PHYS16 – Lecture 30

Fluids: Bernoulli’s Principle November 12, 2010

On a windy day in 1735, a new wig gives Bernoulli an idea.

Outline for Fluids

• Pressure and Pascal’s Principle• Buoyant Force and Archimedes’ Principle• Fluid dynamics– Ideal Fluids– Equation of Continuity– Bernoulli’s Equation

Revisiting Buoyant Force…

Archimedes’ Principle

• Buoyant force = the weight of the water displaced

gVgmFFmgFF

underwaterobjectfluiddisplacedfluidB

B

21

http://www.open2.net/open2static/source/file/root/0/30/19/124156/pressure_cube_b.jpg

Sink or Float?

• Floating requires buoyant force to equal gravity

gVgVFF

objectobjectunderwaterobjectfluid

GB

http://mcat-review.org/fluids-solids.php

Questions…

1) Does the buoyant force change as you go deeper underwater?

2) Does the buoyant force change as you go higher in the atmosphere?

3) Is buoyant force on Diet Coke vs. Coke different? Will Diet Coke or Coke float higher?

No, assume constant density

Yes, changing density

Buoyant force is the same, gravitational force is differet so Diet Coke floats higher…

Demo…

• Rock in boat• Sinking boat• Inverting weight + Styrofoam system• Copper ball vs. wood ball

Ideal Fluids

Ideal Fluids

• Incompressible – density is a constant • Nonviscous – ignore frictional effects• Irrotational – doesn’t rotate• Laminar – no acceleration

Streamlines represent fluid flow

Ideal Fluids

• Mass is conserved• Energy is conserved• Momentum is conserved• Continuum hypothesis is true – properties

defined at infinitesimal points (density, pressure, temperature, etc.)

• Water – can be turbulent (waterfall not ideal, ideal in a slow moving river)

• Air – compressible (piston not ideal, ideal in a laminar wind)

• Honey – viscous fluid such that drag forces can’t be neglected (Not usually ideal)

• Blood – pulsatile flow, filled with proteins/cells (ideal in large arteries or veins, not capillaries)

Which fluids are ideal?

• Water

• Air

• Honey

• Blood

Fluid Dynamics

Equation of Continuity

• For an ideal fluid flowing in a pipe, the volume flow rate through the pipe is constant

2211

constant

vAvA

AvtV

Narrower sectionLarger speed

Wider sectionSmaller speed

Example: Water out of faucet

• Why does the stream of water flowing from a faucet often get more narrow as the water falls?

Gravity accelerates water so velocity increases. If velocity goes up, then area goes down…

http://thegoldenspiral.org/wp-content/uploads/2008/10/faucet_waterglass.jpg

Example: Arterial branching

• An artery branches into two smaller arteries, each with half the diameter of the first. What is the velocity in the smaller artery compared to the larger artery?

A) HalfB) SameC) TwiceD) Four times

http://cardiovascres.oxfordjournals.org/content/65/3/619/F4.small.gif

Bernoulli’s Equation

• For an ideal fluid flowing in a pipe, pressure in the pipe is related to the velocity and height of fluid

2222

2111 2

121 vghpvghp

Example: Two sheets in the wind?

• What happens if I take two sheets of paper, separate them by 1” and blow between them?

A) sheets will move apartB) sheets will come togetherC) sheets will stay at same spots

http://www.practicalphysics.org/imageLibrary/jpeg273/735.jpg

Example: Blood Pressure

• What would happen if the doctor took a blood pressure reading at the wrist instead of on the bicep?

A) Blood pressure would be higherB) Blood pressure would be lowerC) Blood pressure would be the same

http://www.omron.com

Example: Aneurysm

• In an aneurysm the arterial wall weakens and the diameter increases. Why does this increase the chance of rupture?

http://www.nlm.nih.gov/medlineplus/ency/images/ency/fullsize/18072.jpg

A increases, v decreases, P increases

Example: Water jets out of a bottle

• Which jet will have the largest range?

12345

Main Points

• Buoyant force

• Ideal fluid is incompressible, laminar, nonviscous, and irrotational

• Equation of continuity

• Bernoulli’s Equation

gVF underwaterobjectfluidB

constant Av

constant 21 2 vghp