Fluids mechanics Lecture 12

1. Fluids mechanics Lecture 12

2. Content • Density. Pressure. • Pascal’s principle. • Archimedes` principle. • The equation of continuity. • Bernoulli`s theorem.

3. A fluid is defined as any substance that can flow, and hence liquids and gases are both considered to be fluids Microscopic approach deals with molecules Macroscopic approach analyze the fluid in terms of its large-scale characteristics such as its density, pressure, its distribution in space Fluids mechanics: fluid statics or hydrostatics and fluid dynamics or hydrodynamics

4. Density The density of a substance is defined as a amount of mass in a unit volume of that substance In SI

5. Pressure Pressure is defined as the magnitude of the normal force acting per unit surface area. It is a scalar quantity. in SI (French mathematician, physicist Blaise Pascal (1623-1662) in BES The conversion factor

6. The pressure exerted by a fluid The hydrostatic equation The pressure of fluid at any depth h is given by the product of the density of the fluid, the acceleration due to gravity g, and the depth h In general, the total or atmospheric pressure observed at the depth h in the pool is the sum of the atmospheric pressure plus the pressure of the water itself, that is

7. Atmospheric pressure =1013mb A mercury barometer (Evanglista Torricelli (1608-1647) Aneroid barometer, altimeter

8. Moreover, the pressure isn’t affected by the shape of the vessel, as shown in This photograph illustrates the fact that the pressure in a liquid is the same at all points lying at the same elevation. For example, the pressure is the same at points A, B, C, and D. Note that the shape of the vessel does not affect the pressure.

9. Why you get tired by the end of the day. The top of your Professor’s head is approximately circular with a radius of 3.50 inches. What force is exerted on the top of the Professor’s head by normal atmospheric pressure.

10. The area of the top of the student’s head is found from A = πr2 =π (3.50 in)2 = 38.5 in.2 We find the magnitude of the force exerted on the top of the Professor’s head by using the following Equation F = pA Hence, F = (14.7 lb/in.2) (38.5 in.2) = 566 Lb

11. This is a rather large force to have exerted on our heads all day long.

12. However, we don’t notice this enormous force because when we breathe air into our nose or mouth that air is exerting the same force upward inside our head. Thus, the difference in force between the top of the head and the inside of the head is zero.

13. After an exciting but exhausting lecture, a physics professor stretches out for a nap on a bed of nails, as in Figure, suffering no injury and only moderate discomfort. How is this possible?

14. Explanation If you try to support your entire weighton a single nail, the pressure on your body is yourweight divided by the very small area of the end of the nail. The resulting pressure is large enough to penetrate the skin. If you distribute your weight over several hundred nails, however, as demonstrated by the professor, the pressure is considerably reduced because the area that supports your weight is the total area of all nails in contact with your body. (Why is lying on a bed of nails more comfortable than standing on a bed of nails without shoes? )

15. Absolute Pressure and Gauge Pressure If the pressure inside a car tire is equal to atmospheric pressure, the tire is flat. The pressure has to be greater than atmospheric to support the car, so the significant quantity is the difference between the inside and outside pressures. When we say that the pressure in a car tire is "32 pounds" (actually 32Ib/in2, equal to 220 kPa or 2.2 X 105 Pa), we mean that it is greater than atmospheric pressure (l4.7Ib/in2 or 1.01 X 105 Pa) by this amount. The total pressure in the tire is then 47 lb/in2 or 320 kPa. The excess pressure above atmospheric pressure is usually called gauge pressure, and the total pressure is called absolute pressure. Engineers use the abbreviations psig and psia for "pounds per square inch gauge“ and "pounds per square inch absolute," respectively. If the pressure is less than atmospheric, as in a partial vacuum, the gauge pressure is negative. Example A storage tank 12.0 m deep is filled with water. The top of the tank is open to the air. What is the absolute pressure at the bottom of the tank? The gauge pressure?

16. the absolute pressure is The gauge pressure is

18. Pascal`s principle Pascal`s principle states that if the pressure at any point in an enclosed fluid at rest is changed (p), the pressure changes by an equal amount (p), at all points in the fluid The hydraulic lift is a device that is capable of multiplying forces The work done From the law of conservation of energy Since A is much greater than a that y1 must be greater than y2

19. The rear-wheel hydraulic brake system of a front-wheel-drive automobile(Fig.) is an application of Pascals principle. When the driver pushes the brake pedal, the pressure on the piston in the master cylinder is transmitted through the brake fluid to the two pistons in the brake cylinder. This transmitted pressure then forces the brake-cylinder pistons to push the brake shoes against the brake drum and stop the automobile. Releasing the brake pedal releases the pressure on the pistons in the brake cylinder. The spring pulls the brake shoes away from the brake drum, which allows the wheels to turn freely again.

