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Transcript
MATERIALS 1.0 FLUID The physics of fluids, or commonly known as fluid mechanics is the basis of hydraulic engineering, a denomination of engineering that is applied in various fields. A medical physicist or engineer might study the fluid flow in the arteries of an aging patient. A nuclear physicist or engineer might be highly interested in the investigations of the fluid flow in the hydraulic system of an aging nuclear reactor, while an environmental engineer might be constructing a logistical plan to create efficient irrigation of agricultural areas and resolve the drainage problems from waste sites. An aeronautical engineer is meticulously working on improving and designing the hydraulic systems controlling the wing flaps that allow a jet airplane to land. As high as the sky as in the previous example, we now also have to look into the deep sea where we peek into the role of naval engineer in supervising deep sea diving activities or escape from malfunctioned submarine, which potentially pose great dangers to her crew members. Not surprisingly, the knowledge and applications of fluid mechanics are even implemented in many Las Vegas and Broadway performance, where huge sets are quickly put up and brought down by hydraulic systems. Before we can even venture into the realms of the physics of fluid , we need to face an issue that confronts us all, “What exactly is fluid?” 1.1 WHAT IS A FLUID? Fluid is a term to describe a substance, such as liquid or gas, whose molecules flow freely so that it has no fixed shape and little withstanding resistance to shearing stress regardless of how small the applied stress is. In sharp contrast to a solid., fluid readily yields or conforms to the boundaries of any container in which we place them. Ever wonder why we collectively call liquids and gases as fluid? After all, you might argue, isn’t liquid as distinct from gas as it is from solid? In reality, it is not. Solid, a crystalline structure, has its constituent atoms arranged in a fairly rigid threedimensional array called a crystalline lattice. We won’t exactly witness such orderly arrangement in any liquids and gases Pascal’s Law Pascal’s law, developed by French mathematician Blaise Pascal, states that the pressure on a fluid is equal in all directions and in all parts of the container. As liquid flows into the large container at the bottom of this illustration, pressure pushes the liquid equally up into the tubes above the container. The liquid rises to the same level in all of the tubes, reguardless of the shape or angle of the tube. 1.2 DENSITY & PRESSURE 1.2.1 Density : Introduction When we are dealing with fluids, what academically stimulates us more is the idea of variation of properties in different points of the fluid, hence it’s imperative to speak of density and pressure instead of mass and force, the two terminology widely used in mechanics of solids. Let us examine Petaling Jaya and Ipoh, Perak. The number of people living in PJ per area is higher compared to the number of people living in Ipoh, Perak. Such example gives us the idea of the ratio of the number of people to area. Density, however, is a measure of the ratio of mass to volume. Density = Mass / Volume or ρ = m/V ρ is a Greek letter rh. Common units are g cm-3 or kg m-3 1 g cm-3 = 1000 kg m-3 A substance could be (i) denser in large mass and smaller volume (ii) less dense in lower mass and larger volume. Density is not a vector quantity as it does not have direction. Take notice that the density of gas changes considerably with pressure, where as the density of a liquid doesn’t. This is due to the higher compressibility in gases than in liquids. 1.2.2 MEASURING DENSITY Measure the masses in kilograms of a wide range of solid objects. Then measure their voumes. Use a micrometer or vernier callipers to measure their dimensions if they are regular, or use the method of displacement of a volume of water by a solid with irregular shape. The density can be calculated by dividing the mass by volume. As for finding the density of liquid, first calculate the mass of a measuring cylinder, then add liquid into it, getting its total mass. The mass of the liquid alone is obtained by subtracting the mass of the cylinder from the total mass. Again, get the ratio of mass to volume of the liquid as its density. Find the density of a cube of ice. Then observe its melting process in a measuring cylinder and measure its density again. Perform the same experiment with a cube of wax. How much, in general, do substances change in volume when they make a transition from solids to liquids? Measure the volume and mass of a small cube of solid carbon dioxide. Drop it into a plastic bag and seal the bag immediately. Estimate the volume when all the carbon dioxide has become gas (process of sublimation). Calculate the density of solid carbon dioxide and gaseous carbon dioxide. 