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

1

121212

1

222111

fluid of piece""

on iprelationshwork -Energy

PgyvPgyv

gyvPPgyv

VM

MgyMv

VPPMgyMv

UKWUK

UKWUK

M

fffiii

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