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

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

Physics 1501: Lecture 33, Pg 4 l Recall the standard work-energy relation  Apply the principle to a section of flowing fluid with volume  V and mass  m =  V (here W is work done on fluid) VV Conservation of Energy for Ideal Fluid Bernoulli Equation

Physics 1501: Lecture 33, Pg 5 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. What is the pressure in the 1/2” pipe relative to the 1” pipe? a) smallerb) samec) larger v1v1 v 1/2

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

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

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

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

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

Physics 1501: Lecture 33, Pg 11 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. Example: Efflux Speed

Physics 1501: Lecture 33, Pg 12 Solution

Physics 1501: Lecture 33, Pg 13 v h y A B C O 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. Example Fluid dynamics l 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 y max ?

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

Physics 1501: Lecture 33, Pg 15 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 16 Example: Solution Fluid dynamics l Solve for v and y max in term of symbols. çLet us first find v=v B çWe use the points O and B where : P 0 =P atm =1 atm and v 0 =0 and y 0 =0 where: P B =P atm =1 atm and v B =v and y B =-h çSolving for v v h y A B C O

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

Physics 1501: Lecture 33, Pg 18 l Solve for v and y max in term of numbers. çh = 1.00 m and use g=10 m/s 2 çP atm =1 atm =  10 5 Pa (1 Pa = 1 N/m 2 ) çdensity of water  water = 1.00 g/cm 3 = 1000 kg/m 2 Example: Solution Fluid dynamics

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

Physics 1501: Lecture 33, Pg 20 l In ideal fluids mechanical energy is conserved (Bernoulli) l In real fluids, there is dissipation (or conversion to heat) of mechanical energy due to viscosity (internal friction of fluid) Real Fluids: Viscosity Viscosity measures the force required to shear the fluid: 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. area A

Physics 1501: Lecture 33, Pg 21 l Viscosity arises from particle collisions in the fluid ças particles in the top layer diffuse downward they transfer some of their momentum to lower layers Real Fluids: Viscosity       Viscosity (Pa-s) oilairglycerinH2OH2O     area A  lower layers get pulled along (F =  p/  t)

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

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