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Lesson Opener:
How does a plane fly?How does a perfume spray work?Why does a cricket ball curve?
Daniel Bernoulli (1700 – 1782)
Derivation and Applications of the
Bernoulli PrincipalNIS TaldykorganGrade 11 Physics
Lesson Objective:1.To apply Bernoulli’s equation to solve problems
2.To describe Bernoulli’s principle and to derive his formula in terms of conservation of energy
3.To present applications of the Bernoulli principle
Bernoulli’s PrincipleBernoulli’s PrincipleAs the speed of a fluid goes up, its
pressure goes down!
The pressure in a fast moving stream of fluid is less than the pressure in a slower stream
Fast stream = low air pressureFast stream = low air pressure
Slow stream = High air pressureSlow stream = High air pressure
v small v smallv large
p large p large
p small
A1 A2
v1v2
A1
v1Low speedLow KEHigh pressure
high speedhigh KElow pressure
Low speedLow KEHigh pressure
Equation of Continuity
Bernoulli’s Equation in terms of Fluid Energy“for any point along a flow tube or streamline”
P + ½ v2 + g h = constant
Each term has the dimensions of energy / volume or energy density.
½ v 2 KE of bulk motion of fluid
g h GPE for location of fluid
P pressure energy density arising from internal forces within moving fluid (similar to energy stored in a spring) Transformation of SI Units to Joule/meter3= energy/volume:
P [Pa] = [N m-2] = [N m m-3] = [J m-3]
½ v2 [kg m-3 m2 s-2] = [kg m-1 s-2] = [N m m-3] = [J m-3]
g h [kg m-3 m s-2 m] = [kg m s-2 m m-3] = [N m m-3] = [J m-3]
Deriving Bernoulli’s starting with the law of continuity
2211
2211
2211
21
fluid of massgiven aConsider
(constant)
:fluid ible"incompress"an For
vAvA
tvAtvAM
xAxAM
VVM
Bernoulli’s EquationFor steady flow, the velocity, pressure, and elevation of an incompressible and nonviscous fluid are related by an equation discovered by Daniel Bernoulli (1700–1782).
Deriving Bernoulli’s equation as Conservation of Energy
22
222
111
212
1
2222
1121
212
1
2222
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121212
1
222111
fluid of piece""
on iprelationshwork -Energy
PgyvPgyv
gyvPPgyv
VM
MgyMv
VPPMgyMv
UKWUK
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Bernoulli’s equation:
22222
111
212
1 PgyvPgyv
21
2
1
2222
1
2222
11
212
1
221121
1
PPA
Av
PvPv
vAvAyy
BERNOULLI’S EQUATION
2
222
21
1
22
2
Vp
Vp
Vp
• In a moving fluid p+½V2 = constant everywhere• An increase in velocity of the fluid results in a decrease in
pressure• Bernoulli’s equation is an extension of F=ma for fluid
flows and aerodynamics
Constant
HOW DOES A WING GENERATE LIFT?
• An imbalance of pressure over the top and bottom surfaces of the wing.– If the pressure above is lower than the pressure on
bottom surface, lift is generated
Airplane Wing is curved on top
HOW DOES A CURVED WING GENERATE LIFT?
Flow velocity over the top of wing is faster than over bottom surface– Air over wing is squashed to smaller cross-
sectional area– Mass continuity AV=constant, velocity must
increase
high speedlow pressure
force
force
What happens when two ships or trucks pass alongside each other?
velocity increasedpressure decreased
low pressurehigh pressure
(patm)
VENTURI EFFECT
artery
External forces causes artery to collapse
Flow speeds up at constrictionPressure is lowerInternal force acting on artery wall is reduced
Arteriosclerosis and vascular flutter
References and links:• Bernoulli Activity:• http://mitchellscience.com/
bernoulli_principle_discussion_nomath• Steve Spangler and Hydrogen Hexafluoride: • http://www.youtube.com/watch?v=GRLOgmmz_EU• Phet Colorado Fluid Pressure and Flow simulation:• http://phet.colorado.edu/en/simulation/fluid-pressure-and-flow • Types of Fluids:
http://mechteacher.com/fluid/#ixzz2fcgGwLbq• Flowing Fluids,Laminar Flow and stream lines:• http://www.youtube.com/watch?v=_aWdeXby7CA