15. Fluid Motion

1. 15. Fluid Motion Density & Pressure Hydrostatic Equilibrium Archimedes’ Principle & Buoyancy Fluid Dynamics Applications of Fluid Dynamics Viscosity & Turbulence

2. Why is only the “tip of the iceberg ” above water? iceis only slightly less than water.

3. Fluid = matter that flows under external forces = liquid & gas. • Examples of fluid motion: • Tornadoes. • Airflight: plane supported by pressure on wings. • Gas from giant star being sucked into a black hole. • Brake fluid in a car’s braking system. • Breathing: air into lung & blood stream.

4. 15.1. Density & Pressure thousands of molecules Avogadro’s number NA = 6.022  1023 / mol . 1 mole = amount of substance containing NA basic elements. ( with NA = number of atoms in 12 g of 12C ). Fluid: average position of molecules not fixed. Macroscopic viewpoint: deformable continuum. dV fluid point dV  0 Density  = mass / vol, [  ] = kg / m3 . Incompressible = density unchanged under pressure Liquid is nearly incompressible (molecules in contact). Gas is compressible.

5. Pressure Pressure = normal force per unit area Pressure is a scalar, equally exerted in all directions

6. 15.2. Hydrostatic Equilibrium Hydrostatic equilibrium : Fnet = 0 everywhere in fluid Fluid is at rest. Fext 0 gives rise to pressure differences. P is a scalar

7. Hydrostatic Equilibrium with Gravity Fluid element: area A, thickness dh, mass dm. Net pressure force on fluid element: Gravitational force on fluid element: Hydrostatic Equilibrium :  Liquid (~incompressible):

8. Example 15.1. Ocean Depths • At what water depth is the pressure twice the atmospheric pressure? • What’s the pressure at the bottom of the 11-km-deep Marianas Trench, the deepest point in the ocean? • Take 1 atm = 100 kPa & water = 1000 kg/m3 . (a)  (b) Pressure increases by 1 atm per 10 m depth increment. 

9. Measuring Pressure Barometer = device for measuring atmospheric pressure vacuum inside tube: For p = 1 atm = 101.3 kPa : Cf. h = 10 m for a water barometer

10. Manometer Manometer = U-shaped tube filled with liquid to measure pressure differences. equal p Gauge pressure = excess pressure above atmospheric. Used in tires, sport equipments, etc. E.g., tire gauge pressure = 30 psi  tire pressure = 44.7 psi Pascal’s law: An external pressure applied to a fluid in a closed vessel is uniformly transmitted throughout the fluid.

11. Example 15.2. Hydraulic Lift In a hydraulic lift, a large piston supports a car. The total mass of car & piston is 3200 kg. What force must be applied to the smaller piston to support the car? Pascal’s law 

12. 15.3. Archimedes’ Principle & Buoyancy fluid element in equilibrium Buoyancy force: Upward force felt by an object in a fluid Archimedes’ Principle: The buoyancy force on an object is equal to the weight of the fluid it displaces. Neutral buoyancy : average density of object is the same as that of fluid. Fb unchanged after replacement

13. Example 15.3. Working under Water To set up a raft, you need to move a 60-kg block of concrete on the lake bottom. What’s the apparent weight of the block as you lift it underwater? The density of concrete is 2200 kg / m3 . ~ ½ weight on land

14. Example 15.4. Tip of the Iceberg Average density of a typical iceberg is 0.86 that of seawater. What fraction of an iceberg’s volume is submerged?

15. GOT IT? 15.1. Arctic sea ice is melting as a result of global warming. Does this process contribute to a rise in sea level? Explain. No. Volume of the melted ice (which becomes water) is the same as that displaced by the floating ice.

16. Center of Buoyancy Buoyancy force acts at the center of buoyancy (CB), which coincides with the CM of the displaced water. CM must be lower than CB to be stable.