1.2.3 THE RANGE OF DENSITIES Solids and liquids are having small difference of volume when one phase makes a transition to the other phase. Hence, their densities do not differ much. However, there’s a distinct increase in volume for substance in gaseous state, where the density becomes very much smaller, compared to solids and liquids. 1.3 COMPRESSIBILITY Let’s try to compress, one at a time, a plastic syringe containing air, water and a piece of dowel. The compressibility is increasing in that order. A fairly small force, which has no noticeable effect on a liquid or a solid, will cause a large fractional reduction in the volume of a gas. Such observation gives us an oppurtunity to look into the physical microscopic properties of solid, liquid and gas. 1.4 PRESSURE Snowshoes prevent the person from sinking into the soft snow because the force on the snow is spread over a larger area, reducing the pressure on the snow’s surface. After an exciting but exhausting lecture, a physics professor stretches out for a nap on a bed of nails, suffering no injury and only moderate discomfort. How is this possible? Pressure is defined as force over an area in on which the force is acting upon. In the above examples, the snowshoes provide a large area to contribute to less pressure on the ground. The same can be said for the nail bed. High heels for example, is a very treasured item for a woman who has very high level expectation of beauty and class, despite the pain inflicted on the heel due to its small area. Can you name other applications of pressure in our daily lives? Example A water bed is 2.00 m on a side (a square) and 30.0 cm deep. (a) Find its weight. (b) Find the pressure that the water bed exerts on the floor. Assume that the entire lower surface of the bed makes contact with the floor. Calculate the pressure exerted by the water bed on the floor if the bed rests on its side. At any point on the surface of a submerged object in the container that constrain them, the force exerted by the fluid is perpendicular to the surface area of the object. The force exerted by the fluid on the walls of the container is perpendicular to the walls at all points. 1.3.1 PRESSURE IN FLUIDS When a fluid is at rest in a container, every part of the fluid must be in static equilibrium. This means the pressure, which is defined as is acting perpendicular at any point in the fluid from all direction. Besides, the pressure is the same for all points at the same depth of fluid. If it’s different, there would be a flow of fluid from higher pressure region to lower pressure region. Look at the following example in diagram (a). If the pressure on the left side is larger than the pressure on the right side, there would be a net force and the block would accelerate to the right because F1 would be greater than F2. However, as we inspect diagram (b), the depth is different. There are three forces acting on the block, Mg (downward), P1A (downward) and P2A (upward). The net force is zero as it is in equilibrium, therefore P2A - P1A – Mg = 0 Since M = ρV = ρAh Therefore P2A - P1A – (ρAh)g = 0 Eliminating the A, P2 - P1 – ρgh = 0 If the upper surface of the block is at the water surface, exposed to atmospheric pressure, we will get the following equation P2 = P1 + ρgh Where P1 = Atmospheric pressure above the liquid surface P2 = Total pressure on a point in the liquid ρgh = Pressure on a point due to depth of liquid EXAMPLE In a huge oil tanker, salt water has flooded an oil tank to a depth of 5.00 m. On top of the water is a layer of oil 8.00 m deep, as in the cross-sectional view of the tank in figure below. The oil has a density of 0.700 g/cm3. Find the pressure at the bottom of the tank. (Take 1 025 kg/m3 as the density of salt water.) Strategy First, use it to calculate the pressure P1 at the bottom of the oil layer. Then use this pressure in place of P0 and calculate the pressure Pbot at the bottom of the water layer. Solution Remark The weight of the atmosphere results in P0 at the surface of the oil layer. Then the weight of the oil and the weight of the water combine to create the pressure at the bottom. EXERCISE 1. Calculate the pressure on the top lid of a chest buried under 4.00 meters of mud with density 1.75 X 103 kg/m3 at the bottom of a 10.0-m-deep lake. (2.68 X 105 Pa) 2. Estimate the net force exerted on your eardrum due to the water above when you are swimming at the bottom of a pool that is 5.0 m deep. (Estimating the area of the eardrum as 1 cm2. (4.9 Pa) 3. An aeroplane launches off at sea level and elevate to a height of 425 m. Estimate the net outward force on a passenger’s eardrum assuming the density of air is approximately constant at 1.3 kg/m3 and that the inner ear pressure hasn’t been equalized. (0.54 N) 4. An office window has dimensions 3.4 m by 2.1 m. As a result of the passage of a storm, the outside air pressure drops to 0.96 atm, but inside the pressure is held at 1.0 atm. What net force pushes out on the window? 