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2. Chapter 16 Properties of Fluids

3. Main Points of Chapter 16 States of matter: solid, liquid, gas Density and pressure Pressure in a fluid at rest Buoyancy and Archimedes’ principle Fluids in motion Equation of continuity Bernoulli’s equation and applications Real fluids: viscosity, turbulence

4. 16-1 States of Matter Matter has three states: solid, liquid, and gas In a gas, the molecules are far apart and the forces between them are very small

5. 16-1 States of Matter Matter has three states: solid, liquid, and gas In a solid, the molecules are very close together, and the form of the solid depends on the details of the forces between them; that form is often a lattice. Solids resist changes in shape.

6. 16-1 States of Matter Matter has three states: solid, liquid, and gas In a liquid, the molecules are also close together and resist changes in density, but not in shape.

7. 16-1 States of Matter Qualitative behavior of the force between a pair of molecules as a function of their separation, r: A substance with no resistance to shear is called a fluid (it flows). This includes gases and liquids.

8. 16-2 Density and Pressure Definition of density is mass per unit volume: (16-1) Specific gravity of a substance is the density of that substance divided by the density of water. Definition of pressure is force per unit area: (16-2)

9. 16-2 Density and Pressure SI units of pressure: pascals (16-4) Other units used: pounds per square inch (psi), atmospheres (atm), bar, and torr. The pressure in a static fluid at any point must be the same in all directions.

10. 16-3 Pressure in a Fluid at Rest Pressure increases with depth, due to the force of gravity: Here, the upper surface of the imaginary cylinder has the pressure from the rest of the fluid on it; the lower surface has that also, but in addition has the weight of the imaginary cylinder. From this, we find how the pressure varies with depth: (16-6)

11. 16-3 Pressure in a Fluid at Rest Since the pressure varies only with depth (assuming an incompressible fluid), the connected tubes at right all have fluid in them up to the same level.

12. 16-4 Buoyancy and Archimedes’ Principle Assume block below is in equilibrium. Then upward forces must equal downward forces. Upward force: pressure from fluid Downward force: atmospheric pressure plus weight Therefore, (16-8)

13. 16-4 Buoyancy and Archimedes’ Principle In this case, the object is less dense than the fluid, and it floats Here, the object has the same density as the fluid, and is in neutral equilibrium Now, the object is more dense than the fluid, and it sinks

14. 16-4 Buoyancy and Archimedes’ Principle In all cases, the upward force on the object is called the buoyancy force: (16-11) Again, we have the three cases of object density:

15. 16-4 Buoyancy and Archimedes’ Principle Archimedes’ principle: The buoyant force on an immersed object equals the weight of displaced fluid. The picture below shows an object in the air, partially submerged, and completely submerged.

16. 16-5 Fluids in Motion Assumptions for Sections 16-6 and 16-7: 1. The fluid is incompressible. 2. The temperature does not vary. 3. The flow is steady, so that the velocity and pressure do not depend on time. 4. The flow is laminar rather than turbulent. 5. The flow is irrotational, so there is no circulation. 6. There is no viscosity in the fluid. Example of fluids where there is circulation:

17. The equation of continuity says that if a flow is contained within streamlines, whatever goes in one end has to come out the other end without piling up anywhere. 16-6 The Equation of Continuity

18. For an incompressible fluid, where the density is constant, we have the conservation of flow or flux: 16-6 The Equation of Continuity (16-14) (16-15) Therefore: The definition of flux: (16-16)

19. 16-7 Bernoulli’s Equation Along a streamline: (16-22) If a fluid is incompressible and has no viscosity, energy is conserved. Therefore, the work done on a “piece” of fluid is equal to the change in its kinetic energy. This is Bernoulli’s equation.

20. 16-8 Applications of Bernoulli’s Equation Therefore, higher speed means lower pressure. This is evident when blowing across a light sheet of paper (below), and is also responsible for air flow in chimneys and in prairie dog burrows. For flow at constant height: (16-24)

21. 16-8 Applications of Bernoulli’s Equation For flow from holes in a tank, the pressure just outside the hole is the same as the pressure on the upper surface. The only variables are the speed of the flow (assumed zero at the top) and the height of the fluid. Therefore: (16-26)

22. 16-8 Applications of Bernoulli’s Equation Lift is a result of the Bernoulli effect – where the air travels faster, the pressure is lower, resulting in a net upward force on the wing: (16-28) Here, L is the lift, w the width of the wing, S the wingspan, ρ the air density, K a constant depending on the geometry of the wing, and v the air speed.

23. 16-9 Real Fluids Viscosity acts like a drag or frictional force; it slows flow speed relative to a surface. Top: an ideal fluid without viscosity; this is called a velocity profile. Bottom: the same, with viscosity.

24. 16-9 Real Fluids The viscosity of a fluid is described by its coefficient of viscosity; this varies widely among fluids, and also with temperature.

25. 16-9 Real Fluids Turbulence is a complex and chaotic movement of a fluid. Streamlines can no longer be defined. Turbulent flow is much more difficult to analyze than laminar flow.

26. Summary of Chapter 16 Liquids and gases are fluids – they deform in response to external forces, and flow. Liquids and solids both have closely-spaced molecules, but in solids they form a lattice. If a fluid is incompressible, its density is constant. Pressure is defined as force per unit area. Pressure increases with depth in a static fluid.

27. Summary of Chapter 16, cont. An upward buoyant force acts on an object immersed in a fluid: (16-11) (16-14) The equation of continuity is a statement of the conservation of mass: A fluid moving under the influence of gravity follows Bernoulli’s equation: (16-22)