Objectives 184 CHAPTER 4 RESISTANCE

Transcription

1 Objectives Define drag. Explain the difference between laminar and turbulent flow. Explain the difference between frictional drag and pressure drag. Define viscosity and explain how it can be measured. Explain why a freely falling object has a terminal speed. Use Stokes law and Poiseuille s law to solve problems involving fluid resistance. When one solid object slides against another, a force of friction opposes the motion. The direction of the force is opposite the direction of the object s velocity. When a solid object moves through a fluid, there is also a force that opposes the motion. This force is called drag. When a boat moves through water or when an airplane moves through air, a drag force is exerted in the direction opposite the velocity of the boat or airplane. To find out more about resistance in fluid systems, follow the links at You can feel a drag force when you stand in a high wind or when you put your hand out the window of a moving car. In the first case, you are stationary and the fluid is moving past you; in the second case you are moving past the fluid. Drag occurs only when there is relative movement between an object and a fluid. 184 CHAPTER 4 RESISTANCE

2 Laminar (Streamlined) and Turbulent Flow The drag exerted on an object by a fluid depends on many factors. The most important are the speed of the object (or fluid), the size and shape of the object, and the physical properties of the fluid. These factors make it difficult to calculate drag exactly, but you can make approximations. The simplest approximation is to ignore drag forces when they are small. For example, you can usually ignore drag for an object moving slowly in fluids such as air or water. But even very slow speeds produce significant drag in fluids such as molasses and motor oil. When drag forces cannot be ignored, you can make two approximations about the fluid flow the flow can be laminar or turbulent. Laminar, or streamlined flow is a slow, smooth flow over a surface, in which the paths of individual particles do not cross. Each path is called a streamline, as illustrated in Figure The fluid speed at the surface is zero, and fluid moves in theoretical layers, or laminates, with increasing speed away from the surface. Drag is produced by the friction between successive layers of fluid. This is called frictional drag. (a) Laminar flow (b) Turbulent flow (c) Turbulent flow Figure 4.10 Laminar and turbulent flows around obstacles. A turbulent wake can be created by a different shape or a higher fluid speed. Turbulent flow is irregular flow with eddies and whorls causing fluid to move in different directions. Turbulence is produced by high speeds, by shapes that are not streamlined, and by sharp bends in the path of a fluid. Turbulence produces the visible wake behind a moving boat and an invisible wake behind a moving airplane or car. Changing the direction of the fluid into eddies and whorls requires work. When the fluid does work, the pressure drops. (Remember, W = VΔP.) Thus, the fluid pressure in the wake is less than the fluid pressure in the streamlined flow. This pressure difference causes a force to act on the object in the direction opposite its relative velocity. This force is called pressure drag. Figure 4.11 Turbulence creates a region of low pressure in the fluid. The higher pressure over the front surface area of the object causes a drag force. SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 185

3 Frictional drag and pressure drag both increase as speed increases. Figure 4.12 shows how the total drag force on an automobile increases as its speed increases. (The axes do not have a scale because numerical values of drag depend on the aerodynamic design of the automobile. But the linear and nonlinear shape of the graph is similar for all designs.) Figure 4.12 The drag force on a car increases as the car s speed increases. At low speeds, the drag force on the car is frictional drag. The force increases linearly with speed. This means that doubling the speed doubles the frictional drag force. At higher speeds, turbulence and pressure drag are more and more important. This force increases as the square of the speed. Doubling the speed increases the pressure drag by a factor of four. Viscosity Friction between two solid surfaces causes a resistance to movement between the surfaces. On a microscopic level, the resistance is due to electrical forces between atoms and molecules in the surfaces of the solids. Electrical forces also exist between atoms and molecules of a fluid. These forces create internal friction in the fluid, which causes a resistance to movement. Viscosity is the property of a fluid that describes this internal friction. We use the Greek letter η (eta) to represent viscosity. Molasses and bubble gum have high resistance to internal movement, and high viscosities. Air and water have much lower viscosities. 186 CHAPTER 4 RESISTANCE