5. Three liquids that will not mix are poured into a cylindrical container. The volumes and densities of the liquids are 0.50 L, 2.6 g/cm3 ; 0.25 L, 1.0 g/cm3 and 0.40 L, 0.80 g/cm3. What is the force on the bottom of the container due to these liquids? One liter = 1 L = 1000 cm3. (Ignore the contribution due to the atmosphere.) 6. Find the pressure increase in the fluid in a syringe when a nurse applies a force of 42 N to the syringe’s circular piston, which has a radius of 1.1 cm. 7. A partially evacuated airtight container has a tight-fitting lid of surface area 77 m2 and negligible mass. If the force required to remove the lid is 480 N and the atmospheric pressure is 1.0 x 10 Pa, what is the air pressure in the container before it is opened? Pascal’s Principle What important contribution to fluids did Blaise Pascal make in 1647? Known as Pascal’s Principle, the law that Blaise Pascal develop states that any force applied to an enclosed fluid is transmitted in all directions to the walls of the container. The principle is extremely important in the field of hydrostatics and in the development of hydraulics. For example, if a piston pushes against the liquid in a closed cylinder, the force applied by the piston will translate into pressure on the walls of the cylinder. This occurs because liquids cannot be compressed as gases can. What is hydraulics? Hydraulics is the use of a liquid that, when moved from one place to another, accomplishes by its motion some type of function. The liquid used in hydraulic mechanism us usually water or oil. Hydraulic engineers design such things as pumps, lifts, faucets, cranes, shock absorbers, and many other devices. How does a hydraulic lift work? The basic theory behind a hydraulic lift is to multiply forces and give a device mechanical advantage. An automobile lift, used in many automotive repair shops, allows the operator to use very little force to lift an automobile off the ground, by pushing liquid from a small-diameter cyclinder and piston through a thin tube that expands into a largerdiameter cyclinder and piston, which is located beneath the vehicle to be lifted. Since the liquid cannot be compressed like air, the liquid from the small cyclinder is pushed into the large cylinder, forcing the large piston to move upward. Although this is very simplistic view of how hydraulic lift works, Pascal’s Principle states that if a small-area piston pushes a large-area piston, the mechanical advantage can be quite great. Example 1 In a car lift used in a service station, compressed air exerts a force on a small piston that has a circular cross section and a radius of 5.00 cm. This pressure is transmitted by a liquid to a piston that has a radius of 15.0 cm. What force must the compressed air exert to lift a car weighing 13 300 N? What air pressure produces this force? Because the pressure exerted by the compressed air is transmitted undiminished throughout the liquid, we have The air pressure that produces this force is This pressure is approximately twice atmospheric pressure. Eureka! What major discovery did Archimedes make when he stepped into bath in the third century B.C.? To his astonishment, when Archimedes stepped into a tub of water, the water rose! Of course, this wasn’t the first time water rose when Archimedes sat in the tub, but it was the first time he would consider the reasons why. He proceeded to conduct an experiment with gold and silver crowns that he immersed in the tub, measuring the water …… Why does a small clump of steel sink, while a 50000-ton steel ship can float? In order to remain afloat, a ship needs to displace an amount of fluid equal to its own weight. Therefore, if a clump of steel is placed in water, it will sink because its size wouldn’t allow it to displace an amount of water equal to its own weight. In this case, there is no way that the water could apply enough upward force to keep it afloat. A 50000-ton steel ship can easily stay afloat as long as it can displace 50000 tons of water. It can do this by widening the hull of the ship and increasing its volume. Buoyant Force Have you ever attempted to push a basketball into the water? It’s actually a difficult thing to achieve as the upward force exerted by the water on the ball is extremely high. This sort of force exerted by the liquid on any immersed object is known as buoyant force. When an object is immersed in a fluid (either a gas or a liquid), it experiences