Fluids in Motion
Outline
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Archimedes (review)
Flow Rate
Equation of Continuity
Bernoulli’s Equation
Example: Tip of the Iceberg
The density of ice is 920 kg/m3, and that of seawater
is 1030 kg/m3. What fraction of the total volume of
an iceberg is exposed?
Apply Newton’s 2nd Law:
0
Fy  ma y
+y
B  mg  0
mg
 wVs g  iVi g  0
B
Volume
submerged
Example: (cont.)
 wVs g  iVi g  0
Vs  i

Vi  w
Vs 920
 0.90  90%

Vi 1030
Since 90% of an iceberg is submerged, only 10% is
above the sea.
“Ideal” Fluid
An “ideal” fluid with the following properties is
easier to study.
• no viscosity
• incompressible
• fluid motion is steady (constant in time)
• no turbulence (no whirlpools or eddies)
Flow Rate
Flow Rate: the volume of fluid passing by a point
in a tube or pipe per unit time
Flow Rate  Av
fluid speed
cross sectional area
Units:
m m
m  
s
s
2
Also: gallon/min
3
Demo: Water speed
Determine the flow rate of a faucet by timing how long
it takes to fill a beaker of known volume.
Measure the diameter of the faucet to get the
2
cross-sectional area:
D
A  
2
Divide flow rate by cross-sectional area to get
the speed of the water.
Equation of Continuity:
A1v1  A2v2
The flow rate does not change even if the pipe
diameter changes, since the speed of the fluid
changes to compensate.
Demo:
slow
fast
(Pearls simulation)
Examples: Equation of
Continuity
• hypodermic needle
• nozzle on garden hose
• rivers (slow current to white-water)
• toll booths, train yards, etc...
Bernoulli’s Equation
2
flow
1
P1  v  gy1  P2  v  gy2
1
2
2
1
P = pressure (gauge or absolute)
 = fluid density
v = fluid speed
y = vertical position of fluid
1
2
2
2
Example: Venturi Tube
Continuity Equation:
Thus,
A1v1  A2v2
v2 > v1
A1 v2

A2 v1
Example: Venturi Tube
Bernoulli’s Equation:
0
P1  12 v  gy1  P2  12 v  gy2
2
1
2
2
P1  P2  v  v
2
2
1
2

2
1
1
2
P1  P2   v  v
1
2
2
2
2
1
Thus,

P1 > P2
0
positive since
v2 > v 1
Bernoulli’s Principle:
Swiftly moving fluids exert less pressure than
do slowly moving fluids.
slow
fast
High P
Low P
Demo: Dynamic Lift
Fast moving air above ball
has a lower pressure than
still air below the ball, causing
a “dynamic lift”.
When you blow through the
hole, the fast moving air is
at the bottom. The low pressure
keeps the ball near the hole.
fast
fast