Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Two-body problem in general relativity wikipedia , lookup
Differential equation wikipedia , lookup
Exact solutions in general relativity wikipedia , lookup
Partial differential equation wikipedia , lookup
Equations of motion wikipedia , lookup
Schwarzschild geodesics wikipedia , lookup
Euler equations (fluid dynamics) wikipedia , lookup
PHYS16 – Lecture 33 On a windy day in 1735, a new wig gives Bernoulli an idea. Fluids: Bernoulli’s Principle Fluids pre-question • You are watering some flowers with a garden hose. You want to water some flowers in the very back of the garden. Assume that the flowers you want to water are at the same level as the hose. What do you have to do to the opening of the hose to increase the range by 4? A) Decrease the area by a factor of 16 B) Decrease the area by a factor of 4 C) Decrease the area by a factor of 2 D) Increase the area by a factor of 2 E) Increase the area by a factor of 4 Outline for Fluids • Fluid Statics – Pressure and Pascal’s Principle – Buoyant Force and Archimedes’ Principle • Fluid Dynamics – Ideal Fluids – Equation of Continuity – Bernoulli’s Equation Buoyant Force and anchors… • If I have a boat in a pond and I toss out an anchor what happens to the water level in the pond? Fluid Dynamics: 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 – 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) What happens if fluid is not ideal? Poiseuille's Law • When frictional forces dominate velocity decreases – Viscous fluids – Small Diameters Ideal – larger diameters v 2p With Friction – small diameters r p v 8L 2 Fluid Dynamics: Equation of Continuity Equation of Continuity • For an ideal fluid flowing in a pipe, the volume flow rate through the pipe is constant V Av constant t A1v1 A2v2 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? A) B) C) D) Half Same Twice Four times http://cardiovascres.oxfordjournals.org/content/65/3/619/F4.small.gif Fluid Dynamics: Bernoulli’s Equation Bernoulli’s Equation • For an ideal fluid flowing in a pipe, pressure in the pipe is related to the velocity and height of fluid 1 2 1 2 p1 gh1 v1 p2 gh2 v2 2 2 Discussion: 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 Main Points • Buoyant force FB fluidVobject underwater g • Ideal fluid is incompressible, laminar, nonviscous, and irrotational • Equation of continuity • Bernoulli’s Equation Av constant 1 2 p gh v constant 2