4 Viscosity for fluid resistance is similar to the coefficients of friction for mechanical resistance, but viscosity is not a simple coefficient. For example, Figure 4.13 shows a layer of fluid of thickness Δy between two plates. The bottom plate is held in place, and the top plate (of area A) is pulled to the right at a constant speed v. The fluid in contact with the top plate moves with the plate at speed v, and the fluid in contact with the bottom plate remains motionless. The speed of the fluid between the top and bottom varies linearly. Electrical forces between layers of fluid resist this variation in motion between layers. (The top plate drags layers of fluid with it. This is the source of the word that describes the resistance.) The force F is required to overcome the resistance and keep the plate moving at constant speed. Figure 4.13 The viscosity η of a fluid can be measured by pulling a plate at constant speed across a layer of the fluid. Δ η = F y Av When the plate moves to the right at constant speed, no net force is acting on the plate. Therefore, the fluid exerts a force of friction, or drag force F drag on the plate to the left, opposing motion. The magnitude of the drag force equals F. As long as the plate speed v is not so large that turbulence occurs, the fluid flow between the plates is laminar. The force F required to maintain a constant speed for most fluids in laminar flow is found to be: proportional to A and v, and inversely proportional to the thickness of the fluid layer, Δy. The proportionality constant is the viscosity of the fluid. Av Fdrag = F = η Δy Viscosity has units of (pressure) (time). The SI units for viscosity are lb or Pa s. The English units are or psi s. 2 s ft N m 2 s SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 187

5 Different fluids resist motion differently, and therefore have different viscosities. Table 4.2 lists values of viscosity for several fluids. Table 4.2 Viscosities of Common Fluids Fluid Temperature C Gases Air Viscosity Pa s Water Vapor Liquids Water Blood Cooking Oil Motor Oil 20 1 Corn Syrup 20 8 Molten Lava The viscosity of most liquids decreases as the temperature increases. For example, cold honey is very thick and difficult to pour (high viscosity). But if you heat honey in a microwave oven it becomes watery (lower viscosity). As the temperature increases, the molecules in the honey become less and less tightly bound to each other. Thus, the force required to separate the molecules also becomes less. On the other hand, the viscosity of most gases increases with temperature. Forces between gas molecules are exerted only during collisions, and there are more collisions per second with higher temperature. Motor oil is rated by viscosity by the Society of Automotive Engineers (SAE). Oil with an SAE10 rating has lower viscosity than SAE40 oil (at the same temperature). Oil rated as 10W40 has a viscosity at low engine temperatures equivalent to SAE10 (at low temperatures) and a viscosity at higher engine temperatures equivalent to SAE40 (at higher temperatures). 188 CHAPTER 4 RESISTANCE Stokes Law In 1845, the Irish mathematician and physicist George Stokes used viscosity and the equations of fluid flow to predict the drag force on a sphere moving through a fluid. The result is called Stokes law. It applies to objects moving at low enough speeds that the flow of fluid around the objects is streamlined, or laminar. In these cases, there is no turbulence and the only drag force on the objects is due to frictional drag.

6 Figure 4.14 The drag force on a sphere moving through a fluid opposes the sphere s velocity. The drag force acts in the direction opposite the object s velocity (it opposes motion). According to Stokes law, the drag force equals the product of a constant (6π for a sphere), the radius r of the object, the speed v of the object (or the relative speed between the object and fluid), and the fluid s viscosity η: F drag = 6πrvη Terminal Speed When an object moves through a fluid, the drag force on the object increases as the speed increases. If you drop a baseball from a high tower, it has a very low speed and very small drag at first. The force of gravity (weight) acting downward is greater than the drag force acting upward. Therefore, a net force acts downward on the baseball and it accelerates downward. As the speed increases the drag increases, until at some point the upward drag equals the weight. At this point the forces acting on the baseball are balanced and it no longer accelerates. The speed becomes constant. The terminal speed of a falling object is the constant speed that occurs when the drag force equals the gravitational force. The terminal speed of a baseball is about 40 m/s, but the terminal speed of a basketball is only about 20 m/s. Which ball has a greater drag force at any given speed? Example 4.5 Terminal Speed of a Dust Particle When volcanoes erupt, rocks, dust, and ash are thrown into the atmosphere. These particles fall back to Earth at terminal speeds that depend on their size. The terminal speed of a particle in a volcanic dust cloud can be estimated using a sphere to model the particle. Estimate the terminal speed of a 50-μm-diameter particle if the density of the volcanic material is 2500 kg/m 3. SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 189

