Fluid Mechanics

Lesson 9-1
Fluids and Buoyant
Force

Define a Fluid
 Matter is normally classified in one of three
groups
 Solid, Liquid, Gas
 Liquids share a commonality with gases in
that they can flow and alter their shape as
they flow
 Solids do not share this property, they cannot
flow nor can they readily change their shape;
solids are not fluids

Definite Volume
 Even though both liquids and gases are
fluids, they have a distinct difference
 Liquids have a definite volume
 Say you have a one gallon gasoline can full of
fuel
 If you pour the entire can into a lawn mower fuel
tank, there will still be one gallon of gasoline

Indefinite Volume
 Gases have neither definite volume or shape
 When a gas is poured, not only does it
change its shape to fit the new container, the
gas expands to fill the new container

Density and Buoyant Force
 Have you ever felt uncomfortable in a
crowded room? Probably because there
were too many people for the amount of
space; the density of people was too high
 Density is how much there is of a quantity in
a given amount of space
 The quantity can be anything from people to cars
to mass or energy

Density and Buoyant Force
 Mass density is mass per unit volume of a
substance
 When we talk about a fluid’s density, we are
really talking about mass density
 Mass density is represented by the Greek
letter rho (ρ)
 capital V for volume
 The SI unit for mass density is kg/m3
ρ =
m
V

Pressure
 Solids and liquids tend to be incompressible
 Their density change very little with changes
in pressure
 Gases are not completely incompressible
meaning their volumes are greatly affected by
pressure
 That is why there is no standard density for
gas as there is for solids and liquids

Buoyant Force
 Have you noticed that heavy objects seem
lighter under water
 It is much easier to pick up a person in a
swimming pool than on dry ground
 That is because the water exerts an upward force
on objects that are partially or completely
submerged
 The upward force is called the buoyant force

Buoyant Force
 Have you ever laid on a raft in a pool
 You and the raft experience a buoyant force,
which keeps both you and the raft afloat
 Because the buoyant force acts in the opposite
direction as the force of gravity, objects
submerged in a fluid such as water have a net
force on them that is smaller than their weight

Buoyant Force
 This means they appear to weigh less in
water
 The weight of an object immersed in a fluid is the
object’s apparent weight

Archimedes’ Principle
 Imagine you fill a bucket to the top and drop
in a brick
 What happens? Why?
 The total volume of water that overflows is the
displaced volume of water. The volume of the
water is equal to the volume of the portion of the
brick that is underwater

Archimedes’ Principle
 The magnitude of the buoyant force acting on
the brick is known as Archimedes’ Principle
 Any object completely or partially submerged in a
fluid experiences an upward buoyant force equal
in magnitude to the weight of fluid displaced by
the object
 Archimedes’ principle can be written as
F m gB f=

Archimedes’ principle
 Whether an object will float or sink depends
of the net force acting on it
 This net force is the object’s apparent weight
and can be calculated as follows
F F F objectnet B g= − ( )

 Now apply Archimedes Principle, using mo to
represent the mass of the object
 Rewrite using that idea
F m g m gnet f o= −
F V V gnet f f o o= −( )ρ ρ

 A floating object cannot be denser than the
liquid in which it floats
 The expression for the net force on an object
floating on the surface gives the following
result
 Or:
F V V gnet f f o o= = −0 ( )ρ ρ
ρ
ρ
f
o
o
f
V
V
=

 We know the displaced volume of fluid can
never be greater than the volume of the
object
 That means for an object to float the object’s
density must be less than the displaced
fluid’s density
 If the volume of the object is equal to the
volume of displaced liquid, the entire object is
submerged

Floating Objects
 For floating objects, the buoyant force equals
the object’s weight
 Imagine a raft with cargo floating on a river
 Two forces act on the raft and the cargo
 The downward force of gravity
 The upward buoyant force
 Because the raft is floating, the raft is in
equilibrium and the two forces are balanced

Floating Objects
 This means for floating objects the force of
gravity is equal to the buoyant force
 As in
 Notice for a floating object, the buoyant force
can be found by using the first condition of
equilibrium, Archimedes’ principle would be
overkill in such a situation
F m gB o=

Apparent Weight
 The apparent weight of a submerged object
depends of density
 Think of or raft from earlier
 Imagine a hole is punched in the raft
 The raft and cargo eventually sink below the water’s
surface
 The net force on the raft and cargo is the difference
between the buoyant force and the weight of the raft
and cargo

Apparent Weight
 As the volume of the raft decreases, the volume of
displaced water also decreases, as does the
magnitude of the buoyant force
 This is shown with
 After the raft becomes completely submerged, the two
volumes are equal
 Notice that both the direction and the magnitude of the
net force depend on the difference between the
density of the object and the density of the fluid in
which it is immersed
F V V gnet f f o o= −( )ρ ρ
F Vgnet f o= −( )ρ ρ

Apparent Weight
 If the object’s density is greater than the fluid density,
the net force is negative (downward) and the object
sinks
 If the object’s density is less than the fluid density, the
net force is positive and the object rises to the surface
and floats
 If the densities are the same, the object floats, but
underwater
 A simple relationship between the weight of a
submerged object and the buoyant force on the object
can be found by considering their ratios as follows
m g
F
o
B
o
f
=
ρ
ρ

Lesson 9-2
Fluid Pressure and
Temperature

Pressure
 You experience pressure everyday
 When you dive to the bottom of a swimming pool
 When you drive up a large hill
 When you ride in a airplane

Pressure
 Pressure is force per unit area
 The fluids above are exerting force against your
eardrums, so your ears want to ‘pop’ to adjust for
the pressure change
 Pressure is measure of how much force is applied
of a given area. It can be written as
P
F
A
=

Pressure
 The SI unit of pressure is the N/m2 which is
called a Pascal (Pa)
 A Pascal is a very small amount of pressure.
 Air pressure at sea level is about 105 Pa which is
1 atm
 The total air pressure in a typical automobile tire
is about 300000 Pa or 3 atm.