an upward buoyancy force because the pressure at the bottom of the object is greater than the pressure on the top. The great Greek scientist Archimedes (287-212 B.C.) made the following craeful observation, now called Archimedes Principle. Any object completely or partially immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid. In order to verify this, consider a small portion of water in a beaker of water. The downward forces are weight and force P (due to pressure of fluid at the top of the object multiply the area). The upward force is solely force Q (due to pressure of fluid at the bottom of the object multiply the area). In equilibrium, the upward force balances the downward forces. FP + W = FQ The net upward force due to the fluid is called the buoyancy force, FB = FQ - FP FB = FQ - FP = W FB = W This small portion of water is replaced by an object having similar shape and size. The object will feel the same upward buoyant force. A fully immersed object indicates that the object’s density is the same as the density of the fluid. The same cannot be said of the object immersed partially, where it’s density is less than the water. In this discussion about buoyant force, let’s remember two things: #1 In a floating or submerged object, assuming ideal system, the weight downward, W of an object is the same as the buoyant force upward, FB, giving us the following, FB = WOBJ #2 According to Archimedes’ Principle, the magnitude of the buoyant force always equals the weight of the fluid displaced by the object, giving us the following, FB =WFLUID Therefore, we can establish that the weight of an object is equals to the weight of the fluid being displaced. WOBJ = WFLUID Archimedes's Principle An object is subject to an upward force when it is immersed in liquid. The force is equal to the weight of the liquid displaced. The apparent weight of a block of aluminium (1) immersed in water is reduced by an amount equal to the weight of water displaced. If a block of wood (2) is completely immersed in water, the upward force is greater than the weight of the wood. (Wood is less dense than water, so the weight of the block of wood is less than that of the same volume of water.) So the block rises and partly emerges to displace less water until the upward force exactly equals the weight of the block. Apparent Weight When we attach an object to a spring balance, the weight reading is the actual weight of the object. However, when we immerse the object in a pool of water, the reading is less, why? The upthrust force (buoyant force) is acting upward on the object, resulting in a less-than-actual apparent weight. Apparent Weight = Actual Weight - Upthrust Let’s say, Gina has weight 500 N, when she goes into the swimming pool, the upthrust, 100 N, causes her to feel the apparent weight of 400 N only. Determination of Object’s Density Archimedes’ principle also makes possible the determination of the density of an object that is so irregular in shape that its volume cannot be measured directly. If the object is weighed first in air (ACTUAL WEIGHT) and then in water (APPARENT WEIGHT), the difference in weights will equal the weight of the volume of the water displaced (equivalent to UPTHRUST), which is the same as the volume of the object. Thus the density of the object (mass divided by volume) can readily be determined. In very high precision weighing, both in air and in water, the displaced weight of both the air and water has to be accounted for in arriving at the correct volume and density. In the simplest language possible, an object 500 N in air, then immersed in water where you get 400 N reading on the spring balance. The 100 N is the difference between the two weights, equivalent to the buoyant force, which is also equal to the weight of the water being displaced. WH20 = ρVg 100 = 1000 (V )(10) V = 0.01 m3 The volume of the water being displaced 1 m3 , then we shall conclude that 1 m3 is also the volume for the irregular object. Using the 50 kg as the mass of the irregular object (assuming g is 10 m s-2), we use this value and divide by 0.01 m3, you will get a density of 5000 kg m-3 for the irregular object. Applications of Archimedes’ Principle A submarine applies the Archimedes’ Principle to enable it to float and sink. The ballast tanks are special compartments in a submarine. When the compressed air forces the water out of the ballast tank, the submarine rises. This occur because the buoyant force is greater than the weight of the submarine. Meanwhile, when water is allowed to enter the ballast tank, the submarine sinks because the buoyant force is less than the weight of the submarine. In hot air balloon, the fire heats up the helium gas inside the balloon up to 100°C.. As the balloon expands, it displaces a large amount