Chapter 8:Fluid Mechanics

Learning Goal

• To define a fluid.• To distinguish a gas from a liquid

States of Matter

• Solids – definite volume, definite shape

• Liquids – definite volume, indefinite shape

• Gases – indefinite volume, indefinite shape

• (Also plasma and Bose-Einstein condensates but we don’t need to worry about those.)

What state of matter is glass?

Solid

Liquid

Gas

0% 0%0%

1. Solid

2. Liquid

3. Gas

What state of matter is honey?

Solid

Liquid

Gas

0% 0%0%

1. Solid

2. Liquid

3. Gas

The Nature of Fluids

Fluids:• Liquids and Gases comprise the

category of what we call fluids.• Fluids exhibit certain characteristics

that solids do not – they flow when subjected to shear stress

PROPERTIES OF STATIC FLUIDS

Learning Goal

• To use density to describe a fluid.• To apply buoyant force to explain why some

objects float or sink in a fluid.

Static Fluid Properties

• Density () = mass / volume• Viscosity = internal resistance to flow

Note: Atmospheric pressure and temperature influence a fluid’s density and viscosity

Density

The density of an object is represented by:

Density = mass / volume

While this formula is familiar to us, we will use it in subsequent derivations.

Specific Gravity

• In order to have a constant comparison, we use specific gravity instead of density sometimes.

• Since water has a density of 1 g/mL or 1 x 103 kg/m3, we eliminate the units and call the number specific gravity.

• Ex. For iron which has a density of 7.86 g/mL, the specific gravity is 7.86 (or 7.86 as dense as water).

Which is more dense, a pound of feathers or a pound of

bricks?

A pound of bric

ks

A pound of feath

ers

They are th

e same

0% 0%0%

1. A pound of bricks

2. A pound of feathers

3. They are the same

Common Density Misconceptions

• Let’s expel some common misconceptions about density.

• Refer to your worksheet for the following Turning Point questions about whether the object will float or sink.

A. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

B. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

C. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

D. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

E. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

F. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

G. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

H. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

I. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

J. (Refer to worksheet)

Sin

k

Flo

at

0%0%

1. Sink

2. Float

Buoyancy

• The upward force present when an object floats in a fluid, or feels lighter, is the buoyant force on the object.

• The weight of an object immersed in a fluid is the apparent weight of the object (versus the actual weight).

• Apparent weight = FG - FB (when sinking)

Floating Objects• If, and only if, an object is

floating on the surface:– The buoyant force exerted by

the fluid that is displaced is equal in magnitude to the weight of the floating object

• This is because when an object is floating, it is not moving up or down– therefore the net force is zero

and the buoyant force must equal the weight

objectgB FF ,

Only if floating

Archimedes’ Principle

Any object completely or partially submerged in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object gmFF fluidfluidgB ,

Weig

ht of

the hot air ballo

on

Bu

oya

nt

forc

e of

di

spla

ced

air

The hot air balloon rises because of the large volume of

air that it displaces

Apparent Weight

• The apparent weight of an object is the net weight between the force of gravity and the buoyant force.

Apparent Weight= Fnet = FG – FB

The Red line

Volume Displaced

A boat has a mass of 8450kg. What is the minimum volume of water it will need to displace in order to float on the surface of pure water without sinking?

This is something you will have to think about with your cardboard boats!

If an object is sinking to the bottom of a glass of water, the buoyant force

must be?

Equal

to th

e N...

Less

than Fg

More

than Fg

Equal

to Fg

0% 0%0%0%

1. Equal to the Net Force

2. Less than Fg

3. More than Fg

4. Equal to Fg

What must be true for the buoyant force to be greater than gravitational force?

Object

is floa..

.

Object

is floa..

.

Object

is sin

k...

0% 0%0%

1. Object is floating continuously upward

2. Object is floating at the top of the fluid

3. Object is sinking

If a rock is completely submerged in a fluid, what must be true?

The volume of ..

.

The weigh

t of ..

.

Both 1 and 2

None of t

he ab...

0% 0%0%0%

1. The volume of the displaced fluid = the volume of the rock

2. The weight of the rock = weight of the fluid that was displaced.

3. Both 1 and 2

4. None of the above

The apparent weight of an object in a fluid, FB – Fg , could also be called

what?

Net F

orce

Tensional F

orc...

Buoyant F

orce

Actual W

eight

0% 0%0%0%

1. Net Force

2. Tensional Force

3. Buoyant Force

4. Actual Weight

If a raft is floating and is partially submerged in a fluid, what must be

true?

The volume of ..

.

The weigh

t of ..

.

Both 1 and 2

None of t

he ab...

