Fluid Mechanics
• Liquids and gases have the ability to flow
• They are called fluids
• There are a variety of “LAWS” that fluids
obey
Density
• Regardless of form (solid, liquid, gas) we
can define how much mass is squeezed into
a particular space
density mass
volume
Mass Density
2 kg, 4000 cm3
Wood
177 cm3
45.2 kg
; mass m
Densityvolume V
Lead: 11,300 kg/m3
Wood: 500 kg/m3
4000 cm3
Lead
Same volume
2 kg Lead
Same mass
Gases
• The primary difference between a liquid
and a gas is the distance between the
molecules
• In a gas, the molecules are so widely
separated, that there is little interaction
between the individual molecules
Boyle’s Law • Pressure depends on density of the gas
• Pressure is just the force per unit area exerted by
the molecules as they collide with the walls of the
container
• Remember:
• Pressure is measured in pascal units (Pa)
• 1Pa = 1 Newton / m2 (force/area)
• At sea level, 1atm = 101.3 kPa or 101,300 N per
square meter
• Double the density, double the number of collisions
with the wall and this doubles the pressure
Boyle’s Law
Density is mass
divided by
volume.
Cut the volume
in half and you
double the
density and thus
the pressure.
Boyle’s Law
• At a given temperature for a given quantity
of gas, the product of the pressure and the
volume is a constant
P1V1 P2V2
Pressure
Pressure is the ratio of a force F to the area A over
which it is applied:
Pressure ; Force F
PArea A
A = 2 cm2
1.5 kg
2
-4 2
(1.5 kg)(9.8 m/s )
2 x 10 m
FP
A
P = 73,500 N/m2
The Unit of Pressure (Pascal):
A pressure of one pascal (1 Pa) is defined as a force of
one newton (1 N) applied to an area of one square meter
(1 m2).
21 Pa = 1 N/mPascal:
In the previous example the pressure was 73,500 N/m2.
This should be expressed as:
P = 73,500 Pa
Pressure / Density Example
Tofu
Cookbook
Schmedrick uses his 6 lb tofu recipe book to teach his little brother
Poindexter about density and pressure. He sets the book on the table
and calculates the pressure on the table, which depends on the book’s
orientation. The book’s density is 6 lb / (9” · 14” · 3”) = 0.0159 lb / in 3.
Tofu Cookbook
14”
3” 9”
P = 6 lb / (9” · 14” )
= 0.0476 lb / in 2
P = 6 lb / (9” · 3” )
= 0.222 lb / in 2
P = 6 lb / (3” · 14” )
= 0.143 lb / in 2
Pressure in a Fluid
• The pressure is just the weight of all the
fluid around the object
• Atmospheric pressure is just the weight of
all the air above on an area on the surface of
the earth
• In a swimming pool the pressure on your
body surface is just the weight of the water
above you (plus the air pressure above the
water)
Fluid Pressure
Fluid exerts forces in many directions. Try to submerse a rubber
ball in water to see that an upward force acts on the float.
• Fluids exert pressure in
all directions. F
Pressure in a Fluid
• So, the only thing that counts in fluid pressure is the
gravitational force acting on the mass ABOVE you
• The deeper you go, the more weight above you and
the more pressure
• Go to a mountaintop and the air pressure is lower
• Pressure in a fluid is the result of the forces exerted
by molecules as they bounce off each other in all
directions. Therefore, at a given depth in a liquid or
gas, the pressure is the same and acts in every
direction
Pressure vs. Depth in Fluid
Pressure = force/area
; ; mg
P m V V AhA
Vg AhgP
A A
h
mg Area
• Pressure at any point in a
fluid is directly proportional
to the density of the fluid
and to the depth in the fluid. P = gh
Fluid Pressure:
Independence of Shape and Area.
Water seeks its own level,
indicating that fluid pressure
is independent of area and
shape of its container.
• At any depth h below the surface of the water
in any column, the pressure P is the same.
The shape and area are not factors.
*Properties of Fluid Pressure*
• The forces exerted by a fluid on the walls of its
container are always perpendicular.
• The fluid pressure is directly proportional to the
depth of the fluid and to its density.
• At any particular depth, the fluid pressure is the
same in all directions.
• Fluid pressure is independent of the shape or area
of its container.
Barometers
• The height of the mercury
column in a barometer directly
measures air pressure.
• The weight of the column of
mercury is balanced by the
force exerted at the bottom due
to the air pressure.
• Normal air pressure is 760mm
or 760 torr
• Since mercury is 13.6 times
heavier than water, a water
barometer would have to be
13.6 times longer.
Pascal’s Principle
• Pressure applied to a fluid is transmitted
throughout the fluid. • Ex) squeezing tube of toothpaste
• Hydraulic machines work using Pascal’s
principle.
