#### Transcript Fluid Mechanics (97-03)

A continuous, amorphous substance whose molecules move freely past one another and that has the tendency to assume the shape of its container; a liquid or gas ~The American Heritage® Dictionary of the English Language, Fourth Edition. 1. 2. Fluid Statics – study of fluids at rest Fluid Dynamics – study of fluids in motion Mass per unit Volume Symbols: ρ and occasionally D Units: kg/m3, g/cm3, etc. Object Air Density (kg/m3) 1.2 Osmium metal* 22,500 White-dwarf Stars ~ 1010 Neutron Stars ~ 1018 Density relative to water. For example: Force per unit Area The surface area of this room is approximately 1200 ft2 (111.5 m2) and the height of the ceilings are ?. If this room was completely filled with only air (ρ = 1.2 kg/m3) how much mass would be in the room as a result of the air? If the air pressure in the room is 1 atm (1.013 x 105 Pa) what would be the total force on the floor from the weight of the air? Fluids have mass that cannot be ignored so pressure increases with depth. Po ρ This equation only applies to situations when the density of the fluid stays relatively constant. It only applies to gases (which are compressible) over short vertical distances. h P The world record for the “No-limit” free dive is 214 m set by Herbert Nitsch in June 2007. As a result Herbie is known as the “Deepest Man on Earth.” If the density of ocean water is ρ = 1025 kg/m3 what is the pressure that Herbie experienced at the bottom of his dive? [Assume the pressure at the surface of the water is 1 atm = 101,300 Pa] A tank open to the atmosphere (with atmospheric pressure p) is filled to a height L with a liquid of density ρ as shown in the diagram. A block of density D (D < ρ) and dimensions x, y, and z is attached to the bottom of the tank by a string so that its top surface is a distance h from the surface of the liquid. What is the force due to pressure on the a) top of the block? b) bottom of the block? c) front of the block? A mercury barometer consists of a long glass tube, closed at one end, that has been filled with mercury (ρ = 13.6 x 103 kg/m3) and then inverted in a dish of mercury. The space above the mercury column is almost a perfect vacuum. Compute the atmospheric pressure on a day when the height of mercury in a barometer is 76.0 cm. [A barometer is a device used to measure air pressure.] h=0 Po = 0 h mercury P = Patm Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. The pressure depends only on depth; the shape of the container does not matter. Increasing Po in the cylinder will increase the pressure throughout the cylinder. Po ρ h P Suppose a hydraulic lift has a small cylindrical piston with radius 5 cm and a larger piston with radius 20 cm. The mass of a car placed on the larger piston is 1000 kg. a) If the two pistons are at the same height, what force must be applied to the small piston to keep the car in the air? b) How far must the small piston move down to lift the car through a height of 0.10 m? If the pressure in a car tire is atmospheric pressure, the tire is flat. The pressure has to be greater than atmospheric pressure to support the car. When you check the pressure and your gauge reads 32 psi (220 kPa) that means the tire pressure is 32 psi greater than atmospheric, or really 47 lb/in2 (psi). [Remember atmospheric pressure is ~15 psi] A solar water-heating system uses solar panels on the roof, 12 m above the storage tank. The pressure at the level of the panels is 1 atm. What is the absolute pressure at the top of the tank? [1 atm = 1.01 x 105 Pa] What is the gauge pressure at the top of the tank? Follow-up question: At what distance below the panels is the gauge pressure equal to 1 atm? When an object is completely or partially immersed in a fluid, the fluid exerts an upward force on the object equal to the weight of the fluid that is displaced by the object. This upward force is called the buoyant force. Fbuoy x mg A tank open to the atmosphere (with atmospheric pressure p) is filled to a height L with a liquid of density ρ as shown in the diagram. A block of density D (D < ρ) and dimensions x, y, and z is attached to the bottom of the tank by a string so that its top surface is a distance h from the surface of the liquid. a) What is the total force due to pressure on the block? b) What is the tension in the string? c) If the string is cut, how long would it take for the top of the block to reach the surface? A hollow plastic sphere is held below the