Fluid Dynamics
Two Parts
1. Fluid Flow
2. Bernoulli’s Equation and Applications
Assumptions for Fluid Flow:
1. Non-viscous. (isn’t “sticky”)
2. Incompressible (constant ρ)
3. All particles in cross section travel at the same speed
(flow rate)
4. Flow is laminar (no turbulence)
Streamline flow
Turbulent flow
Laminar Flow
Laminar flow, type of fluid (gas or liquid) flow in which the fluid travels
smoothly or in regular paths
Laminar flow over a horizontal surface may be thought of as consisting
of thin layers, or laminae, all parallel to each other.
Laminar Flow
• Video:
Flow Rate
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)
Rate of Flow
V  Avt
A
vt
Volume = A(vt)
Avt
R
 vA
t
Rate of flow = velocity x area
Since A1 > A2…
For an incompressible, frictionless fluid, the velocity increases
when the cross-section decreases:
R  v1 A1  v2 A2
v1 < v2
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
PHet
• Fluid Flow
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?
Venturi Meter
The higher the velocity in the constriction at Region-2, the
lower the pressure... Wait what?
Venturi Effect
Venturi Effect
Airplane Wings
Airplane Wings
How do Plane’s Fly
Video
The Physics of Sailing
Video
http://science.kqed.org/quest/video/thephysics-of-sailing/
Question
A small ranger vehicle has a
soft, ragtop roof. When the car
is at rest the roof is flat. When
the car is cruising at highway
speeds with its windows rolled
up, does the roof
a. bow upward
b. remain flat
c. bow downward?
Question
A small ranger vehicle has a
soft, ragtop roof. When the car
is at rest the roof is flat. When
the car is cruising at highway
speeds with its windows rolled
up, does the roof
a. bow upward
b. remain flat
c. bow downward?
Fluid Flow Questions
1. MC - 4,14,21,42,47
2. Homework: Watch Bernoulli Video
3. MOST IMPORTANTLY: Paper Airplane Competition next class
Go to: http://www.funpaperairplanes.com/index.html
a.
b.
c.
d.
Pick a plane and build it for the start of class
Make TWO of the same design
Planes will be thrown in players hall
Winner will be determined by displacement from initial throw
Sports Science
• Record Paper Airplane
Conservation of Energy of Fluids within a Pipe
Bernoulli's Principle
PRESSURE plus ENERGY is CONSTANT!
1. P + E = P + E
2. P + U + K = P + U + K
3. P + ρgh + ½ρν² = P + ρgh + ½ρν²
This hold at ANY point!
P1 + ρgh1 + ½ρν1² = P2 + ρgh2 + ½ρν2²
Bernuolli Effect
1. High Velocity: _____ Pressure
2. Low Velocity: _____ Pressure
Bernuolli Effect
1. High Velocity: LOW Pressure
2. Low Velocity: HIGH Pressure
Special Case #1 – Horizontal Pipe
P1   gh1  ½v12  P2   gh2  ½v22
Horizontal Pipe (h1 = h2)
Horizontal Pipe
P   gh  ½ v  ½ v
2
2
2
1
Question
Suppose the pressure in
the fire hose is 350 kPa.
What is the pressure in
the nozzle?
ν1 = 1.3 m/s
ν2 = 19.17 m/s
Special Case #2 – Constant Velocity
P1   gh1  ½v  P2   gh2  ½v
2
1
2
2
Constant velocity (ν1 = ν2)
Notice how a difficult problem becomes easier when we remove constants!
Question
Water flows with constant speed
through a garden hose that goes
up a step 20.0 cm high. If the
water pressure is 143 kPa at the
bottom of the step, what is its
pressure at the top of the step?
ν1 = ν2
Special Case #3 – Fluids at Rest
P1   gh1  ½v12  P2   gh2  ½v22
P1 - P2 = gh2 - gh1
P = g(h2 - h1)
We have already seen this!
Special Case #4 – No Change in Pressure
Know as Torricelli’s Theorem
P1   gh1  ½v12  P2   gh2  ½v22
v2  0
h2 h
h1
v  2gh
Torricelli’s theorem:
v  2gh
Question:
A dam springs a leak at a point
20.0 m below the surface.
What is the emergent velocity?
h
v = 19.8 m/s2
v  2gh
Summary of Hydrodynamics
Streamline Fluid Flow in Pipe:
R  v1 A1  v2 A2
Fluid at Rest:
PA - PB = gh
Horizontal Pipe (h1 = h2)
P1  P2  ½ v22  ½ v12
Bernoulli’s Theorem:
P1   gh1  ½ v12  Constant
Torricelli’s theorem:
v  2gh
Bernoulli’s Principal
1. MC: 5,13,22,25,27,28,33,36,37,44
2. Homework: Review Free Response Questions Posted
on Website
3. Next Class: Hydrodynamics Quiz