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Fluid Dynamics
AP Physics 2
Fluid Flow
Up till now, we have pretty much focused on fluids at
rest. Now let's look at fluids in motion
Characteristics of an Ideal Fluid

The fluid is nonviscous


The fluid is incompressible

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Its density is constant
The fluid motion is steady

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There is no internal friction between adjacent layers
Its velocity, density, and pressure do not change in time
The fluid moves without turbulence
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No eddy currents are present
The elements have zero angular velocity about its center
Fluids in Motion:
Streamline Flow
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A fluid’s motion can be said to be streamline,
or laminar.
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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.
Fluids in Motion:
Streamline Flow

Streamline flow
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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

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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
Fluid Flow: Viscosity

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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
Laminar Flow
Laminar Flow
ESSENTIALLY: Laminar flow, type of fluid (gas or liquid) flow in
which the fluid travels smoothly or in regular paths
Fluids in Motion:
Turbulent Flow

The flow becomes irregular

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
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
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)
R
Avt
 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
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
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.
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.
Bernoulli’s Equation

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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, steadystate manner
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  gy  constant
2
Bernoulli's Equation Derivation
X=L
F1 on 2
Let’s look at this principle
mathematically.
-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.
Bernoulli's Equation Derivation
v2
A2
y2
L1=v1t
L2=v2t
v1
y1
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.
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!
Bernoulli's Equation Derivation
Consider that Density = Mass per unit
Volume AND that VOLUME is
equal to AREA time LENGTH
Bernoulli's Equation Derviation
We can now cancel out the AREA and LENGTH
Leaving:
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.
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
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?
A1v1  A1v2
1 atm = 1x105 Pa
r12v1  r22v2
(0.04) 2 0.50  (0.026) 2 v2
v2  1.183 m/s
1 2
1
vo  gho  P  v 2  gh
2
2
1
1
3x105  (1000)(0.50) 2  (1000)(9.8)(0)  P  (1000)(1.183) 2  (1000)(9.8)(5)
2
2
P  2.5x105 Pa(N/m2) or 2.5 atm
Po 
Application – Golf Ball
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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
Airplane Wings - Application
Application – Airplane Wing

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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

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Called lift
Other factors are also
involved
Applications of Bernoulli’s
Principle: Venturi Tube

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Shows fluid flowing
through a horizontal
constricted pipe
Speed changes as
diameter changes
Can be used to
measure the speed of
the fluid flow
Venturi Meter
The higher the velocity in the constriction at Region-2,
the lower the pressure... Wait why?
Venturi Effect
Venturi Effect
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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