Fluid Mechanics. Liquids and gases have the ability to flow They are called fluids There are a variety of LAWS that fluids obey

Transcription

1 Fluid Mechanics

2 Fluid Mechanics Liquids and gases have the ability to flow They are called fluids There are a variety of LAWS that fluids obey

3 Density Regardless of form (solid, liquid, gas) we can define how much mass is squeezed into a particular space density mass volume

4 Mass Density Wood Density mass volume ; m V 2 kg, 4000 cm 3 Lead: 11,300 kg/m 3 Wood: 500 kg/m 3 Lead 4000 cm kg Lead 177 cm 3 2 kg Same volume Same mass

5 Gases The primary difference between a liquid and a gas is the distance between the molecules In a gas, the molecules are so widely separated, that there is little interaction between the individual molecules

6 Boyle s Law

7 Boyle s Law Pressure depends on density of the gas Pressure is just the force per unit area exerted by the molecules as they collide with the walls of the container Remember: Pressure is measured in pascal units (Pa) 1Pa = 1 Newton / m 2 (force/area) At sea level, 1atm = kpa or 101,300 N per square meter Double the density, double the number of collisions with the wall and this doubles the pressure

8 Boyle s Law Density is mass divided by volume. Cut the volume in half and you double the density and thus the pressure.

9 Boyle s Law At a given temperature for a given quantity of gas, the product of the pressure and the volume is a constant P 1 V 1 P 2 V 2

10 Pressure A measure of the amount of force exerted on a surface area pressure force area

11 Pressure Pressure is the ratio of a force F to the area A over which it is applied: Force Pressure ; Area P F A A = 2 cm kg P F A 2 (1.5 kg)(9.8 m/s ) 2 x 10 m -4 2 P = 73,500 N/m 2

12 The Unit of Pressure (Pascal): A pressure of one pascal (1 Pa) is defined as a force of one newton (1 N) applied to an area of one square meter (1 m 2 ). Pascal: 2 1 Pa = 1 N/m In the previous example the pressure was 73,500 N/m 2. This should be expressed as: P = 73,500 Pa

13 Pressure / Density Example Schmedrick uses his 6 lb tofu recipe book to teach his little brother Poindexter about density and pressure. He sets the book on the table and calculates the pressure on the table, which depends on the book s orientation. The book s density is 6 lb / ( ) = lb / in 3. P = 6 lb / (9 3 ) = lb / in 2 P = 6 lb / (3 14 ) = lb / in 2 P = 6 lb / (9 14 ) = lb / in 2 9 Tofu Cookbook 14 3 Tofu Cookbook

14 Pressure in a Fluid The pressure is just the weight of all the fluid around the object Atmospheric pressure is just the weight of all the air above on an area on the surface of the earth In a swimming pool the pressure on your body surface is just the weight of the water above you (plus the air pressure above the water)

15 Fluid Pressure Fluid exerts forces in many directions. Try to submerse a rubber ball in water to see that an upward force acts on the float. Fluids exert pressure in all directions. F

16 Pressure in a Fluid So, the only thing that counts in fluid pressure is the gravitational force acting on the mass ABOVE you The deeper you go, the more weight above you and the more pressure Go to a mountaintop and the air pressure is lower Pressure in a fluid is the result of the forces exerted by molecules as they bounce off each other in all directions. Therefore, at a given depth in a liquid or gas, the pressure is the same and acts in every direction

17 Pressure in a Fluid Pressure acts perpendicular to the surface and increases at greater depth.

18 Pressure vs. Depth in Fluid Pressure = force/area mg P ; m V ; V Ah A Vg Ahg P A A Area mg h Pressure at any point in a fluid is directly proportional to the density of the fluid and to the depth in the fluid. Fluid Pressure: P = gh

19 Independence of Shape and Area. Water seeks its own level, indicating that fluid pressure is independent of area and shape of its container. At any depth h below the surface of the water in any column, the pressure P is the same. The shape and area are not factors.

20 *Properties of Fluid Pressure* The forces exerted by a fluid on the walls of its container are always perpendicular. The fluid pressure is directly proportional to the depth of the fluid and to its density. At any particular depth, the fluid pressure is the same in all directions. Fluid pressure is independent of the shape or area of its container.

21 Pressure in a Fluid

22 Barometers The height of the mercury column in a barometer directly measures air pressure. The weight of the column of mercury is balanced by the force exerted at the bottom due to the air pressure. Normal air pressure is 760mm or 760 torr Since mercury is 13.6 times heavier than water, a water barometer would have to be 13.6 times longer.

23 Pascal s Principle Pressure applied to a fluid is transmitted throughout the fluid. Ex) squeezing tube of toothpaste Hydraulic machines work using Pascal s principle.

