1. Fluid Dynamics
AP Physics 2

2. Fluid Flow
Up till now, we have pretty much focused on fluids at rest. Now let's look at fluids in motion

3. Characteristics of an Ideal Fluid
The fluid is nonviscous
There is no internal friction between adjacent layers
The fluid is incompressible
Its density is constant
The fluid motion is steady
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

4. Fluids in Motion: Streamline Flow
A fluid’s motion can be said to be streamline, or laminar.
The path itself is called streamline.
By laminar we mean that every particle moves exactly along the smooth path as every other particle that follows it.
If the fluid does not have laminar flow, it has turbulent flow.

5. 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

6. Streamline Flow, Example
Streamline flow shown around an auto in a wind tunnel

7. Fluid Flow: Viscosity
Viscosity is the degree of internal friction in the fluid
The internal friction is associated with the resistance between two adjacent layers of the fluid moving relative to each other

8. Laminar Flow

9. Laminar Flow
ESSENTIALLY: Laminar flow, type of fluid (gas or liquid) flow in which the fluid travels smoothly or in regular paths

10. 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

11. 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

12. Flow Rate ƒ = Aν = (m2)(m/s)
Flow Rate (ƒ): Volume of fluid that passes a particular point in a given time
Units used to measure Flow Rate = m³/sec
Equation for: Flow Rate
ƒ = Aν = (m2)(m/s)
(A = cross sectional area)
(ν = velocity of fluid)

13. Rate of Flow vt
Volume = A(vt) A
Rate of flow = velocity x area

14. Since A1 > A2…
For an incompressible, frictionless fluid, the velocity increases when the cross-section decreases:

15. ƒ₁ = ƒ₂ A₁ν₁ = A₂ν₂ Continuity Equation
Flow rates are the same at all points along a closed pipe
Continuity Equation:
ƒ₁ = ƒ₂
A₁ν₁ = A₂ν₂
Reminder: the equation for Area of a circle: A = πr²
Emphasive misconception with Area!

16. Question:
Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out through a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle?
ANS: 19 m/s

17. Bernoulli's Principle
The Swiss Physicist Daniel Bernoulli, was interested in how the velocity changes as the fluid moves through a pipe of different area. He especially wanted to incorporate pressure into his idea as well. Conceptually, his principle is stated as: " If the velocity of a fluid increases, the pressure decreases and vice versa."
The velocity can be increased by pushing the air over or through a CONSTRICTION
A change in pressure results in a NET FORCE towards the low pressure region.

18. Bernoulli's Principle in Action
The constriction in the Subclavian artery causes the blood in the region to speed up and thus produces low pressure. The blood moving UP the LVA is then pushed DOWN instead of up causing a lack of blood flow to the brain. This condition is called TIA (transient ischemic attack) or “Subclavian Steal Syndrome.
One end of a gopher hole is higher than the other causing a constriction and low pressure region. Thus the air is constantly sucked out of the higher hole by the wind. The air enters the lower hole providing a sort of air re-circulating system effect to prevent suffocation.

19. Bernoulli’s Equation Relates pressure to fluid speed and elevation
Bernoulli’s equation is a consequence of Conservation of Energy applied to an ideal fluid
Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steady-state manner

20. 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

21. Bernoulli's Equation Derivation
X = L
Let’s look at this principle mathematically.
F1 on 2
-F2 on 1
Work is done by a section of water applying a force on a second section in front of it over a displacement. According to Newton’s 3rd law, the second section of water applies an equal and opposite force back on the first. Thus is does negative work as the water still moves FORWARD. Pressure*Area is substituted for Force.

22. Bernoulli's Equation Derivationy2
L1=v1t
L2=v2t y1 v1 A1
ground
Work is also done by GRAVITY as the water travels a vertical displacement UPWARD. As the water moves UP the force due to gravity is DOWN. So the work is NEGATIVE.

23. Bernoulli's Equation Derivation
Now let’s find the NET WORK done by gravity and the water acting on itself.
WHAT DOES THE NET WORK EQUAL TO? A CHANGE IN KINETIC
ENERGY!

24. Bernoulli's Equation Derivation
Consider that Density = Mass per unit Volume AND that VOLUME is
equal to AREA time LENGTH

25. Bernoulli's Equation Derviation
We can now cancel out the AREA and LENGTH
Leaving:

26. Bernoulli's Equation Derivation
Moving everything related to one side results in:
What this basically shows is that Conservation of Energy holds true within a fluid and that if you add the PRESSURE, the KINETIC ENERGY (in terms of density) and POTENTIAL ENERGY (in terms of density) you get the SAME VALUE anywhere along a streamline.

27. An Object 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

28. Example
Water circulates throughout the house in a hot-water heating system. If the water is pumped at a speed of 0.50 m/s through a 4.0 cm diameter pipe in the basement under a pressure of 3.0 atm, what will be the flow speed and pressure in a 2.6 cm-diameter pipe on the second floor 5.0 m above?
1 atm = 1x105 Pa
1.183 m/s
2.5x105 Pa(N/m2) or 2.5 atm

29. Application – Golf Ball
The dimples in the golf ball help move air along its surface
The ball pushes the air down
Newton’s Third Law tells us the air must push up on the ball
The spinning ball travels farther than if it were not spinning

30. Airplane Wings - Application

31. 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
Called lift
Other factors are also involved

32. 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

33. Venturi Meter
Law of Conservation of Energy ~ Bernoulli’s Equation!
Energy due to pressure gets converted into energy due to velocity (kinetic energy)
So higher the velocity the lower the pressure
The higher the velocity in the constriction at Region-2, the lower the pressure... Wait why?

34. Venturi Effect
High Pressure required to move MORE fluids

35. Venturi Effect Law of Conservation of Energy ~ Bernoulli’s Equation!
Energy due to pressure gets converted into energy due to velocity (kinetic energy)
So higher the velocity the lower the pressure