of the surrounding air, thus the upthrust is very huge. The density inside is getting less and the weight is decreased. The balloon starts to rise when the upthrust is larger than the weight of the balloon. If the hot air ballon is neither going up nor down, Fup = WBALLOON = WATMOSPHERIC AIR Example A helium-filled hot air balloon is rising. The density of surrounding air is 1.3 kg m-3 The density and volume of helium gas is 0.18 kg m-3 and 0.06 m3 respectively. A weight, W is attached to the balloon. (a) What is the weight of the helium gas inside the balloon? (b) Calculate the upthrust acting upward upon the balloon. (c) How much W is needed to enable the balloon to be staticly floating in air? The helium gas contained inside the balloon is considered contributing downward weight to the balloon and MUST NEVER BE CONFUSED with the outside surrounding air displaced by the balloon (equivalent to the upthrust). Solution: How Science Works The Plimsoll Line What happenes to the buoyancy of the ship when cargo and passengers are added? Ship builders must always consider the floatation level of the ship when cargo and passengers are added to it. This increases the weight of the ship. The ship will float as long as the total weight of the ship plus contents is less then the weight of the water it displaces. When the weight of the ship and contents exceed this value, the ship sinks. The amount of ship lowers in the water as a result of cargo and passengers can be critical for navigation and maneuverability. Large cargo and cruise ships have numbers on the bow of the ship that indicate how far the ship is submerged. If the ship has a twenty-foot draft and the water is only eighteen feet deep, cargo and passengers must be unloaded to allow the ship to rise. When a ship floats on the surface of the ocean, two forces are acting on the ship, namely, the weight of the ship and the upthrust. The weight of the ship changes when the total load in the ship varies. Upthrust is proportional with the density and the volume of the displaced ocean water. Whenever the load increases, the weight of the ship increases. The ship will submerge a little deeper into the ocean to displace more water. In this way, the ship will gain more upthrust to support the increasing weight of the ship. Ships that navigate to places throughout the world will sail through various densities of ocean water. Plimsoll Line is drawn on the side of the ship to give indication on the maximum load allowed for the ship. If the ocean level is directly on the line, the ship has carried the maximum permissible load. The International Load Line (LR: Lloyd's Register of Shipping) LTF Lumber, Tropical, Fresh TF Tropical Fresh Water Mark LF Lumber, Fresh F Fresh Water Mark LT Lumber, Tropical T Tropical Load Line LS Lumber, Summer S Summer Load Line LW Lumber, Winter W Winter Load Line LWNA Lumber, Winter, North Atlantic WNA Winter, North Atlantic FLUID MECHANICS The mechanics of fluid flow are complex and produce many unexpected effects, some of which are still not fully understood. The onset of turbulence when smooth flow is suddenly disrupted by complex motions like vortices and eddies is very difficult to predict or model effectively. In some situations turbulence is desirable; for example, to provide rapid mixing of fuel and oxygen inside a jet engine. In other situations it is disastrous. Turbulent airflow over a wing destroys lift. The motion of real fluids is very complicated and not yet fully understood. Instead, the discussion of ideal fluid is proposed instead. The following are four assumptions abpout ideal fluid, which are concerned with flow: Steady or laminar flow: The velocity of the moving fluid any fixed point does not change with time, either in magnitude or in direction. The gentle flow of water near the center of a quiet stream is steady; the flow in a chain of rapids is not. Assume flow is smooth (laminar) and not turbulent. Fluid flow is described using streamlines. These are arrows that represent the velocity of the fluid at each point. In steady flow they correspond to lines of motion which follow the paths of the particles. Incompressible flow: We assume that the ideal fluid is incompressible at rest. In other words, it means the density has a uniform and constant value. Nonviscous flow: Ignore fluid friction, that is, viscosity. Viscosity is the measure of how resistive the fluid is to flow. Hgher viscosity means that the velocity gradient is lower; lower viscosity allows the velocity to increase rapidly away from the surface. For instance, compare water and raw petroleum. The organic fluid is more resistive to flowing than the water, so the raw petroleum is considered more viscous than water. Using the same analogy as the solid-friction interaction, we