Chapter 10
Fluids
Fluids
A fluid is a gas or a liquid.
A gas expands to fill any container
A liquid (at fixed pressure and
temperature), has a fixed volume, but
deforms to the shape of its container.
The atoms in a liquid are closely packed while those
in a gas are separated by much larger distances.
Gas have a density ~ 1/1000 x liquid density
Density and Pressure
 The
density of a substance of uniform
composition is defined as its mass per unit
volume:
m
 
V
are kg/m3 (SI) or g/cm3 (cgs)
 1 g/cm3 = 1000 kg/m3
 Units
Density, cont.
 The
densities of most liquids and solids
vary slightly with changes in temperature
and pressure
 Densities of gases vary greatly with
changes in temperature and pressure
Density

Density = Mass/Volume



 = M/V
SI unit: [kg/m3]
Densities of some common things (kg/m3)










Water
ice
blood
lead
Copper
Mercury
Aluminum
Wood
air
Helium
1000
917 (floats on water)
1060 (sinks in water)
11,300
8890
13,600
2700
550
1.29
0.18
SPECIFIC GRAVITY
 The
specific gravity of a substance is the
ratio of its density to the density of water at
4° C

The density of water at 4° C is 1000 kg/m3
 Specific
gravity is a unitless ratio
Pressure
Pressure P is the amount
of force F per unit area
By the Action-Reaction principle,
A:
F
Pressure is the inward force per unit area
P
that the container exerts on the fluid.
A
Pressure is the outward force per unit
area that the fluid exerts on its container.
A1
F1
F2
A2
PRESSURE

The force exerted
by a fluid on a
submerged object
at any point if
perpendicular to
the surface of the
object
F
N
P
in P a  2
A
m
A woman’s high heels sink into the soft ground,
but the larger shoes of the much bigger man do not.
Pressure = force/area
The pressure exerted on the piston extends uniformly
throughout the fluid, causing it to push outward with equal
force per unit area on the walls and bottom of the cylinder.
MEASURING PRESSURE

The spring is
calibrated by a
known force
 The force the fluid
exerts on the
piston is then
measured
ConcepTest 10.3
You are walking out on
a frozen lake and you
begin to hear the ice
cracking beneath you.
What is your best
strategy for getting off
the ice safely?
On a Frozen Lake
1) stand absolutely still and don’t move a muscle
2) jump up and down to lessen your contact time with
the ice
3) try to leap in one bound to the bank of the lake
4) shuffle your feet (without lifting them) to move
towards shore
5) lie down flat on the ice and crawl toward shore
ConcepTest 10.3
You are walking out
on a frozen lake and
you begin to hear the
ice cracking beneath
you. What is your
best strategy for
getting off the ice
safely?
On a Frozen Lake
1) stand absolutely still and don’t move a muscle
2) jump up and down to lessen your contact time
with the ice
3) try to leap in one bound to the bank of the lake
4) shuffle your feet (without lifting them) to move
towards shore
5) lie down flat on the ice and crawl toward shore
As long as you are on the ice, your weight is pushing down. What is
important is not the net force on the ice, but the force exerted on a
given small area of ice (i.e., the pressure!). By lying down flat, you
distribute your weight over the widest possible area, thus reducing the
force per unit area.
Atmospheric Pressure
Atmospheric pressure comes from the weight of the
column of air above us. At sea level, atmospheric
pressure is:
Pat = 1.01  105 N/m2
= 1.01  105 Pa 1 Pascal= 1 N/m2
= 14.7 lb/in2 (psi)
= 1 bar (tire pressure gauges in Europe read 1, 2,..bar)
Hurricane Rita 2005: P = 882 millibar = 0.882 bar
F=Mg
F=PA
Pressure examples
Estimate the force of the atmosphere on the top of your
head.
1.
•
•
•
•
A = (10cm)(15cm)=0.015m2
F=PA = [1.01  105 N/m2 ][0.015 m2] = 1.5 kN
A = (4in)(6in)=24 in2
F=PA = [15 lb/in2][24in2] = 360 lb.
Is atmospheric pressure on top of a mountain greater or
less than at sea level?
2.
•
Less. At higher altitude, there is less mass above.
Pressure
 Example
VARIATION OF PRESSURE WITH DEPTH

If a fluid is at rest in a container, all
portions of the fluid must be in static
equilibrium
 All points at the same depth must be at
the same pressure


Otherwise, the fluid would not be in
equilibrium
The fluid would flow from the higher
pressure region to the lower pressure region
PRESSURE AND DEPTH

Examine the darker
region, assumed to be
a fluid



It has a crosssectional area A
Extends to a depth h
below the surface
Three external forces
act on the region
PRESSURE AND DEPTH EQUATION
P