Fluids Exert Pressure
 When you use a bicycle pump to put air into a
tire, you apply force on the piston, which
exerts a force on the gas inside the tire.
 The gas pushes back on the piston and on the
walls of the tire.
 The pressure is the same throughout the volume
of the gas

Hydraulic Pressure
 An important use of fluid pressure is the
hydraulic press
 Garages can use a motor to generate a large
force over a small area to provide a force to equal
a large area, such as lifting a car
F
A
F
A
1
1
2
2
=

Pressure and Depth
 Pressure varies with depth in a fluid
 As a submarine dives into the water the pressure
of the water against the hull of the sub increases.
 Water pressure increases with depth because the
water at a given depth must support the weight of
the water.
 The weight of the entire column of water above an
object exerts force on the object.

Pressure and Depth
 The column of water exerting force has a
volume equal to Ah were A is the cross
sectional area and h is the depth.
 The pressure at a depth caused by the
weight of a volume of water can be calculated
as
P
F
A
mg
A
Vg
A
Ahg
A
hg= = = = =
ρ ρ
ρ

Gauge Pressure
 This pressure is referred to as gauge
pressure. It is NOT the total pressure at this
depth because atmospheric pressure is
applying pressure at the surface.
 Absolute pressure P is calculated
 Po is atmospheric pressure
P P hgo= + ρ

Atmospheric Pressure
 Atmospheric pressure is pressure from above
 The weight of the air pushing down on the earth
and the bodies on earth is known as atmospheric
pressure.
 Atmospheric pressure is actually quite large,
assuming a SA of 2 m2, ATM is 200,000 N.
 How can our bodies withstand such an
incredible force without being crushed?
 Bodies are in equilibrium, fluids inside push back
with the same force creating a state of balance

Ideal Fluid
 Ideal fluid model simplifies fluid flow analysis
 Many fluid features are considered by studying an
ideal fluid
 No real world fluid is an ideal fluid, but an ideal
fluid does showcase many properties of real world
fluids

Idea Fluid
 We assume an ideal fluid is incompressible,
meaning the density always remains constant
 We also assume an ideal fluid is nonviscous
 Viscosity refers to the amount of internal friction in
a fluid
 A fluid with a high viscosity tries to bond with its
container and flows more slowly than a low
viscous fluid

Ideal Fluid
 Another property of an ideal fluid is ideal flow
 We assume the velocity, density and pressure at
every point in the fluid is constant.
 This is known as non-turbulent flow, there can be
no undertows or rip currents present in the
moving fluid

Conservation Laws of Fluids
 If a fluid flows into a pipe, the mass that flows
into the pipe must equal the mass exiting the
pipe, even if the diameter of the pipe changes
x2
x1
V2
V1
A2
A1

Conservation Laws in Fluids
 This can be shown as
 Recall and
 and recall

 Both the time interval and density remain
constant though (ideal fluid), so they are
cancelled and we are left with
m m1 2=
m V= ρ V A x= ∆
ρ ρ1 1 1 2 2 2A x A x∆ ∆= v
d
t
=
ρ ρ1 1 1 1 2 2 2 2A v t A v t=
A v A v1 1 2 2=

 This is referred to as the continuity equation
where A shows two different cross sectional
areas and v shows two different velocities
A v A v1 1 2 2=

 As the cross sectional area increases, the
velocity slows, and as the cross sectional
area decreases, the velocity increases to flow
the same volume of water
 The flow rate remains constant regardless
the diameter of the pipe

 The expressions for conservation of energy in
fluids differs slightly from our previous form
studied in chapter 5
 The reason is that fluids also exert pressure,
so the conservation equation must take into
account the pressure of the fluids
 A change in pressure can be related to the
transfer of energy into or out of the volume.
We must account for this energy.

 As a fluid moves through a pipe of varying
cross sectional area and elevation, the
pressure and speed can change, but the total
energy does not.
 Bernoulli’s equation explains such a situation
 constant, that is to say conservedP v gh+ + =
1
2
2
ρ ρ

 We can set up a version of the equation to
compare the energy of a fluid at two different
points in a pipe
P v gh P v gh1 1
2
1 2 2
2
2
1
2
1
2
+ + = + +ρ ρ ρ ρ

Special Case 1
 Lets say the fluid is not moving and the initial
height is zero
 a pressure function of depth
P P gh1 2 2= + ρ

Special Case 2
 Lets say a fluid is flowing through a horizontal
pipe with restriction
 Since
 This expression implies that if at some
point in the flow, then
P v P v1 1
2
2 2
21
2
1
2
+ = +ρ ρ
h h1 2=
v v2 1>
P P2 1<

Bernoulli’s Principle
 Swiftly moving fluids exert less pressure than
slowly moving ones.
 Reason why a curve ball curves, a plane can
fly, a soccer player can ‘bend it like Beckham’
and why homes are ripped to shreds during
hurricanes and tornadoes

Fluid Mechanics

Fluid Mechanics

More Related Content

What's hot

Viewers also liked

Similar to Fluid Mechanics

More from JPoilek

Recently uploaded

Fluid Mechanics