# Lecture 27 (Walker: ) Fluid Dynamics Nov. 9, 2009

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1 Physics 111 Lecture 27 (Walker: ) Fluid Dynamics Nov. 9, 2009 Midterm #2 - Monday Nov. 16 Chap. 7,Chap. 8 (not 8.5) Chap. 9 (not 9.6, 9.8) Chap. 10, Chap. 11 (not ) Chap. 13 (not ) Chap. 15 (not ) Midterm Review - Friday, 3:10 p.m., TH230 Lecture 27 1/32 Fluids in Motion: Flow and Continuity If the flow of a fluid is smooth, it is called streamline or laminar flow (a). We will work with laminar flow. Above a certain speed, the flow becomes turbulent (b). Turbulent flow has eddies; the viscosity (friction) of the fluid is much greater when eddies are present. Lecture 27 2/32 Flow Rate and the Equation of Continuity Look at fluid flowing in a tube or pipe. The mass flow rate is the mass of fluid that passes a given point per unit time. The mass flow rates at any two points in a pipe or tube must be equal, as long as no fluid is being added or taken away. What goes in must come out Lecture 27 3/32 Volume Flow Rate V 1, Volume of fluid flowing through A 1 in time t, is A 1 l 1 where l 1 is the distance fluid flows in t. If fluid in region 1 moving at speed v 1, l 1 = v 1 t and V 1 = A 1 v 1 t. Volume flow rate V 1 / t = A 1 v 1. Lecture 27 4/32

2 Mass Flow Rate The mass flow rate at point 1 is M 1 / t = ρ 1 V 1 / t = ρ 1 A 1 v 1 Flow Rate and the Equation of Continuity If the density doesn t change typical for liquids this simplifies to. Where the pipe is wider, the flow is slower. Equation of Continuity for Liquids. Must have same mass flow rate at point 2: ρ 1 A 1 v 1 = ρ 2 A 2 v 2 General equation of continuity. Lecture 27 5/32 Lecture 27 6/32 Water Pipe Water is flowing continuously in the pipe shown below. Where is the velocity of the water greatest? A) B) C) D) equal everywhere A B C Four-lane highway merges to two-lane. Officer in police car observes 8 cars passing per second, at 30 mph. How many cars does officer on motorcycle observe passing per second? A) 4 B) 8 C) 16 How fast must cars in two-lane section be going? Lecture 27 7/32 Lecture 27 8/32 A) 15 mph B) 30 mph C) 60 mph

3 Example A horizontal pipe contains water at a pressure of 110 kpa flowing with a speed of 1.4 m/s. When the pipe narrows to one-half its original radius, what is the speed? Bernoulli s Equation Bernoulli related the pressure in a fluid to fluid kinetic and potential energies. Consider a volume element V of fluid moving through a pipe. For now, we neglect fluid friction (we assume fluid has no viscosity). v f A f = v i A i v f = v i A i /A f = v i [πr i2 ]/[π(r i /2) 2 ] = 4v i v f = 5.6 m/s Lecture 27 9/32 Lecture 27 10/32 Bernoulli s Equation When a fluid moves from a wider area of pipe to narrower one, its speed increases; therefore, work has been done on it. Bernoulli s Equation - Const. Height The kinetic energy of a fluid element is: Equating the work done on fluid element to its increase in kinetic energy gives: Lecture 27 11/32 This is Bernoulli s Equation for a pipe at constant height. Lecture 27 12/32

4 Bernoulli s Equation - Const. Diameter If a fluid flows in a pipe of constant diameter, but changing height, there is also work done on the fluid against the force of gravity. General Bernoulli s Equation The general case, where both height and speed may change, is described by Bernoulli s equation: Equating the work done with the change in potential energy gives: This equation is essentially a statement of conservation of energy in a (frictionless) fluid. Lecture 27 13/32 Lecture 27 14/32 Water Pipe Water is flowing continuously in the pipe shown below. The velocity of the water is greatest at B. P + (1/2)ρv 2 = constant. Where is the pressure in the water greatest? A) B) C) D) equal everywhere A B C Example: Breathing When a person inhales, air moves down the bronchus (windpipe) at 0.15 m/s. The average flow speed of the air doubles through a constriction in the bronchus. Assuming incompressible and frictionless flow, determine the pressure drop in the constriction. We apply Bernoulli s Eqn. and neglect the small change in vertical position. Lecture 27 15/32 Lecture 27 16/32