can describe the “friction” of the fluids in terms of viscosity. Both contribute to the changing of kinetic energy to thermal energy. If the fluid is in the state of nonviscous flow, it would not experience viscous drag force. The British scientist Lord Rayleigh noted that in an ideal fluid a ship’s propeller would not work, however, on the other hand, in an ideal fluid a ship (once set into motion it would not need a propeller!) Irrotational flow: The particle of dust in an experiment in fluid may rotate in a circular path, but it will not rotate at its own axis. Take the example of Ferris wheel, the passenger doesn’t rotate, but their bodies rotate according to the axis of the wheeel! LAMINAR FLOW/STREAMLINE FLOW The steady flow of a fluid around a cylinder, as revealed by a dye tracer that was injected into the fluid upstream of the cylinder. In laminar flow, sometimes known as streamline flow through a pipe, adjacent layers of a liquid move parallel to one another, without disruption between the layers. It is opposite of turbulent flow. In layman language, laminar flow is “smooth” while turbulent flow is “rough” Laminar and Turbulent Motion At low velocities, fluids flow in a streamlined pattern called laminar motion. Laminar motion can be described mathematically by equations derived by Claude Navier and Sir George Stokes in the mid 1800s. At high velocities, fluids flow in a complex pattern called turbulent motion. For fluids flowing in pipes, the transition from laminar to turbulent motion depends on the diameter of the pipe and the velocity, density, and viscosity of the fluid. The larger the diameter of the pipe, the higher the velocity and density of the fluid, and the lower its viscosity, the more likely the flow is to be turbulent. Let’s look at the smoking man photo. We can observe that the smoke is rising up off the cigarette in a still air surrounding. It will, initially rise up, increasing in its speed in a vertical and smooth manner for some distance (laminar flow). After some time, the smoke will start undulating into a turbulent, non laminar flow. The whole process is changing from the steady to non-steady. The Equation of Continuity Imagine an ideal fluid flowing through a frictionless pipe which becomes narrower and narrower. The fluid is incompressible, so as much fluid enters the pipe each second as leaves it. If the cross-sectional areas and velocities at entry and exit are A1 and A2 and v1 and v2 respectively then: volume entering per second = A1 v1 = volume leaving per second = A2v2 This is the equation of continuity and leads to: The flow speed is greater in the thinner part of the pipe. The flow within any tube of flows obeys the equation of continuity: Rv = Av = constant In which Rv is the volume flow rate A is the cross-sectional area of the tube of flow at any point, and v is the speed of the fluid at that point. How Science Works : Continuity Why does a river’s current run faster when the river is narrow? When water flows down a river, the current represents the amount of water that passes by a section of the river in a unit of time. For example, if the current of a river is 2000 L/min, this means that, assuming the slope of the river is constant, every minute 2000 liters pass by every section of the river. If a section of the river narrows, the 2000 liters of water still must pass in one minute beacause the water from behind does not let up in its desire to flow downriver. Since the river is narrower, the water needs to speed up in order to accomplish this task. The principle behind this phenomenon is called continuity. Example 1 Water is flowing out from a water tap. The cross-sectional areas at the starting point is A0 = 1.2 cm2 and a point 45 mm below the starting point is A = 0.35 cm2. What is the initial velocity of the flowing water? What is the flow rate Rv? Example 2 A garden hose with an internal diameter of 1.9 cm is connected to a (stationary) lawn sprinkler that consists merely of a container with 24 holes, each 0.13 cm in diameter. If the water in the hose has a speed of 0.91 m s-1, at what speed does it leave the sprinkler holes. Solution 2 We use the equation of continuity. Let v1 be the speed of the water in the hose and v2 be its speed as it leaves one of the holes. A1 = πR2 is the cross-sectional area of the hose. If there are N holes and A2 is the area of a single hole, then the equation of continuity becomes where R is the radius of the hose and r is the radius of a hole. Noting that R/r = D/d (the ratio of diameters) we find Example 3 The water flowing through a 1.9 cm (inside diameter) pipe flows out through three 1.3 cm diameter pipes. (a) If the flow rates in the three smaller pipes are 26, 19 and 11 L/min, what is the flow rate in the 1.9 cm diameter pipe? (b) What is