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Fluids: Pressure and Fluid Dynamics

This lecture covers Chapter 15 on fluids, including incompressible fluids, pressure calculations, and fluid dynamics. Homework assignment due on Nov. 15.

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Fluids: Pressure and Fluid Dynamics

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  1. Lecture 21 Goals: • Chapter 15, fluids • Assignment • HW-8 due Tuesday, Nov 15 • Wednesday: Read through Chapter 16

  2. Fluids • Another parameter • Pressure (force per unit area) P=F/A SI unit for pressure is 1 Pascal = 1 N/m2 • The atmospheric pressure at sea-level is 1 atm = 1.013 x105 Pa = 1013 mbar = 760 Torr = 14.7 lb/ in2 (=PSI)

  3. Incompressible fluids (liquids) • What is the pressure at the bottom of the container? F=Mg=ρVg F=ρAyg Pressure=F/A=ρyg y P=ρgy Area=A

  4. What if there is outside gas? Pressure=P0 F=P0A+Mg P0A P=P0+ρgy y Area=A

  5. P1=P0+ρgy1 Pressure=P0 P2=P0+ρgy2 y1 y2 P2-P1=ρg(y2-y1) Area=A

  6. What is the pressure 10m down? P=P0+ρgy =P0+(1000 kg/m3)(10 m/s2) (10 m) =P0+105 N/m2 = approximately 2 atm • Home exercise: what is the pressure 6 miles down?

  7. Consider the open, connected container shown below. How would the two heights compare? • y1<y2 • y1=y2 • y1>y2 y1 y2

  8. Pressure vs. Depth • In a connected liquid, the pressure is the same at all points through a horizontal line. p

  9. Pressure Measurements: Barometer • Invented by Torricelli • A long closed tube is filled with mercury and inverted in a dish of mercury • The closed end is nearly a vacuum • Measures atmospheric pressure as 1 atm = 0.760 m (of Hg) P0=ρgh

  10. W2? W1 c) W1 > W2 a) W1 < W2 b) W1 = W2 Archimedes’ Principle • Suppose we weigh an object in air (1) and in water (2). How do these weights compare?

  11. Buoyancy F2=P2 Area F1=P1 Area F2-F1=(P2-P1) Area =ρg(y2-y1) Area =ρ g Vobject =weight of the fluid displaced by the object y1 F1 y2 F2

  12. Float or sink? • If we immerse the object completely in the liquid: float weight of the object < bouyant force ρobject Vobject< ρfluid Vobject float ρobject < ρfluid float • How does a steel ship float? • ρsteel < ρwater • overall density of the ship < ρwater • none of the above

  13. Float • If the object floats, then we can find the portion of the object that will be immersed in the fluid FB=mg Vimmersed ρfluid g =Vobject ρobject g FB Vimmersed ρfluid =Vobject ρobject

  14. 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. Pressure=P0 y P=P0+ρgy

  15. Pascal’s Principle in action: Hydraulics, a force amplifier • Consider the system shown: • A downward force F1 is applied to the piston of area A1. • This force is transmitted through the liquid to create an upward force F2. • Pascal’s Principle says that increased pressure from F1 (F1/A1) is transmitted throughout the liquid. P1 = P2 F1 / A1 = F2 / A2 A2 / A1 = F2 / F1

  16. Fluid dynamics • To describe fluid motion, we need something that describes flow: • Velocity v • Ideal fluid model: • Incompressible fluid. • No viscosity (no friction). • Steady flow

  17. Types of Fluid Flow

  18. Streamlines • Keep track of a small portion of the fluid:

  19. A 2 A 1 v 1 v 2 Continuity equation A1v1 : units of m2 m/s = volume/s A2v2 : units of m2 m/s = volume/s A1v1=A2v2

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