Physics 1501 Lecture 33 Today s Agenda l Homework

Physics 1501: Lecture 33 Today’s Agenda l Homework #11 (due Friday Dec. 2) l Midterm 2: graded by Dec. 2 l Topics: ç Fluid dynamics ç Bernouilli’s equation ç Example of applications Physics 1501: Lecture 33, Pg 1

Pascal and Archimedes’ Principles l Pascal’s Principle Any change in the pressure applied to an enclosed fluid is transmitted to every portion of the fluid and to the walls of the containing vessel. l Archimedes’ principle The buoyant force is equal to the weight of the liquid displaced. çObject is in equilibrium Physics 1501: Lecture 33, Pg 2

Ideal Fluids l l l Fluid dynamics is very complicated in general (turbulence, vortices, etc. ) Consider the simplest case first: the Ideal Fluid çno “viscosity” - no flow resistance (no internal friction) çincompressible - density constant in space and time streamlines do not meet or cross velocity vector is tangent to streamline volume of fluid follows a tube of flow bounded by streamlines Flow obeys continuity equation ç volume flow rate Q = A·v is constant along flow tube: ç follows streamline A 1 v 1 = A 2 v 2 from mass conservation if flow is incompressible. Physics 1501: Lecture 33, Pg 3

Conservation of Energy for Ideal Fluid l Recall the standard work-energy relation çApply the principle to a section of flowing fluid with volume d. V and mass dm = r d. V (here W is work done on fluid) d. V Bernoulli Equation Physics 1501: Lecture 33, Pg 4

Lecture 33 Act 1 Bernoulli’s Principle l A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v 1/2 What is the pressure in the 1/2” pipe relative to the 1” pipe? a) smaller b) same c) larger Physics 1501: Lecture 33, Pg 5

Some applications l Lift for airplane wing l Enhance sport performance l More complex phenomena: ex. turbulence Physics 1501: Lecture 33, Pg 6

More applications l Vortices: ex. Hurricanes l And much more … Physics 1501: Lecture 33, Pg 7

Ideal Fluid: Bernoulli Applications l Bernoulli says: high velocities go with low pressure l Airplane wing ç shape leads to lower pressure on top of wing ç faster flow lower pressure lift » air moves downward at downstream edge wing moves up Physics 1501: Lecture 33, Pg 8

Ideal Fluid: Bernoulli Applications l Warning: the explanations in text books are generally oversimplified! l Curve ball (baseball), slice or topspin (golf) ç ball drags air around (viscosity) ç air speed near ball fast at “top” (left side) ç lower pressure force sideways acceleration or lift Physics 1501: Lecture 33, Pg 9

Ideal Fluid: Bernoulli Applications l Bernoulli says: high velocities go with low pressure l “Atomizer” ç moving air ‘sweeps’ air away from top of tube ç pressure is lowered inside the tube ç air pressure inside the jar drives liquid up into tube Physics 1501: Lecture 33, Pg 10

Example: Efflux Speed The tank is open to the atmosphere at the top. Find and expression for the speed of the liquid leaving the pipe at the bottom. Physics 1501: Lecture 33, Pg 11

Solution Physics 1501: Lecture 33, Pg 12

Example Fluid dynamics l l A siphon is used to drain water from a tank (beside). The siphon has a uniform diameter. Assume steady flow without friction, and h=1. 00 m. You want to find the speed v of the outflow at the end of the siphon, and the maximum possible height y above the water surface. C y O A v h B Use the 5 step method ç Draw a diagram that includes all the relevant quantities for this problem. What quantities do you need to find v and ymax ? Physics 1501: Lecture 33, Pg 13

Example: Solution Fluid dynamics l Draw a diagram that includes all the relevant quantities for this problem. What quantities do you need to find v and ymax ? çNeed P and v values at points O, A, B, C to find v and ymax çAt O: P 0=Patm and v 0=0 çAt A: PA and v. A C çAt B: PB=Patm and v 0=v çAt C: PC and v. C y O A çFor y set P =0 max C v h B Physics 1501: Lecture 33, Pg 14

Example: Solution Fluid dynamics l What concepts and equations will you use to solve this problem? çWe have fluid in motion: fluid dynamics çFluid is water: incompressible fluid çWe therefore use Bernouilli’s equation çAlso continuity equation Physics 1501: Lecture 33, Pg 15

Example: Solution Fluid dynamics l Solve for v and ymax in term of symbols. çLet us first find v=v. B C O A y v h çWe use the points O and B B where : P 0=Patm=1 atm and v 0=0 and y 0=0 where: PB=Patm=1 atm and v. B=v and y. B=-h çSolving for v Physics 1501: Lecture 33, Pg 16

Example: Solution Fluid dynamics l C O A Solve for v and ymax in term of symbols. çIncompressible fluid: Av =constant çA is the same throughout the pipe v. A= v. B= v. C = v çTo get ymax , use the points C and B (could also use A) y v h B where: PB=Patm=1 atm and v. B=v and y. B=-h set : PC=0 (cannot be negative) and v. C=v and y. C= ymax çSolving for ymax Physics 1501: Lecture 33, Pg 17

Example: Solution Fluid dynamics l Solve for v and ymax in term of numbers. çh = 1. 00 m and use g=10 m/s 2 çPatm=1 atm = 1. 013 105 Pa (1 Pa = 1 N/m 2 ) çdensity of water = 1. 00 g/cm 3 = 1000 kg/m 2 Physics 1501: Lecture 33, Pg 18

Example: Solution Fluid dynamics l Verify the units, and verify if your values are plausible. ç[v] = L/T and [ymax] = L so units are OK çv of a few m/s and ymax of a few meters seem OK » Not too big, not too small l Note on approximation çSame as saying C PA= PO =Patm or v. A=0 y çi. e. neglecting the flow in the pipe at point A O A v h B Physics 1501: Lecture 33, Pg 19

Real Fluids: Viscosity l l In ideal fluids mechanical energy is conserved (Bernoulli) In real fluids, there is dissipation (or conversion to heat) of mechanical energy due to viscosity (internal friction of fluid) Viscosity measures the force required to shear the fluid: area A where F is the force required to move a fluid lamina (thin layer) of area A at the speed v when the fluid is in contact with a stationary surface a perpendicular distance y away. Physics 1501: Lecture 33, Pg 20

Real Fluids: Viscosity l Viscosity arises from particle collisions in the fluid ç as particles in the top layer diffuse downward they transfer some of their momentum to lower layers area A ç lower layers get pulled along (F = Dp/Dt) air H 2 O oil glycerin Viscosity (Pa-s) Physics 1501: Lecture 33, Pg 21

Real Fluids: Viscous Flow l How fast can viscous fluid flow through a pipe? ç Poiseuille’s Law L p+Dp Q r p R l Because friction is involved, we know that mechanical energy is not being conserved - work is being done by the fluid. l Power is dissipated when viscous fluid flows: P = v·F = Q ·Dp the velocity of the fluid remains constant power goes into heating the fluid: increasing its entropy Physics 1501: Lecture 33, Pg 22

l Lecture 33 Act 2 Viscous flow Consider again the 1 inch diameter pipe and the 1/2 inch diameter pipe. L/2 1) Given that water is viscous, what is the ratio of the flow rates, Q 1/2, in pipes of these sizes if the pressure drop per meter of pipe is the same in the two cases? a) 3/2 b) 2 c) 4 Physics 1501: Lecture 33, Pg 23