Physics 101: Lecture 25
Fluids in Motion: Bernoulli’s Equation
z
Today’s lecture will cover Textbook Sections 11.7-11.10
Î Fluids in motion: Continuity & Bernoulli’s equation
Note: Everything we do assumes fluid
is non-viscous and incompressible.
Physics 101: Lecture 25, Pg 1
Physics 101: Lecture 24
Archimedes Principle (summary)
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Buoyant Force (FB)
ÎFB=weight of fluid displaced
ÎFB = ρfluid Vdispl g
ÎW = Mg = ρobject Vobject g
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If object floats….
ÎFB=W
ÎTherefore ρfluid g Vdispl. = ρobject g Vobject
ÎTherefore Vdispl./Vobject = ρobject / ρfluid
Physics 101: Lecture 25, Pg 2
Concept Question
Suppose you float a large ice-cube in a glass of water, and that after
you place the ice in the glass the level of the water is at the very brim.
When the ice melts, the level of the water in the glass will:
1. Go up, causing the water to spill out of the glass.
2. Go down.
3. Stay the same.
CORRECT
FB = ρW g
Vdisplaced
W = ρice g Vice
Vdisplaced = Vice under water = Vice ρice/ρW
Physics 101: Lecture 25, Pg 3
Fluids in Motion
Consider an ideal fluid (incompressible and nonviscous)
that flows steadily.
z Steady Flow:
Every fluid particle passing trough the same point in the
stream has the same velocity.
Streamlines are used to visualize the trajectory of fluid
particles in motion. The velocity vector of the fluid
particle is tangent to the streamline.
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The fluid velocity can vary from point to point along a
streamline but at a given point the velocity is constant in
time.
Physics 101: Lecture 25, Pg 4
Equation of Continuity
Mass is conserved as the fluid flows.
If a certain mass of fluid enters a pipe at one end at a
certain rate, the same mass exits at the same rate
at the other end of the tube (if nothing gets lost in
between through holes, for instance).
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Mass flow rate at position 1 = Mass flow rate at position 2
ρ1 A1 v1 = ρ2 A2 v2
ρ A v = constant along a tube that has a single entry
and a single exit point for fluid flow.
Physics 101: Lecture 25, Pg 5
Concept Question
A stream of water gets narrower as it falls from a faucet (try it & see).
This phenomenon can be explained using the equation of continuity
The water's velocity is increasing as it flows down, so in
order to compensate for the increase in velocity, the area
must be decreased because the density*area*speed must
be conserved
A1
V1
V2
A2
Physics 101: Lecture 25, Pg 6
Bernoulli’s Equation
Work-Energy Theorem : Wnc = change of total mechanical energy
applied to fluid flow :
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Difference in pressure => net force is not zero => fluid accelerates
Pressure is due to collisional forces which is a nonconservative force:
Wnc = (P2-P1) V
Consider a fluid moving from height h1 to h2. Its total mechanical
energy is given by the sum of kinetic and potential energy. Thus,
Wnc = Etot,1 –Etot,2 = ½ m v12+m g h1 –( ½ m v22+m g h2)
Physics 101: Lecture 25, Pg 7
Fluid Flow (summary)
A1 ρ1 v1
A2 ρ2 v2
• Mass flow rate: ρAv (kg/s)
• Continuity: ρ1A1 v1 = ρ2A2 v2
i.e., mass flow rate the same everywhere
e.g., flow of river
For fluid flow without friction (nonviscous):
• Bernoulli:
P1 + 1/2 ρv12 + ρgh1 = P2 + 1/2 ρv22 + ρgh2
Physics 101: Lecture 25, Pg 8