17. 15.4. Fluid Dynamics Moving fluid is described by its flow velocity v( r, t ). Streamlines = Lines with tangents everywhere parallel to v( r, t ). Spacing of streamlines is inversely proportional to the flow speed. slow fast Small particles (e.g., dyes) in fluid move along streamlines. Steady flow: e.g., calm river. Example of unsteady flow: blood in arteries ( pumped by heart ). Fluid dynamics: Newton’s law + diffusing viscosity  Navier-Stokes equations

18. GOT IT? 15.2. • Photo shows smoke particles tracing streamlines in a test of a car’s aerodynamic properties. • Is the flow speed greater • over the top, or • at the back?

19. Conservation of Mass: The Continuity Equation Steady flow Flow tube: small region with sides tangent, & end faces perpendicular, to streamlines.  flow tubes do not cross streamlines. Mass entering tube: Mass leaving tube: Conservation of mass: Equation of continuityfor steady flow: Mass flow rate = [  v A ] = kg / s Liquid: Volume flow rate = [ v A ] = m3 / s Liquid & gas with v < vs : flows faster in constricted area. Gas with v > vs : flows slower in constricted area.

20. Steady flow Mass entering tube: Mass leaving tube: Conservation of mass: Equation of continuityfor steady flow: Mass flow rate = [  v A ] = kg / s Liquid: Volume flow rate = [ v A ] = m3 / s Liquid : flows faster in constricted area. Gas with v < vs ound: flows faster in constricted area. Gas with v > vsound : flows slower in constricted area.

21. Example 15.5. Ausable Chasm The Ausable river in NY is about 40 m wide. In summer, it’s usually 2.2 m deep & flows at 4.5 m/s. Just before it reaches Lake Champlain, it enters Ausable Chasm, a deep gorge only 3.7 m wide. If the flow rate in the gorge is 6.0 m/s, how deep is the river at this point?? Assume a rectangular cross section with uniform flow speed. 

22. Conservation of Energy: Bernoulli’s Equation Same fluid element enters & leaves tube: Work done by pressure upon its entering tube: Work done by pressure upon its leaving tube: Work done by gravity during the trip: W-E theorem:  Incompressible fluid: Bernoulli’s Equation  Viscosity & other works neglected

23. 15.5. Applications of Fluid Dynamics • Strategy • Identify a flow tube. • Draw a sketch of the situation, showing the flow tube. • Determine two points on your sketch. • Apply the continuity equation and Bernoulli’s equation.

24. Example 15.6. Draining a Tank A large open tank is filled to height h with liquid of density . Find the speed of liquid emerging from a small hole at the base of the tank. At top surface : At hole: 

25. Venturi Flows Venturi : a tube with constricted central region C. Eq. of continuity  v larger in C. Bernoulli eq.  p lower in C.  Venturi flow meter measures flow speed by measuring pressure drop in C.

26. Example 15.7. Venturi Flowmeter Find the flow speed in the unconstricted pipe of a Venturi flowmeter. Bernoulli’s eq. Continuity eq.  

27. Bernoulli Effect Bernoulli Effect: p   v  • Example: Prairie dog’s hole • Dirt mound forces wind to accelerate over hole • low pressure above hole • natural ventilation A ping-pong ball supported by downward-flowing air. High-velocity flow is inside the narrow part of the funnel.

28. GOT IT? 15.3. A large tank is filled with liquid to level h1 . It drains through a small pipe whose diameter varies. Emerging from each section of pipe are vertical tubes open to the atmosphere. Although the picture shows the same liquid level in each pipe, they really won’t be the same. Rank order the levels h1 through h4. ordered inversely by flow speed

29. Flight & Lift Air pushes up (3rd law) Aerodynamic lift Air pushes  (3rd law) Blade pushes down on air Wing deflects air  Top view on a straight ball : no spin Top view on a curved ball : spin

30. Application: Wind Energy A chunk of air, of speed v & density , passing thru a turbine of area A in time t, has kinetic energy  available power per unit area = Better analysis  For Present tech gives 80% of this.

31. 15.6. Viscosity & Turbulence flow with no viscosity Viscosity: friction due to momentum transfer between adjacent fluid layers or between fluid & wall. • B.C.: v = 0 at wall • drag on moving object. • provide 3rd law force on propellers. • stabilize flow. flow with viscosity Smooth flow becomes turbulent.