5 Applications of Bernoulli s Principle: Using Bernoulli s principle, we can find the speed of fluid coming from a spigot on an open tank: Take P 2 = P 1 = P at ; the Bernoulli Eqn becomes ½ρv 12 +ρgy 1 = ½ρv 22 +ρgy 2 v 1 2 = 2gy 2-2gy 1 How high can you suck water up using a straw? (or pump water uphill?) In order to draw water (or any fluid) upward you must lower the pressure difference between the pressure inside the straw and the outside environment (usually the atmosphere). 1. Is there a limit to how low you can make the pressure inside a straw? 2. How high can you suck water? Lecture 27 17/32 Lecture 27 18/32 Pumping Up Water P 1 +½ρv 12 +ρgy 1 = P 2 + ½ρv 22 +ρgy 2 Take v 1 = v 2 0 Take y 1 (top of water) = 0 P 1 = P at (top of water); P 2 = 0 P at = ρgy 2 y 2 = P at /(ρg) = 10 5 Pa/[(10 3 kg/m 3 )g] y 2 10m Applications of Bernoulli s Principle: Airplane Lift on an airplane wing is due to the different air speeds and thus pressures on the two surfaces of the wing. Higher speed Lecture 27 19/32 Lecture 27 20/32

6 Applications of Bernoulli s Principle: Sailboat A sailboat can move against the wind, using the pressure differences on each side of the sail, and using the keel to keep from going sideways. Applications of Bernoulli s Principle: Baseball A ball s path will curve due to its spin, which results in the air speeds on the two sides of the ball not being equal. Lecture 27 21/32 Lecture 27 22/32 Applications of Bernoulli s Principle: TIA A person with constricted arteries will find that they may experience a temporary lack of blood to the brain (TIA) as blood speeds up to get past the constriction, thereby reducing the pressure. Applications of Bernoulli s Principle A venturi meter can be used to measure fluid flow speed by measuring pressure differences. Lecture 27 23/32 Lecture 27 24/32

7 Fluid Dynamics Summary Equation of continuity for liquids: Viscosity Viscosity is a form of friction felt by fluids as they flow along surfaces. We have been dealing with nonviscous fluids, but real fluids have some viscosity. Bernoulli s Eqn. for non-viscous fluids: Lecture 27 25/32 A viscous fluid flowing in a tube will have zero velocity next to the walls and maximum velocity in the center. Lecture 27 26/32 Viscous Flow It takes a force to maintain a viscous flow, just as it takes a force to maintain motion in the presence of friction. The force causing viscous fluid to flow through a tube is provided by the pressure difference P 1 -P 2 across a length L of the tube. A fluid is characterized by its coefficient of viscosity, η, defined so that the pressure difference needed to get an average velocity v in a tube of length L and cross-sectional area A is given by: Coefficient of Viscosity η The SI units of η are N s/m 2 Also called dekapoise Common unit: poise = 0.1 N s/m 2 Some viscosity values: Fluid η (N s/m 2 ) Temp( C) Glycerine wt motor oil wt motor oil Whole blood Water Water Water Lecture 27 27/32 Lecture 27 28/32

8 Poiseuille s Equation* For a fluid of viscosity η flowing in a tube of radius r and length L with pressure difference P 1 -P 2 : End of Lecture 27 For Friday, Nov. 13, read Walker ; Homework Assignment 15b is due at 11:00 PM on Thursday, Nov. 12. Midterm Review - Friday, Nov. 13, 3:10 p.m., TH230 Note the dependence on the fourth power of the radius of the tube! Lecture 27 29/32 Lecture 27 30/32

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