7 Solution: When the particle moves at terminal speed, no net force is acting on it. The force of gravity equals the drag force. Use Stokes law to calculate the drag force. The radius r of the particle is one-half the diameter: 50 μm r = = 25 μ m or m 2 The mass m of the particle is the product of the density ρ and 4 volume V. The volume of a spherical particle is πr 3 : 3 m = ρv = kg π m 3 m 3 (. ) 10 = kg The weight of the particle is F g : F g 10 = mg = ( kg) (9.80 m/s ) 2 9 = kg m/s or N Use Stokes law, and equate the drag to the weight. Solve for the speed v. Use the value of viscosity from Table 4.2 for air at 20 C: 2 9 F drag = F g 6πrvη= N v = 6πη r The terminal speed of a 50-μm particle of volcanic dust is about 18 centimeters per second. 9 9 N N = 6π( m) = 018. m/s or 18 cm/s 5 5 N 2 m s 190 CHAPTER 4 RESISTANCE

8 Poiseuille s Law Poiseuille s law gives the volume flow rate of a fluid flowing through a tube or pipe. Like Stokes law, Poiseuille s law applies to laminar flow. Figure 4.15 illustrates laminar flow in a pipe. The fluid layer at the center moves the fastest, and layers nearer the wall move more slowly. (Fluid in contact with the wall does not move.) Jean Louis Poiseuille was a physician who was also trained as a physicist and mathematician. In the mid-1840s, he experimented with water flowing through glass capillary tubes as a simulation of blood flowing through small blood vessels. Poiseuille learned that the rate at which fluid flows through a tube increases proportionately to the pressure applied and to the fourth power of the radius of the tube. According to Poiseuille s law, the volume flow rate (m 3 /s) of a fluid of viscosity η through a tube or pipe of radius r and length L is: In this equation, ΔP is the change in pressure of the fluid as it flows the length L. The internal friction of the fluid causes the pressure to decrease as the fluid flows. Therefore, ΔP = P 2 P 1 is negative because P 2 < P 1. With a negative ΔP, V is positive. 4 π r ΔP V = 8 ηl V Figure 4.15 Cross section through the center of a pipe, showing laminar flow. The fluid at the center of the pipe has the highest speed. The speed decreases closer to the wall of the pipe. Figure 4.16 A fluid s viscosity, or internal friction, causes the pressure to drop along the direction of flow. SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 191

9 Example 4.6 Blood Flow Rate The flow rate of blood through a vein is 3.2 cm 3 /s. The diameter of the vein is 3.6 mm. If the pressure drops by 1100 Pa between two points in the vein, estimate the length of the vein between the points. Solution: Convert units of and r to SI: V = 3.2 cm 1 m s 10 cm = m /s 6 3 r = d/2 = (3.6 mm)/2 = 1.8 mm = m Solve the equation for Poiseuille s law for the length L. Use the value of viscosity from Table 4.2: V π r ΔP = 8 ηl π r ΔP L = 8 ηv The vein is about 28 cm long. 4 4 V π ( m) ( 1100 Pa) = 8 3 ( Pa s) m s = 028. m or 28 cm 4 Factors Affecting Flow Through a Pipe Resistance decreases the flow rate of fluid through a pipe. Poiseuille s law shows how this resistance depends on three factors: (1) the radius (or crosssectional area) of the pipe, (2) the length of the pipe, and (3) the viscosity of the fluid. These dependencies can be illustrated using graphs of volume flow rate versus pressure drop. For each graph, you can define a fluid resistance R as the ratio of the prime mover to the volume flow rate. The prime mover in fluid systems is pressure change, or in this case pressure drop. Pressure drop is ΔP. (ΔP is negative, so pressure drop and fluid resistance are positive.) R = V pressure drop P volume flow rate = Δ V Therefore, R is the slope of each line graphed on the following pages. A high-resistance pipe has a large slope, and a low-resistance pipe has a small slope. 192 CHAPTER 4 RESISTANCE