Pascal’s Law
Pascal’s Law: An external pressure applied
to an enclosed fluid is transmitted uniformly
throughout the volume of the liquid.
Fout Fin Aout Ain Pressure in = Pressure out
in out
in out
F F
A A
Hydraulic Press
oil
A2 F1
A1
F2
A force F1 is applied to a hydraulic press. This increases the pressure
throughout the oil, lifting the car--Pascal’s principle. This would not
work with air, since air is compressible. The pressure is the same
throughout the oil. The volume of oil pushed down on the left is the
same as the increase on the right. The distance pushed on the left is the
trade off.
h1
h2
Example 3. The smaller and larger pistons of a
hydraulic press have diameters of 4 cm and 12 cm.
What input force is required to lift a 4000 N weight
with the output piston?
Fout Fin Aout Ain ; in out out in
in
in out out
F F F AF
A A A
2
2
(4000 N)( )(2 cm)
(6 cm)inF
2; 2
DR Area R
F = 444 N
Rin= 2 cm; Rout = 6 cm
Floating in Fluids We all know that dense objects sink in fluids of lower density. A
rock sinks in air or water, and oil floats on top of water.
Basements stay cool in the summer because cool air is denser
than warm air. The USS Eisenhower is a 95 000 ton nuclear
powered aircraft carrier made of dense materials like steel, yet it
floats. If you weigh yourself under water, the scale would say
you are lighter than your true weight. All of these facts can be
explained thanks one of the greatest scientists of all time--the
Greek scientist, mathematician, and engineer--Archimedes.
USS Eisenhower Archimedes
Archimedes’ Principle
• An object that is completely or partially submerged in
a fluid experiences an upward buoyant force equal to
the weight of the fluid displaced.
2 lb
2 lb
• The buoyant force is due to the
displaced fluid. The block material
doesn’t matter.
• If the buoyant force on an object is
greater than the force of gravity acting
on the object, the object will float.
• The apparent weight of an object in a
liquid is gravitational force (weight)
minus the buoyant force
Flotation
• A floating object displaces a weight of fluid equal
to its own weight. An object floats if its density is
less than the density of the fluid it is placed in.
Submarines & Blimps A sub is submerged in water, while a
blimp is submerged in air. In each a
buoyant force must balance the weight
of the vessel. Blimps and hot air
balloons must displace huge amounts
of air because air isn’t very dense. The weight of the air a blimp
displaces is equal to the blimp’s weight. Likewise, the weight of
the water a sub displaces is equal to the sub’s weight.
Buoyancy in a Gas
• An object surrounded by air is buoyed up by
a force equal to the weight of the air
displace.
• Exactly the same concept as buoyancy in
water. Just substitute air for water in the
statement
• If the buoyant force is greater than the
weight of the object, it will rise in the air
Buoyancy in a Gas
Since air gets less
dense with altitude,
the buoyant force
decreases with
altitude. So helium
balloons don’t rise
forever!!!
Atmospheric Pressure
• Just the weight of the air above you
• Unlike water, the density of the air
decreases with altitude since air is
compressible and liquids are only very
slightly compressible
• Air pressure at sea level is about 105
newtons/meter2
Bernoulli’s Principle
• When the speed of a fluid increases, the
pressure exerted by the fluid decreases.
Bernoulli’s Principle • Uses: airplanes, hose-end sprayers
• Energy conservation requires that the
pressure be lower in a fluid that is moving
faster
Air is not incompressible, but the Bernoulli principle can
explain, in part, why an airplane flies. The upper surface of
the wing has a smaller radius of curvature than the bottom
surface. Air on top must travel farther, so it moves faster, and
the pressure there is lower, creating lift. Also, because of the
wing’s upward tilt, air is pushed downward. So, the air
pushes back on the wing in the direction of F.
Viscosity
• The resistance to flow by a fluid
• When a container of liquid is tilted to allow
flow, the flowing particles will transfer
energy to the particles that are stationary.
• Increasing temperature of a fluid will
decrease viscosity
Viscosity Different kinds of fluids flow more easily than others. Oil, for
example, flows more easily than molasses. This is because molasses
has a higher viscosity, which is a measure of resistance to fluid flow.
Inside a pipe or tube a very thin layer of fluid right near the walls of
the tube are motionless because they get caught up in the microscopic
ridges of the tube, or microwelds. Layers closer to the center move
faster and the fluid sheers. The middle layer moves the fastest.
The more viscous a fluid is, the more the layers want to cling together,
and the more it resists this shearing. The resistance is due the frictional
forces between the layers as the slides past one another. Note, there is
no friction occurring at the tube’s surface since the fluid there is
essentially still. The friction happens in the fluid and generates heat.
The Bernoulli equation applies to fluids with negligible viscosity.
v = 0