Fluid Mechanics
Chapter 13
Fluid
• Anything that can flow
• A liquid or a gas
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Density
• Mass per unit volume
m
r
V
• Where r (rho) is the density, m is
mass, and V is volume
• A homogeneous material has the
same density throughout
• The SI unit of density is the kg/m3.
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Specific gravity
• Should be called relative density, but
we are stuck with the traditional
term
• The ratio of its density to the density
of water.
r material
specific gravity 
r water
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Pressure
• Pressure is force per unit area,
expressed in Pascals (Pa).
1 Pa = 1 N/m2
F
p
A
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Pressure changes
• Pressure increases with depth
– Atmospheric pressure is greater at sea
level than on top of a mountain
– Water pressure is greater in deeper
water
p2  p1  rg  y2  y1 
• When y2 is greater, p2 is less.
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Open containers
• The pressure at the surface is
atmospheric pressure, or p0. If we
are at a depth, h, below the surface,
p0  p   rgh
p  p0  rgh
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Pascal’s Law
• If we increase the pressure at the
surface, the pressure at any depth
increases by the same amount.
• The pressure is transmitted
throughout the fluid – if it has a
uniform density – this is a fairly safe
assumption for most liquids and for
gases over small distances.
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Pascal’s Law
• Used in hydraulics to use a small
force over a small area to exert a
large force over a large area.
– See page 304
F1 F2
p

A1 A2
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Gauge pressure
• If the pressure in your tire equals
atmospheric pressure, the tire is flat.
• When your pressure gauge reads
32 psi, that means the pressure in
the tire is 32 psi above the
atmospheric pressure.
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Absolute pressure
• Total pressure
pa  p0  p g
• Atmospheric pressure at sea level is
101.3 kPa or 14.7 psi. Also called
1 atm
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Example
• A residential hot water heating
system has an expansion tank in the
attic, 12 m above the boiler. If the
tank is open to the atmosphere,
what is the gauge pressure in the
boiler? What is the absolute
pressure?
• pg= 118 kPa, pa=219 kPa
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Open-tube manometer
• Measures the
pressure in a
container.
p  p0  rgh  p gauge
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Barometers
• Long glass tubes full of mercury used
to measure atmospheric pressure.
p  p0  rgh
0  p0  rgh
h
p0  rgh
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Buoyancy
• When an object is less dense than
water, it floats.
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Archimedes's principle
• When an object is completely or
partially immersed in a fluid, the
fluid exerts an upward force on the
object equal to the weight of the
fluid displaced by the object.
• We call this upward force the
buoyant force.
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Weight of water displaced
m
r
V
m  rV
w  mg
w  rgV
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Example
• A cork has a density of 200 kg/m3.
Find the fraction of the volume of the
cork that is submerged when the
cork floats in water.
Fnet , y  Fbuoyant  wcork  0
wwater  wcork  0
wwater  wcork
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Example continued
r water gVsumberged  r cork gVtotal
r waterVsumberged  r corkVtotal
Vsumberged
Vtotal
Vsumberged
Vtotal
r cork

r water
3
200 kg/m
1


3
1000 kg/m
5
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Example
• An ore sample weighs 14.00 N in air.
When the sample is suspended by a
light cord and totally immersed in
water, the tension in the cord is
9.00 N. Find the total volume and
the density of the sample.
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Example continued
Fnet  T  FB  w  0
FB  w  T
r water gV  w  T
w T
V
r water g
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Example continued
14.00 N  9.00 N
V
1000 kg/m 3 9.8 m/s 2



V  5.10 10 4 m 3
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Example continued
w  rgV
w
r
gV
14.00 N
r
2
4
3
9.8 m/s 5.10 10 m


r  2801 kg/m
Physics Chapter 13
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3
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Fluid Flow
• An ideal fluid is incompressible and
has no internal friction.
• We will only deal with laminar flow,
which has a steady-state pattern.
• We will not deal with turbulent flow,
which is chaotic.
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Continuity equation
• The mass of a moving fluid doesn’t
change as it flows.
rA1v1t  rA2v2t
A1v1  A2v2
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Volume flow rate
• The rate at which volume crosses a
section of the tube:
V
 Av
t
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Example
• Blood flows from an artery of radius
0.3 cm, where it’s speed is 10 cm/s
into a region where the radius has
been reduced to 0.2 cm. What is the
speed of the blood in the narrower
region?
• 22.5 cm/s
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Bernoulli’s Equation
• Relates pressure, flow speed, and
height for flow of ideal fluids.
• Derived in book by applying the work
energy theorem to a flowing fluid
1 2
1 2
p1  rgy1  rv1  p2  rgy2  rv2
2
2
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Bernoulli’s Equation
• Make sure that your units are
consistent.
– Always use pascals, kg/m3, and m/s
• Always use either all absolute
pressures or all gauge pressures.
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Example
• A large tank of
water has a
small hole a
distance h below
the water
surface. Find
the speed of the
water as it flows
from the tank.
• Vb=sqrt(2gh)
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On your own
• Water enters a house through a
pipe with an inside diameter of
2.0 cm at an absolute pressure of
4.0 x 105 Pa. A 1.0 cm diameter
pipe leads to the second-floor
bathroom 5.0 m above. When the
flow speed at the inlet pipe is
1.5 m/s, find the pressure and
volume flow rate in the bathroom.
• 3.3 x 105 Pa
4.7 x 10-4 m3/s
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