the ratio of the speed in the 1.9 cm diameter pipe to that in the pipe carrying 26 L/min? Solution 3 (a) Equation of continuity provides us the initial flow rate to be the same as final flow rate. Giving us (26 + 19 + 11) L/min = 56 L/min in the main pipe (1.9 cm diameter) (b) Using the equation of continuity, Using v = R/A and A = πd2/4, Example 4 The merging of two streams, A and B, forms river C. The following are their respective data: A Width: 8.2 m Depth: 3.4 m Speed: 2.3 m s-1 B Width: 6.8 m Depth: 3.2 m Speed: 2.6 m s-1 C Width: 10.5m Depth: h Speed: 2.9 m s-1 What is the value of h? What is fluid dynamics? Fluid dynamics is the study of fluids in motion. There are several different types of fluid motion: steady flow, where the liquid or gas moves in a constant and predictable manner; unsteady flow, where the fluid makes turns and changes its velocity; and turbulent flow, where the fluid motion is extremely difficult to predict. What makes a fluid flow? As in all of physics, objects move as a result of forces. Just as a basketball dropped in the air falls to the ground because of gravitational force, a fluid flows because there is an unbalanced force acting on the liquid – that is, a difference in pressure between two points; fluid will flow in the direction of decreasing pressure. Why does it always seem windier in the city? The explanation behind this question is not meteorological, but physical. In major cities, there ae skyscrapers and other tall buildings that obstruct the flow of wind. In order to flow past these large obstacle, the wind speed increases in the corridors of the streets and avenues. The same effect can also be found in tunnels and outdoor “breezeways”. It is the continuity of the fluid speed rushing through the narrow corridors of streets and avenues that makes the city such a windy place. Drag Act As we investigate the dynamics of objects in fluid dynamics, we soon encounter a property called drag (also known as fluid resistance). Drag is the force that resist movement In fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a solid object through a fluid (a liquid or gas). The most familiar form of drag is made up of friction forces, which act parallel to the object's surface, plus pressure forces, which act in a direction perpendicular to the object's surface. For a solid object moving through a fluid, the drag is the component of the net aerodynamic or hydrodynamic force acting in the direction of the movement. The component perpendicular to this direction is considered lift. Therefore drag acts to oppose the motion of the object, and in a powered vehicle it is overcome by thrust. Terminal Velocity How Science Works : Stokes’ Law Introduction This investigation involves determining the viscosity and mass density of an unknown fluid using Stokes’ Law. Viscosity is a fluid property that provides an indication of the resistance to shear within a fluid. Specifically, you will be using a fluid column as a viscometer. To obtain the viscometer readings you will use a stopwatch to determine the rate of drop of various spheres within the fluid. You will determine both density and viscosity. 2.Learning Outcomes On completion of this laboratory investigation students will: • Appreciate the engineering science of 'fluid mechanics.' • Understand the concept of fluid 'viscosity.' • Understand the concept of dimensionless parameters, and most specifically the determination of Reynold's Number. • Be able to predict the settling time of spheres in a quiescent fluid. • Be able to calculate the viscosity of an unknown fluid using Stokes' Law and the terminal velocity of a sphere in this fluid. • Be able to correct for the diameter effects of fluid container on the determination of fluid viscosity using a 'falling ball' viscomter 3. Definitions Viscosity – a fluid property that relates the shear stress in a fluid to the angular rate of deformation. Fluid Mechanics – the study of fluid properties. Reynold’s Number – dimensionless parameter that represents the ratio of viscous to inertial forces in a fluid Strength of Materials The Physical Properties of Solids Sugar Crystals This electron microscope image of raw cane sugar reveals the shape of sugar crystals. The crystals form after purified cane juice has been heated and some of the water in the juice has evaporated, leaving behind a cane syrup. Seed crystals added to the syrup make the sugar molecules dissolved in the syrup separate from the liquid to form larger, solid crystals around the seed crystals. Stretching Materials Stress, Strain and the Young Modulus The strength of a wire has direct proportional relationship with its cross-sectional area. Thicker wires are tougher than thinner wires of the similar material. For a given substance, the tension needed