24 Pascal s Law Pascal s Law: An external pressure applied to an enclosed fluid is transmitted uniformly throughout the volume of the liquid. F in A in F out A out Pressure in = Pressure out F A in in F A out out

25 Hydraulic Press A force F 1 is applied to a hydraulic press. This increases the pressure throughout the oil, lifting the car--pascal s principle. This would not work with air, since air is compressible. The pressure is the same throughout the oil. The volume of oil pushed down on the left is the same as the increase on the right. The distance pushed on the left is the trade off. h 2 h 1 F 1 A 2 F 2 A 1 oil

26 Example 3. The smaller and larger pistons of a hydraulic press have diameters of 4 cm and 12 cm. What input force is required to lift a 4000 N weight with the output piston? F F F A ; Fin A A A F in in out out in in out out D R ; Area R 2 (4000 N)( )(2 cm) 2 (6 cm) 2 2 F = 444 N F in A in F out A out R in = 2 cm; R out = 6 cm

27 Floating in Fluids We all know that dense objects sink in fluids of lower density. A rock sinks in air or water, and oil floats on top of water. Basements stay cool in the summer because cool air is denser than warm air. The USS Eisenhower is a ton nuclear powered aircraft carrier made of dense materials like steel, yet it floats. If you weigh yourself under water, the scale would say you are lighter than your true weight. All of these facts can be explained thanks one of the greatest scientists of all time--the Greek scientist, mathematician, and engineer--archimedes. USS Eisenhower Archimedes

28 Archimedes Principle An object that is completely or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. 2 lb 2 lb The buoyant force is due to the displaced fluid. The block material doesn t matter. If the buoyant force on an object is greater than the force of gravity acting on the object, the object will float. The apparent weight of an object in a liquid is gravitational force (weight) minus the buoyant force

29 Buoyancy Net upward force is called the buoyant force!!!

30 Displacement of Water The amount of water displaced is equal to the volume of the rock.

31 Flotation

32 Flotation A floating object displaces a weight of fluid equal to its own weight. An object floats if its density is less than the density of the fluid it is placed in.

33 Submarines & Blimps A sub is submerged in water, while a blimp is submerged in air. In each a buoyant force must balance the weight of the vessel. Blimps and hot air balloons must displace huge amounts of air because air isn t very dense. The weight of the air a blimp displaces is equal to the blimp s weight. Likewise, the weight of the water a sub displaces is equal to the sub s weight.

34 Buoyancy in a Gas An object surrounded by air is buoyed up by a force equal to the weight of the air displace. Exactly the same concept as buoyancy in water. Just substitute air for water in the statement If the buoyant force is greater than the weight of the object, it will rise in the air

35 Buoyancy in a Gas Since air gets less dense with altitude, the buoyant force decreases with altitude. So helium balloons don t rise forever!!!

36 Atmospheric Pressure Just the weight of the air above you Unlike water, the density of the air decreases with altitude since air is compressible and liquids are only very slightly compressible Air pressure at sea level is about 10 5 newtons/meter 2

37 Bernoulli s Principle When the speed of a fluid increases, the pressure exerted by the fluid decreases.

38 Bernoulli s Principle Uses: airplanes, hose-end sprayers Energy conservation requires that the pressure be lower in a fluid that is moving faster

39 Bernoulli s Principle

40 Air is not incompressible, but the Bernoulli principle can explain, in part, why an airplane flies. The upper surface of the wing has a smaller radius of curvature than the bottom surface. Air on top must travel farther, so it moves faster, and the pressure there is lower, creating lift. Also, because of the wing s upward tilt, air is pushed downward. So, the air pushes back on the wing in the direction of F.

41 Viscosity The resistance to flow by a fluid When a container of liquid is tilted to allow flow, the flowing particles will transfer energy to the particles that are stationary. Increasing temperature of a fluid will decrease viscosity

42 Viscosity Different kinds of fluids flow more easily than others. Oil, for example, flows more easily than molasses. This is because molasses has a higher viscosity, which is a measure of resistance to fluid flow. Inside a pipe or tube a very thin layer of fluid right near the walls of the tube are motionless because they get caught up in the microscopic ridges of the tube, or microwelds. Layers closer to the center move faster and the fluid sheers. The middle layer moves the fastest. v = 0 The more viscous a fluid is, the more the layers want to cling together, and the more it resists this shearing. The resistance is due the frictional forces between the layers as the slides past one another. Note, there is no friction occurring at the tube s surface since the fluid there is essentially still. The friction happens in the fluid and generates heat. The Bernoulli equation applies to fluids with negligible viscosity.