1. Chapter 9
Fluid Mechanics

2. Chapter Objectives Define fluid Density Buoyant force
Buoyantly of floating objects
Pressure
Pascal's principle
Pressure and depth
Temperature
Fluid flow continuity equation
Bernoulli's principle
Ideal gas law

3. What is a Fluid?
So far we have studied the causes of motion dealing with solids.
That leaves us with gases and liquids.
Liquids and gases are different phases, but have common properties.
One common property of gases and liquids is their ability to flow and alter their shape on the process.
Materials that exhibit the property to flow are called fluids.

4. Density m ρ = v SI units = Kg/m3
It is a difficult concept to visualize the mass of a fluid because its shape can change.
So a more useful measurement is the density of an object.
The density of an substance is the mass per unit volume of the substance.
Because this uses mass, it is called the mass density.
If it uses weight, it is called weight density. m
ρ =
SI units = Kg/m3 v
rho

5. Densities of Common Substances
kg/m3
Hydrogen
0.0899
Helium
0.179
Steam (100 oC)
0.598
Air
1.29
Oxygen
1.43
Carbon Dioxide
1.98
Ethanol
8.06 x 102
Ice
9.17 x 102
Fresh water
1.00 x 103
Sea water
1.025 x 103
Iron
7.86 x 103
Mercury
13.6 x 103
Gold
19.3 x 103

6. Specific Gravity
The specific gravity of a substance is a ratio of a substance’s density to that of water.
It gives us an easier scale for comparison of whether objects will float or not.
Example: Lead has a specific gravity of 11.4.
Which means that lead is 11.4 times more dense than water.
Water has a density of 1.00 x 103 kg/m3, so lead has a density of 11.4 x 103 kg/m3.
Or water has a density of 1 g/cm3, so lead has a density of 11.4 g/cm3.
The object will the larger specific gravity will sink!

7. Buoyancy
The ability of a substance to float in a liquid is based of the densities of the two substances.
The less dense substance will move to the top, or float.
The force pushing on an object while in a liquid or floating is called the buoyant force.
The buoyant force acts opposite of gravity, and that is why objects seem “lighter” in water

8. Archimedes’ Principle
When an object is placed in water, the total volume of water is raised the same volume as the Portion of the Object that is submerged.
Archimedes Principle states the Buoyant Force is equal to the weight of water displaced.
Use this formula if the object is totally submerged in the fluid.
FB = Fg(displaced fluid) = mfg
Buoyant Force
Mass Fluid = Vf ρf

9. Buoyant Force on Floating Objects
For an object to float, the Buoyant Force must be equal magnitude to the weight of the object.
The density of the object determines the depth of the submersion.
Use the following for an object that is floating on top of the fluid.
The object is not totally submerged.
FB = Fg (object) = mog
Mass of Object

10. Unknown Densities/Objects
When faced with the challenge of identifying an unknown substance, we compare its density to that of water.
To do this, we use the Buoyant Force of water to determine the density of the unknown substance.
This is done by comparing the weight of the object in air and then again in water.
The difference between the two weights would show the Buoyant Force.
Use this when finding identifying an unknown substance by its density
FB = Fg (in air) - Fg (in water)
object
Fg (in air) =
fluid FB

11. Pressure F P = A 105 Pa = 1 atm = 1 bar = 29.92 in Hg = 14.7 psi
Pressure is a measure of how much force is applied over a given area.
The SI Unit for pressure is the Pascal (Pa), which is equal to 1 N/m2.
The air around us pushes with a pressure. This is called atmospheric pressure, which is about 105 Pa.
That amount gives us another unit, the atmosphere (atm).
105 Pa = 1 atm = 1 bar = in Hg = 14.7 psi

12. Pascal’s Principle
When you pump up a bicycle tire, it just doesn’t grow sideways, but also in height.
Pascal’s Principle states that the pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container.
Pinc = F1 = F2
Pressure in a closed container A1 A2

13. Pressure and Depth
Water pressure increases with depth because the water at a given depth has to support the weight of the water above it.
Imagine an object suspended in a fluid. There is an imaginary column that is the same cross-sectional area of the object.
There is water trapped below pushing up on the object. The water above is pushing down on the object.
Since the water is suspended, the two pressures are equal.
If one becomes larger, the object will sink or float.

14. Fluid Pressure Equation
Pressure varies with the depth in a fluid.
That is because there is a larger column of water above the object pushing downward.
We must also account for atmospheric pressure pushing down on top of the water. F mg
ρVg
ρAhg
ρhg
P = = = = = A A A A
And taking atmospheric pressure into account, we get the following.
P =
Po +
ρhg

15. Temperature
Temperature is a measure of the average kinetic energy of the particles in a substance.
There are actually two SI units for temperature, kelvin (K) and degrees Celsius (oC).
To convert, add 273 to the Celsius measurement.
Fahrenheit is the units for temperature in the United States.

16. Ideal Gas Law PV = nRT PV = NkBT
The ideal gas law varies slightly for physics versus chemistry.
That is due to Boltzmann’s Constant (kB).
kB = 1.38 x J/K
Chemistry’s version uses the ideal gas constant (R).
R = 8.31 J/(mol *K)
PV = nRT
PV = NkBT
n is number of moles
P is pressure (N/m2)
V is volume (m3)
N is number of particles
T is temperature (K)

17. 3rd Version of Ideal Gas Law
Assuming the amount of gas (N1= N2) remains constant in a closed container, we can derive a 3rd version of the ideal gas law.
This version will also help us to see the basis for the 3 gas laws from chemistry (Boyle’s Law, Charles’ Law, and Gay-Lussac’s Law).
Notice that
And since N1=N2,
Leaving us with PV
P1V1
P2V2
P1V1
P2V2 N= = =
kBT
kBT1
kBT2 T1 T2
Because kB is constant, it cancels itself out

18. Fluid Flow Fluid flows in one of two ways:
Laminar flow is when every particle of fluid follows the same smooth path.
That path is said to be streamline.
Turbulent flow is when there is irregular flow due to objects in the path or sharp turns in the flowing chamber.
Irregular motions of the fluid are called eddy currents.
Since laminar flow is predictable and easy to model, we will use its characteristics in this book.

19. Continuity Equation
Due to the conservation of mass, the amount of fluid as it flows through a chamber is consider to also be conserved.
So m1= m2
But the mass of a gas is hard to find and we do know the density and the space it takes up.
ρV1= ρV2
But what happens when the chamber changes size?
ρA1Δx1= ρA2Δx2
ρA1v1Δt = ρA2v2Δt
It is hard to measure displacement of a gas, but we can measure the time it takes to travel.
Density of the gas will be constant and the time will be constant, so…
A1v1 = A2v2
Continuity Equation

20. Bernoulli’s Principle
The pressure in a fluid decreases as the fluid’s velocity increases.
This is the principle responsible for lift.
As air flows over the top of the wing, the speed must increase because it travels a longer distance.
Because the speed increased, the pressure then decreases.
Now there is more pressure on the bottom of the wing pushing upward, creating lift!

21. Bernoulli’s Equation P1 + 1/2ρ1v12 + ρ1gh1 = P2 + 1/2ρ2v22 + ρ2gh2
This equation relates pressure to energy in a moving fluid.
Since energy is conserved, Bernoulli’s Equation is set to be a constant.
For our use, we will then set Bernoulli’s Equation equal to itself under initial and final conditions.
P1 + 1/2ρ1v12 + ρ1gh1 = P2 + 1/2ρ2v22 + ρ2gh2
Pressure
Velocity
Height
Density