Fluids: Bernoulli’s Principle November 12, 2010 PHYS16 – Lecture 30 On a windy day in 1735, a new wig gives Bernoulli an idea. Fluids: Bernoulli’s Principle November 12, 2010

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

Questions… Does the buoyant force change as you go deeper underwater? Does the buoyant force change as you go higher in the atmosphere? 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.)

Which fluids are ideal? Water Air Honey Blood 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)

Fluid Dynamics

Equation of Continuity For an ideal fluid flowing in a pipe, the volume flow rate through the pipe is constant Narrower section Larger speed Wider section Smaller 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? Half Same Twice 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

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 apart B) sheets will come together C) 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? Blood pressure would be higher Blood pressure would be lower 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? A increases, v decreases, P increases http://www.nlm.nih.gov/medlineplus/ency/images/ency/fullsize/18072.jpg

Example: Water jets out of a bottle Which jet will have the largest range? 1 2 3 4 5

Main Points Buoyant force Ideal fluid is incompressible, laminar, nonviscous, and irrotational Equation of continuity Bernoulli’s Equation