to break the wire divided by the crosssectional area is constant. Stress, σ is defined as the average amount of tension (force), F exerted on a given area, A The tension divided by cross-sectional area is called the tensile stress, symbol σ (the small Greek letter sigma): tensile stress = tension/cross-sectional area Stress has the same units as pressure, N m-2 or the pascal, Pa. The stress needed to break a material is called the breaking stress or ultimate tensile stress. If you want to compare the strengths of different materials, you compare the breaking stresses, since these do not depend on the cross-sectional area of the sample that you are testing. A strong material like steel has a high ultimate tensile stress; a weak material has a low ultimate tensile stress. A steel wire, cross-sectional area 1.0 mm2 breaks under a tension of 250 N. breaking stress = F/A = 250 N/(1.0 x 10-6 m2) = 250 MPa The shape of a tension—extension graph depends on the dimensions of the sample of wire you are testing. If you have wires of the same length and the same material, the thicker wire will need a larger tension for the same extension. If the wires are of the same material and thickness, but different lengths, the longer wire will have a larger extension for the same tension. If you wish to compare the behaviour of different materials, you need to plot quantities that do not depend on the size of the sample of material tested. Strain is the deformation of materials caused by the action of tensile stress. The difference in the initial displacement of two points, say, A & B and final displacement of two points, A & B after strain is applied is known commonly as extension: Strain, symbol ε (the small Greek letter epsilon), is the extension divided by the original length ε = strain l = length of extension + original length l0 = original length Strain accounts for the fact that samples stretch in proportion to their lengths. Strain is a unitless quantity. The ratio of extended length (meter) to original length (meter) cancels the unit out, therefore it doesn’t have unit of measure. A steel wire 2.0 m long, stretched to just below its breaking point, extends by 2.6 mm. strain = (2.6 X 10-3 m)/(2.0 m) = 1.3 X 10-3 In solid mechanics, Young's modulus (E) is a physical property that describe the stiffness of a material. It is defined as the ratio of stress over strain in the region in which Hooke's Law is obeyed for the material.We can conduct an experiment to acquire the value from the slope of a stress-strain graph drawn during tensile tests conducted on a sample of the material. Based on the graph above (left), the slope of the straight line is called the "Young's Modulus", and has dimensions of force over area. The three different materials have various numerical gradient values, indicating the different measure of “stiffness”, namely the Young’s Modulus. Smaller values (lowest gradient, polystyrene) indicate that less stress is required for more strain. In other words, it experiences longer stretches for a small stretching force. Likewise, larger values of Young's Modulus (largest gradient, steel) indicate that more stress is required for a given small strain. Less stretching is achieved, even though much stretching force ha been applied. In general, the stress versus strain curve will differ for each material for each type of similar stress. The photo on above (right) shows steel cables used in lifting heavy objects. These steel cables break under a tension of 300 kN. Stiffness is important for engineering materials, since for most applications it is important that the shape of a component changes very little when it is under stress. So, most of the time, materials with higher Young’s Modulus such as steel and copper are chosen for engineering projects. On the other hand, stiffness of consumer products is not necessary since the significant changes in shape do not really matter for substance like polystyrene. How Science Works : Climbing Ropes How Science Works : Uncertainties in Measurement Characteristics of Solids How Science Works : The Mohs Hardness Scale In the Mohs scale, named for the German mineralogist Friedrich Mohs who devised it, ten common minerals are arranged in order of increasing hardness and are assigned numbers: 1. talc 2. gypsum 3. calcite 4. fluorite 5. apatite 6. orthoclase (feldspar) 7. quartz 8. topaz 9. corundum 10. diamond The hardness of a mineral specimen is obtained by determining which mineral in the Mohs scale will scratch the specimen. Thus, galena, which has a hardness of 2.5, can scratch gypsum and can be scratched by calcite. The hardness of a mineral largely determines its durability. How Science Works : Materials Selection Charts Materials In The Real World How Science Works : Real World Materials