surface of a freshwater lake ( = 1000 kg/m3) by a cord anchored to the bottom of the bottom of the lake. The sphere has a volume of 0.650 m3 and the tension in the cord is 900 N. a) b) c) Calculate the buoyant force exerted by the water on the sphere. What is the mass of the sphere? The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged? Density is mass per unit volume Pressure is force per unit area Pressure varies with depth Archimedes’ Principe: fluid exerts an upward force on the object equal to the weight of the fluid that is displaced by the object 2 Kinds of Fluid Flow 1. Laminar Flow – every particle that passes a particular point moves exactly along the smooth path followed by the particles that passed that point earlier (stream line) 2. Turbulent Flow – irregular fluid flow (eddy currents) Laminar Flow Video The fluid is nonviscous…viscous means to have relatively high resistance to flow 2. The fluid is incompressible…its density stays constant 3. The fluid motion is in a steady-state meaning that the velocity, density, and pressure at each point in the fluid do not change in time 4. The fluid moves without turbulence…laminar flow 1. In steady state, the rate at which fluid mass moves through a tube with a single entry point and a single exit point remains the same, even if the cross section of the tube varies. A1 v1 A2 v2 Δx2 Δx1 The figure below shows a portion of a pipe for oil with rectangular cross sections. If the flow speed at the bottom is v, what is the flow speed at the top? When a horizontal pipe is constricted, fluid speeds up. This means that there must be some acceleration and therefore some net force. anet vwide Fwide Fwide = PwideA Fnarrow vnarrow > vwide Fnarrow = PnarrowA In a static fluid pressure was dependent on depth. The same holds true for a fluid that is in motion as long as the cross sectional area remains constant. Lower Pressure h Higher Pressure Combining the effects of pressure changes and gravity into the work energy theorem results in a relationship known as Bernoulli’s equation. In the steady-state flow of an ideal fluid of density ρ, the following equation is true along a streamline: When using the equation for two points on the same streamline the equation is: Notice the similarities to conservation of energy. It looks like there are K and U terms in the equation. The figure below shows a pipe with a Venturi U-tube attached to it. Fluid of density ρ1 flows through the pipe from point X to point Y. The cross sectional area of the pipe at point X is A1, and the cross sectional area at Y is A2. The pressure at X is P, and the velocity of the fluid there is v. The U-tube is filled with a fluid of density ρ2. Express your answers in terms of all of the above values and necessary constants. a) Determine the velocity of the fluid at point Y. b) Determine the pressure at point Y. c) Determine the pressure at point W. d) Find h. The lift force that allows an airplane to take flight can be explained by Bernoulli’s equation…or can it? “When a solid body is placed in a fluid flow and a nonsymmetrical situation occurs the direction of the force on the body does not coincide with the direction of the (undisturbed) flow. This principle makes flying possible.” High Pressure v F Low Pressure The curve ball can be explained by Bernoulli’s equation. FoilSimb\Ball.html Water enters a house through a pipe with an inside diameter of 2.0 cm at a gauge pressure of 4.0 x 105 Pa (about 4 atm, or 60 lb/in2). The cold-water pipe leading to the second-floor bathroom 5.0 m above is 1.0 cm in diameter. a) Find the flow speed and gauge pressure in the bathroom when the flow speed at the inlet pipe is 2.0 m/s. b) How much time is required to fill a 100 L bathtub? A cylindrical tank of radius R is filled to a depth L with a fluid of density ρ. A hole of radius r (less than R) is punctured in the side of the tank a distance h from the top. Express your answers in terms of R, r, L, h, ρ, and g. a) What is the ratio of the rate at which the level of the fluid in the tank is decreasing to the speed of the fluid emerging from the hole? b) What is the speed of the fluid emerging from the hole? c) Determine the distance between the base of the cylinder and the point where the fluid strikes the floor. d) Another hole is punctured at a distance h2 from the surface, where h2 is not equal to h. Determine h2 such that the water coming from there lands at the same point as the fluid from the first hole.