0% 0%0%0%

1. The volume of the displaced fluid = the volume of the raft

2. The weight of the raft = weight of the fluid that was displaced.

3. Both 1 and 2

4. None of the above

Archimedes Principle example

• A bargain hunter purchases a “gold” crown at a garage sale. After she gets home, she hangs the crown from a scale and finds its weight to be 7.84 N. She then weighs the crown while it is immersed in water, and the scale reads 6.86N. Is the crown made of pure gold?

Pressure in Fluids

• In solids, pressure is defined as the amount of force per unit area.

P = F/A

• Pressure occurs within fluids due to the constant motion of their molecules but it is more difficult to determine the area.

Common Pressure Units

• For example, standard atmospheric pressure is:

• 14.7 psi (pounds per square inch)• 1.01 x 105 Pa (Pascal) = N/m2

• 760 mmHg (millimeters mercury)• 1 atm (atmosphere)

Pressure as a function of depth

damwater

Which hole will have the water shoot out the furthest?

Top hole

Middle Hole

Bottom Hole

All will

be eq...

0% 0%0%0%

1. Top hole

2. Middle Hole

3. Bottom Hole

4. All will be equal

Absolute and Gauge Pressure

• Absolute pressure = Atmospheric + Gauge

Pressure Pressure

• Atmospheric pressure is the pressure due to the gases in the atmosphere (always present)

• Gauge pressure is the pressure due to a fluid (not counting atmospheric pressure)

• Absolute pressure is the total pressure

Ex. 3

• Calculate the absolute pressure at an ocean depth of 1,000m. Assume that the density of water is 1,025 kg/m3 and that

Po= 1.01 x 105Pa.

What is the gauge pressure as well?

Pascal’s Principle

Pascal’s Principle

• Because force is directly proportional to area, one can vary the cross-sectional area to provide more force.

• Eg. Hydraulic brakes, car jacks, clogging of arteries

In order to use a lesser force to accomplish a difficult task, you should apply the force on the

hydraulic cylinder with

Smalle

r radius

Larg

er rad

ius

Doesn

’t matter

0% 0%0%

1. Smaller radius

2. Larger radius

3. Doesn’t matter

Ex. 2

• A car weighing 12000 N sits on a hydraulic press piston with an area of 0.90 m2. Compressed air exerts a force on a second piston, which has an area of 0.20m2. How large must this force be to support the car?

Laminar versus Turbulent Flow

Laminar flow:–Low velocity relative to fluid medium–Streamline path

Turbulent flow:–High velocity relative to fluid medium– Irregular Flow (Eddy currents)

15-6

Ideal Fluids

• Laminar flow• Nonviscous• Incompressible• Constant density and pressure

• All these characteristics must be true for these equations to hold true. (Hence, the name for the ideal gas laws.)

Fluids in Motion

• Steady, Laminar Flow (Ideal Fluid): -Every fluid particle passing trough the same

point in the stream has the same velocity.

Flow Rate

• Flow rate stays constant (at constant pressure in a closed system)

Flow Rate = Av = V/t

A1v1 = A2v2

A = cross-sectional area (m2)

v = speed (m/s)

V = volume (m3/s)

t = time (s)

How are cross-sectional area and velocity of fluids

proportional?

Inverse

ly

Dire

ctly

No re

lationshi...

0% 0%0%

1. Inversely

2. Directly

3. No relationship

Continuity Equation

• Based on Law of Conservation of Mass – what comes in has gotta come out

What will happen to the yellow foam ball?

It will

stay i..

.

It will

shoot ..

.

It will

explod...

0% 0%0%

1. It will stay in the funnel

2. It will shoot out

3. It will explode into yellow chunks

What will happen to the pop cans when air is blown

between them?

They will

come...

They will

move...

It will

remain...

Pop will

fly o...

0% 0%0%0%

1. They will come together and collide.

2. They will move apart from each other

3. It will remain motionless.

4. Pop will fly out from the openings.

How are pressure and velocity of fluids proportional?

Inverse

ly

Dire

ctly

No re

lationshi...

0% 0%0%

1. Inversely

2. Directly

3. No relationship

Bernoulli’s EquationP1 + ρgh1 + ½ ρv1

2 = P2 + ρgh2 + ½ ρv22

Helpful notes:• P = Patm if either side is open.

• Set bottom height (h2 ) = 0

• If there is a large volume up top, (v1 ) = 0

Bernoulli’s Equation

P + ρgh + ½ ρv2 = constant- Results from conservation of energy.

P = Pressure energy resulting from internal forces within the fluid

ρgh = similar to gravitational potential energy

½ ρv2 = similar to kinetic energy

Bernoulli’s Principle

• Bernoulli’s Principle states that the flow speed (Av) in a constriction must be greater than the flow speed before or after it.

• Also, swiftly moving fluids exert less pressure than do slowly moving fluids.

• Eg. Tornadoes and blown off roofs

Bernoulli’s principle

• Pressure in a fluid varies inversely with the velocity