10 (1) Dependence on Radius Compare the flow of a fluid through two pipes of the same length, but one with a small radius and the other with a large radius. The larger pipe has a greater cross-sectional area and can move a greater volume of fluid per second. This pipe also has a lower resistance to flow than the smaller pipe. According to Poiseuille s law, the volume flow rate increases as the fourth power of the radius. If the radius of the large pipe is twice the radius of the small pipe, the volume flow rate is sixteen times higher (if r 2r, then r 4 16r 4 ). Figure 4.17 shows graphically the effect of increasing the pipe radius. The volume flow rates and pressure drops are shown for water flowing through a 1-m length of pipe for three pipe radii. Which graph has the highest slope? The highest resistance? For a pressure drop of 25 Pa, if you increase the radius of the pipe from 1.0 cm to 1.5 cm, you increase the cross-sectional area, decrease the flow resistance, and increase the volume flow rate from 98.2 cm 3 /s to 497 cm 3 /s. Figure 4.17 Fluid resistance decreases as pipe radius and cross-sectional area increase. (2) Dependence on Length Figure 4.18 shows how fluid resistance changes when the pipe length changes. Longer pipes have higher fluid resistance. If you double the length of a pipe, you double the resistance and the volume flow rate is halved. Volume flow rate is inversely proportional to length. Figure 4.18 Fluid resistance increases as pipe length increases. SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 193

11 (3) Dependence on Viscosity Figure 4.19 shows how resistance varies with the viscosity of the fluid. Which graph has the highest slope? The highest resistance? Volume flow rate is inversely proportional to viscosity. If you use a fluid with half the viscosity, you double the volume flow rate. Figure 4.19 Fluid resistance increases as viscosity increases. If the flow becomes turbulent, resistance increases rapidly. As illustrated in Figure 4.20, bends and Ts in a pipe or air duct cause turbulence. When it is important to maintain laminar flow and reduce resistance, designers use curves with radii as large as possible rather than abrupt changes in the path of a fluid. Figure 4.20 Abrupt changes in the direction of fluid flow can cause turbulence and increase resistance. Obstructions or restrictions also cause turbulence. For example, the grill of a car is an obstruction that causes turbulence, affecting the aerodynamic drag of an automobile. Filters in air ducts are restrictions. Figure 4.21 shows the pressure drops along an air duct containing a filter. There is a small pressure drop from P 1 to P 2 (exaggerated in the graph) along the length of the duct, and there is a much larger drop from P 2 to P 3 because of the filter. If the filter is dirty, the pressure drop is even larger. In fact, a clogged filter can almost stop airflow. 194 CHAPTER 4 RESISTANCE

12 (a) Pressure vs position for a clean filter (b) Pressure vs position for a dirty filter. The volume flow rate for this situation is lower than that for (a). Figure 4.21 Pressure drop along the flow path for an air duct and filter. Note the pressure drop from P 1 to P 2 is exaggerated, as is the drop from P 3 to the right. Summary Drag is the force that opposes the motion of an object moving through a fluid or the force a moving fluid exerts on a stationary object. Laminar flow is slow, smooth flow over a surface, where particles follow streamlines. The streamlines define theoretical layers of fluid that do not mix. The friction between successive layers of fluid is called frictional drag. SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 195