Po is normal
atmospheric
pressure


= Po + ρgh
1.013 x 105 Pa =
14.7 lb/in2 = 1 atm
The pressure does
not depend upon
the shape of the
container
Pressure in a Fluid
Pressure in a fluid depends only on the depth h
below the surface.
P = Pat + gh
 = density of fluid
Weight/Area of fluid
Weight/Area of atmosphere above fluid
IF the density of the fluid is constant and it has
atmospheric pressure (Pat) at its surface.
Mass of fluid above depth h is
(density)(volume) = hA
Force of gravity on fluid above depth h: W= ghA
Pressure under water
To what depth in water must you dive to double the
pressure exerted on your body?
P = Pat + gh
gh = Pat , h= Pat /g
h
[1.01  105 N / m 2 ]
[103 kg / m3 ][9.81m / s 2 ]
 10.3m
Start to feel strong pressure at 3m
Pressure variation in fluid
The variation in pressure at two different depths is
given by:
P2 = P1 + gh
Pressure and Depth
Barometer: a way to measure
atmospheric pressure
p2 = p1 + gh
p1=0
patm = gh
Measure h, determine patm
p2=patm
example--Mercury
 = 13,600 kg/m3
patm = 1.05 x 105 Pa
 h = 0.757 m = 757 mm = 29.80” (for 1 atm)
h
PRESSURE MEASUREMENTS
 Absolute
vs. Gauge Pressure
 The pressure P is called the absolute
pressure

P
Remember, P = Po + gh
– Po = gh is the gauge pressure
PRESSURE MEASUREMENTS:
MANOMETER

One end of the Ushaped tube is
open to the
atmosphere
 The other end is
connected to the
pressure to be
measured
 Pressure at B is
Po+ρgh
PRESSURE VALUES IN VARIOUS UNITS
 One
atmosphere of pressure is defined
as the pressure equivalent to a column
of mercury exactly 0.76 m tall at 0o C
where g = 9.806 65 m/s2
 One atmosphere (1 atm) =



76.0 cm of mercury (760mm = 1 torr)
1.013 x 105 Pa
14.7 lb/in2
PRESSURE
 Example:
PASCAL’S PRINCIPLE
A
change in pressure applied to an
enclosed fluid is transmitted undiminished
to every point of the fluid and to the walls
of the container.

First recognized by Blaise Pascal, a French
scientist (1623 – 1662)
PASCAL’S PRINCIPLE, CONT

The hydraulic press is an
important application of
Pascal’s Principle
F1
F2
P

A1 A 2

Also used in hydraulic
brakes, forklifts, car lifts,
etc.
A small force F1 applied to a piston with a small
area produces a much larger force F2 on the larger
piston. This allows a hydraulic jack to lift heavy objects.
Pascal’s Principle, Force



A external pressure P
applied to any area of a
fluid is transmitted
unchanged to all points in
or on the fluid.
This is just an application
of the Action-Reaction
principle.
Hydraulic Lift
A Force F1 is applied to area A1, displacing the fluid by a distance d1.
The pressure increase in the fluid is P=F1/A1.
The Pressure F1/A1 creates a force on the car F2= A2 (F1/A1) = F1 (A2 /A1).
A small force acting on a small area creates a big force acting over a large area!
Archimedes’ Principle:
The buoyant force acting on an
object fully or partially submerged
in a fluid is equal to the weight of
the fluid displaced by the object.
The weight of a column of water is proportional to
the volume of the column. The volume V is equal
to the area A times the height h.
Equilibrium…
BUOYANT FORCE


The upward force is
called the buoyant
force
The physical cause of
the buoyant force is
the pressure
difference between the
top and the bottom of
the object
BUOYANT FORCE, CONT.
 The
magnitude of the buoyant force
always equals the weight of the displaced
fluid
B   flu id V flu id g  w
 The
flu id
buoyant force is the same for a totally
submerged object of any size, shape, or
density
BUOYANT FORCE, FINAL
 The
buoyant force is exerted by the fluid
 Whether an object sinks or floats depends
on the relationship between the buoyant
force and the weight
ARCHIMEDES’ PRINCIPLE:
TOTALLY SUBMERGED OBJECT
 The
upward buoyant force is B=ρfluidgVobj
 The downward gravitational force is
w=mg=ρobjgVobj
 The net force is B-w=(ρfluid-ρobj)gVobj
TOTALLY SUBMERGED OBJECT

The object is less
dense than the
fluid
 The object
experiences a net
upward force
TOTALLY SUBMERGED OBJECT, 2

The object is more
dense than the
fluid
 The net force is
downward
 The object
accelerates
downward
Question: How do steel ships float if steel is roughly 6 times more dense than water?
ARCHIMEDES’ PRINCIPLE:
FLOATING OBJECT
 The
object is in static equilibrium
 The upward buoyant force is balanced by
the downward force of gravity
 Volume of the fluid displaced corresponds
to the volume of the object beneath the
fluid level
ARCHIMEDES’ PRINCIPLE:
FLOATING OBJECT, CONT

The forces balance

obj
 f luid
Vf luid

Vobj
ARCHIMEDES’S PRINCIPLE
Suppose you float a large ice-cube in
a glass of water, and that after you
place the ice in the glass the level
of the water is at the very brim.
When the ice melts, the level of the
water in the glass will:
1. Go up causing the water to spill.
2. Go down.
3. Stay the same.
Archimedes’ Principle: The buoyant force on an object equals the weight
of the fluid it displaces.
Weight of water displaced = Buoyant force = Weight of ice
When ice melts it will turn into water of same volume
 Example

