1. Subdivisions of matter solidsliquidsgases rigidwill flowwill flow dense dense low density and incompressible and incompressible compressible fluids condensed matter Q: what about thick liquids and soft solids? AP Physics B: Ch.11 - Fluid Mechanics.

2. Fluid mechanics Ordinary mechanics Mass and force identified with objects Fluid mechanics Mass and force “distributed”

3. Density and Pressure Density  for element of fluid mass  M volume  V for uniform density mass M volume V units kg m -3

4. Density and Pressure Pressure p force per unit area for uniform force units N m -2 or pascals (Pa) Atmospheric pressure at sea level p 0 on average 101.3 x10 3 Pa or 101.3 kPa Gauge pressure p g excess pressure above atmospheric p = p g + p 0

5. Density and Pressure Gauge pressure p g p = p g + p 0 pressure in excess of atmospheric typical pressurestotalgauge atmospheric1.0x10 5 Pa0 car tire 3.5x10 5 Pa 2.5x10 5 Pa deepest ocean1.1x10 8 Pa 1.1x10 8 Pa best vacuum10 -12 Pa - 100 kPa atmospheric gauge total

6. Example to pump 30 cms 15 cms The can shown has atmospheric pressure outside. The pump reduces the pressure inside to 1/4 atmospheric What is the gauge pressure inside? What is the net force on one side?

7. Fluids at rest (hydrostatics) Hydrostatic equilibrium laws of mechanical equilibrium pressure just above surface is atmospheric, p 0 hence, pressure just below surface must be same, i.e. p 0 surface is in equilibrium

8. Fluids at rest (hydrostatics) Hydrostatic equilibrium laws of mechanical equilibrium (p+  p)A pA yy element of fluid surface area A height  y pA - (p+  p)A - mg = 0  F y =0  p A -  A  yg = 0 mg =  A  yg  p =-  g  y p = p 0 +  gh at distance h below surface, pressure is larger by  gh Pressure at depth h

9. Question How far below surface of water must one dive for the pressure to increase by one atmosphere? What is total pressure and what is the gauge pressure, at this depth? ?

10. Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point

11. Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Open tube manometer (i)If h=6 cm and the fluid is mercury (  =13600 kg m -3 ) find the gauge pressure in the tank (ii) Find the absolute pressure if p 0 =101.3 kPa

12. Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Barometer Find p 0 if h=758 mm

13. Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point A change in the pressure applied to an enclosed incompressible fluid is transmitted to every point in the fluid Hydraulic press alternative argument based on conservation of energy work out = work in volume moved is same on each side

14. Archimedes’s principle When a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is upward and has a magnitude equal to the weight of fluid displaced. imagine a hole in the water-a buoyancy force exists fill it with fluid of mass m f and equilibrium will exist F b =m f g stone more dense than water so sinks wood less dense than water so floats now the water displaced is less -just enough buoyancy force to balance the weight of the wood F b =F g FbFb FgFg FgFg

15. Example 1 What fraction of an iceberg is submerged? (  ice for sea ice =917 kg m -3 and  sea for sea water = 1024 kg m -3 ) FbFb FgFg Fb=FgFb=Fg  fluid V i g=  V g V i /V =  /  fluid volume immersed V i total volume V Example 2 A “gold” statue weighs 147 N in vacuum and 139 N when immersed in salt water of density 1024 kg m -3. What is the density of the “gold”? Flotation For object of uniform density 

16. Fluid Dynamics The study of fluids in motion. Ideal Fluid 1. Steady flow Velocity of the fluid at any point fixed in space doesn’t change with time. This is called “ laminar flow”, and for such flow the fluid follows “streamlines”. 2. Incompressible We will assume the density is fixed. Accurate for liquids but not so likely for gases. 3. Inviscid “Viscosity” is the frictional resistance to flow. Honey has high viscosity, water has small viscosity. We will assume no viscous losses. Our approach will only be true for low viscosity fluids. laminar turbulent

17. Equation of continuity Streamlines Conservation of mass in tube of flow means mass of fluid entering A 1 in time  t = mass of fluid leaving A 2 in time  t For incompressible fluid this means volume is also conserved. Volume entering and leaving in time  t is  V  V = A 1 v 1  t =A 2 v 2  t Therefore A 1 v 1 = A 2 v 2 Equation of continuity (Streamline rule) tube of flow

18. Bernoulli’s equation (Daniel Bernoulli, 1700-1782) For special case of fluid at rest (Hydrostatics!) For special case of height constant (y 1 =y 2 ) The pressure of a fluid decreases with increasing speed

19. Proof of Bernoulli’s equation Use work energy theorem work done by external force (pressure) =change in KE + change in PE W=  K+  U Work done Change in KE Change in PE Note: same volume  V with mass  m enters A 1 as leaves A 2 in time  t Work done at A 1 in time  t (p 1 A 1 )v 1  t =p 1  V

20. Problem Titanic had a displacement of 43 000 tonnes. It sank in 2.5 hours after being holed 2 m below the waterline. Calculate the total area of the hole which sank Titanic.

21. Examples of Bernoulli’s relation at work Venturi meter Aircraft lift

22. Examples of Bernouilli’s relation at work “spin bowling”