluids are ideal?
• Water – can be turbulent (waterfall not ideal,
ideal in a slow moving river)
• Air – compressible (piston not ideal, ideal in a
laminar wind)
• Honey – viscous fluid such that drag forces
can’t be neglected (Not usually ideal)
• Blood – pulsatile flow, filled with proteins/cells
(ideal in large arteries or veins, not capillaries)
Fluid Dynamics
Equation of Continuity
• For an ideal fluid flowing in a pipe, the volume
flow rate through the pipe is constant
V
 Av  constant
t
A1v1  A2v2
Narrower section
Larger speed
Wider section
Smaller speed
Example: Water out of faucet
• Why does the stream of water flowing from a
faucet often get more narrow as the water
falls?
Gravity accelerates water so
velocity increases. If velocity
goes up, then area goes down…
http://thegoldenspiral.org/wp-content/uploads/2008/10/faucet_waterglass.jpg
Example: Arterial branching
• An artery branches into two smaller arteries,
each with half the diameter of the first. What
is the velocity in the smaller artery compared
to the larger artery?
A)
B)
C)
D)
Half
Same
Twice
Four times
http://cardiovascres.oxfordjournals.org/content/65/3/619/F4.small.gif
Bernoulli’s Equation
• For an ideal fluid flowing in a pipe, pressure in
the pipe is related to the velocity and height
of fluid
1 2
1 2
p1  gh1  v1  p2  gh2  v2
2
2
Example: Two sheets in the wind?
• What happens if I take two sheets of paper,
separate them by 1” and blow between them?
A) sheets will move apart
B) sheets will come together
C) sheets will stay at same spots
http://www.practicalphysics.org/imageLibrary/jpeg273/735.jpg
Example: Blood Pressure
• What would happen if the doctor took a blood
pressure reading at the wrist instead of on the
bicep?
A) Blood pressure would be higher
B) Blood pressure would be lower
C) Blood pressure would be the same
http://www.omron.com
Example: Aneurysm
• In an aneurysm the arterial wall weakens and
the diameter increases. Why does this
increase the chance of rupture?
A increases, v decreases,
P increases
http://www.nlm.nih.gov/medlineplus/ency/images/ency/fullsize/18072.jpg
Example: Water jets out of a bottle
• Which jet will have the largest range?
1
2
3
4
5
Main Points
• Buoyant force
FB   fluidVobject underwaterg
• Ideal fluid is incompressible, laminar,
nonviscous, and irrotational
• Equation of continuity
• Bernoulli’s Equation
Av  constant
1 2
p  gh  v  constant
2