20. Archimedes`s principle The upward force is called the buoyant force and is a consequence of pressure increasing with depth

21. Archimedes`s principle A completely submerged object always displaces a volume of liquid equal to its own volume

22. Archimedes`s principle The relationship between buoyancy up and displaced liquid was first discovered in the third century BC by the Greek philosopher Archimedes. Archimedes` principle: an immersed object is buoyed up a force equal to the weight of the fluid it displaces Principle of flotation When the buoyant force on the body is equal to the weight of the body, the body does not sink in the water but rather floats

23. A block of oak wood 5 cm high, 5 cm wide, and 10 cm long is placed into a tub of water, as shown in figure. The density of the wood is 7.20*102 kg/m3 How far will the block of wood sink before it floats(3.59 cm)?

24. Iron sinks. Repeat example for a block of iron of the same dimensions. Density of iron is 7860 kg/m3 (iron block sinks) But ships are made of iron and they do not sink. Why should the block sink and not the ship? If this same weight of iron is made into thin slabs, these thin slabs could be welded together into a boat structure of some kind. By increasing the size and hence the volume of this iron boat, a greater volume of water can be displaced. An increase in the volume of water displaced increases the buoyant force. If this can be made equal to the weight of the iron boat, then the boat floats.

28. Fluids in motion is the subject matter of hydrodynamics Assumptions: 1) the fluid is incompressible, 2) it flows freely without any turbulence. A fluid, flowing steadily without turbulence, is referred to as being in streamline flow 3) it can be neglected the friction between the various parts of the fluid itself and boundary containing the fluid, such as the walls of a pipe. Such fluid is called a non-viscous fluid. The conservation of mass The equation of continuity Bernoulli`s theorem The conservation of energy

29. The equation of continuity

30. The equation of continuity The law of conservation of mass states that mass is neither created nor destroyed in any ordinary mechanical or chemical process Mass flowing into the pipe=mass flowing out of the pipe The equation of continuity for liquids says that when the cross-sectional area of a pipe gets smaller, the velocity of the fluid must become greater in order that the same amount of mass passes a given point in a given time. Conversely, when the cross-sectional area increases, the velocity of the fluid must decrease. equation of continuity Av=constant

31. Bernoulli`s theorem The work done on the system The work done by the system

32. Bernoulli`s theorem The fluid is incompressible By the law of conservation of energy the net work done on the system produces a change in the energy of the system Bernoulli`s theorem says that the sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any one location of the fluid is equal to the sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume at any other location in the fluid, for a nonviscous, incompressible fluid in streamlined flow

33. Example In figure, the pressure p1 = 2.94* 103 N/m2 whereas the velocity of the water is v1 = 0.322 m/s. The diameter of the pipe at location 1 is 10 cm and it is 5 m above the ground. If the diameter of the pipe at location 2 is 4 cm, and the pipe is 2 m above the ground, find the velocity of the water v2 at position 2 and the pressure p2 of the water at position 2( v2 = 2.01 m/s; 3.04*104 N/m2).

34. Bernoulli`s theorem (a special case) The effect of the decrease in pressure with the increase in speed of the fluid in a horizontal pipe is called the Venturi effect

35. The pressures in pipe 1 and in pipe 2 are Where h01 and h02 are the heights shown in figure Replacing v2 by its value from the continuity equation we get

36. Solving for v12 we have Solving for v1 we get Example Venturi meter reads heights of h01 = 30 cm h02 = 10 cm. Find the velocity of flow v1 in the pipe. The area A1 = 7.85* 10-3 m2 and the area of A1 = 1.26* 10-3 m2.

37. Application of Bernoulli`s theorem The curving baseball By the Venturi principle the pressure of the fluid is smaller where the velocity is greater

38. Application of Bernoulli`s theorem Lift on an airplane wing By the Venturi principle if the velocity is greater at the top of the wing, the pressure must be less there at the bottom of the wing There is a net positive force F2-F1acting upward on the wing, producing lift on the airplane wing