9.9
A raft is constructed of wood having a density of
6.00 x 102 kg/m3. Its surface area is 5.70m2, and
volume is 0.60m3. When the raft is placed in
fresh water, what depth h is the bottom of the
raft submerged?
Concept Question
Which weighs more:
1. A large bathtub filled to the brim with water.
2. A large bathtub filled to the brim with water
with a battle-ship floating in it.
3. They will weigh the same.
CORRECT
Tub of water
Weight of ship = Buoyant force =
Weight of displaced water
Overflowed water
Tub of water + ship
FLUIDS IN MOTION

Streamline flow



Every particle that passes a particular point
moves exactly along the smooth path followed
by particles that passed the point earlier
Also called laminar flow
Streamline is the path


Different streamlines cannot cross each other
The streamline at any point coincides with the
direction of fluid velocity at that point
STREAMLINE FLOW, EXAMPLE
Streamline flow shown around an auto in a wind
tunnel
FLUIDS IN MOTION:
TURBULENT FLOW
 The


flow becomes irregular
exceeds a certain velocity
any condition that causes abrupt changes in
velocity
 Eddy
currents are a characteristic of
turbulent flow
TURBULENT FLOW, EXAMPLE

The rotating blade
(dark area) forms a
vortex in heated air


The wick of the
burner is at the
bottom
Turbulent air flow
occurs on both
sides of the blade
FLUID FLOW: VISCOSITY
 Viscosity
is the degree of internal friction in
the fluid

Measure of a fluid's ability to resist gradual
deformation by shear or tensile stresses
 The
internal friction is associated with the
resistance between two adjacent layers of
the fluid moving relative to each other
FLUID FLOW: VISCOSITY
 Viscous
Liquid!
CHARACTERISTICS OF AN IDEAL FLUID

The fluid is nonviscous


The fluid is incompressible


Its density is constant
The fluid motion is steady


There is no internal friction between adjacent layers
Its velocity, density, and pressure do not change in time
The fluid moves without turbulence


No eddy currents are present
The elements have zero angular velocity about its
center
Equation of Continuity

Mass is conserved as the fluid flows.
If a certain mass of fluid enters a pipe at one end at a
certain rate, the same mass exits at the same rate
at the other end of the tube (if nothing gets lost in
between through holes, for instance).
Mass flow rate at position 1 = Mass flow rate at
position 2
1 A1 v1 = 2 A2 v2
 A v = constant along a tube that has a single entry
and a single exit point for fluid flow.
EQUATION OF CONTINUITY

What goes in comes out!
 If density is constant:


The product of the crosssectional area of a pipe
and the fluid speed is a
constant


A1v1 = A2v2
Speed is high where the
pipe is narrow and
speed is low where the
pipe has a large
diameter
Av is called the flow rate
EQUATION OF CONTINUITY, CONT

The equation is a consequence of conservation
of mass and a steady flow
 A v = constant

This is equivalent to the fact that the volume of fluid
that enters one end of the tube in a given time interval
equals the volume of fluid leaving the tube in the
same interval
• Assumes the fluid is incompressible and there are no leaks
Bernoulli’s Equation
Work-Energy Theorem : Wnc = change of total mechanical energy
applied to fluid flow :

Difference in pressure => net force is not zero => fluid accelerates
Pressure is due to collisional forces which is a nonconservative force:
Wnc = (P2-P1) V
Consider a fluid moving from height h1 to h2. Its total mechanical
energy is given by the sum of kinetic and potential energy. Thus,
Wnc = Etot,1 –Etot,2 = ½ m v12+m g h1 –( ½ m v22+m g h2)
BERNOULLI’S EQUATION, CONT.
 States
that the sum of the pressure, kinetic
energy per unit volume, and the potential
energy per unit volume has the same
value at all points along a streamline
1 2
P   v   g y  co n stan t
2
APPLICATIONS OF BERNOULLI’S
PRINCIPLE: VENTURI TUBE




Shows fluid flowing
through a horizontal
constricted pipe
Speed changes as
diameter changes
Can be used to
measure the speed of
the fluid flow
Swiftly moving fluids
exert less pressure
than do slowly moving
fluids
OTHER APPLICATIONS OF FLUID
DYNAMICS
Objects Moving Through a Fluid

Many common phenomena can be explained by
Bernoulli’s equation


At least partially
In general, an object moving through a fluid is
acted upon by a net upward force as the result
of any effect that causes the fluid to change its
direction as it flows past the object
APPLICATION – AIRPLANE WING

The air speed above the
wing is greater than the
speed below
 The air pressure above
the wing is less than the
air pressure below
 There is a net upward
force


Other factors are also
involved



Called lift
Designed to produce lift
Racecars designed to
produce faster airflow on
the bottom
High velocity implies low
pressure, IN the fluid
APPLICATION – AIRPLANE WING
APPLICATIONS OF FLUID FLOW
 Example:
 A jet of water
squirts out
horizontally from a
hole near the
bottom of the tank
with a velocity of
1.33 m/s. What is
the height of the
water level in the
tank?