13 Turbulent flow is irregular flow with eddies and whorls that mix the fluid. Turbulence causes a wake behind a moving object. The pressure difference between the fluid outside the wake and the fluid inside the wake causes pressure drag. Drag increases with speed. When turbulence is created, pressure drag increases more rapidly than friction drag. Viscosity is the property of a fluid that describes its internal friction. The SI units of viscosity are Pa s. Stokes law can be used to calculate the drag force on a sphere moving at constant speed in a viscous fluid. When the drag equals the gravitational force acting on a falling body, the body falls at a constant speed, called the terminal speed of the body. Poiseuille s law can be used to calculate the volume flow rate or pressure drop of a viscous fluid flowing through a tube or pipe. Exercises 1. Drag on an object moving through a stationary fluid (increases or decreases) as the object s speed increases. 2. If a fluid flows past a stationary object, the drag on the object (increases or decreases) as the fluid speed increases. 3. Fluid resistance increases as flow becomes (laminar or turbulent). 4. Why do downhill ski racers and bicycle racers wear specially formed clothing and helmets, and form their bodies in the shape of an egg? 5. The forces acting on four objects moving through a fluid are shown below. The force vectors are drawn to scale. Which object(s) moves at constant velocity? (a) (b) (c) (d) 196 CHAPTER 4 RESISTANCE

14 6. Two metal plates are separated by a 1.5-mm thickness of motor oil, as shown in Figure The top plate measures 12 cm by 15 cm. (a) What force is required to move the top plate at a constant speed of 0.4 cm/s? Use the value of viscosity in Table 4.2. (b) If the force on the top plate is halved, what is the plate s speed? 7. (a) The plate in Exercise 6 moves to the right at a constant speed of 0.4 cm/s. What are the magnitude and direction of the drag force on the plate? (b) The oil between the plates is replaced with water at 20 C. What is the drag force on the top plate when it moves to the right at a constant 0.4 cm/s? (c) The water is replaced with air at 20 C. What is the drag force for the same motion of the top plate? (d) Suppose you are approximating the drag force on a plate moving through a fluid at 0.4 cm/s. For which fluid oil, water, or air are you most accurate when you say the drag force is approximately zero? 8. A sky diver falls at a terminal speed of 14 ft/s with her parachute open. Her weight plus the weight of her clothing and gear is 168 lb. What are the magnitude and direction of the drag force on the sky diver? 9. Metal particles produced in a milling machine are collected in an oil tank. The particles can be modeled as spheres. One particle is 1.5 mm in diameter and weighs N. What is the terminal speed of the particle in the oil if the oil s viscosity is 2.8 Pa s? 10. The volume flow rates and pressure drops for two pipes are graphed below. (a) Which pipe has the higher resistance? (b) If the pipes are the same length, which one has the larger diameter? (c) If the pipes have the same diameter, which one is longer? SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 197

15 11. (a) Solve the Poiseuille s law equation for fluid resistance. ΔP R = =? V What are the SI units of this quantity? (b) A 40-cm length of copper tubing has an internal diameter of 0.32 cm. Calculate the resistance of the tubing when water at 20 C flows through it. (c) What is the water flow rate through the tubing when there is a pressure drop of 1.5 kpa between the tubing inlet and its outlet? (d) A second, identical 40-cm length of tubing is connected to the outlet of the first length. What is the water flow rate through the combined length if the pressure drop is 1.5 kpa? 12. A 12-inch-diameter pipeline transports crude oil 54.4 miles. The maximum pressure in the pipeline is 950 psi. The pressure drop is 850 psi. The viscosity of the oil is psi s. (a) Find the volume flow rate (in in 3 /s) of crude oil through the pipeline. (b) Calculate the amount of work (in ft lb) that must be done by pumps to operate the pipeline for one second. (c) Suppose the pipeline is replaced with one whose diameter is 14 inches. What is the pressure drop required to produce the same flow rate as in (a)? 13. The two water pipes shown below have the same diameter and length. The water flow rate through the pipes is the same, but the pressure drop is not the same. What could cause the difference in pressure drop? Explain. 198 CHAPTER 4 RESISTANCE

16 14. The piston in an air compressor has a diameter of 12.5 cm and a height of 15.0 cm. There is a cm space between the piston and cylinder wall filled with lubricating oil. The oil has a viscosity of 0.35 Pa s. If the average speed of the piston is 43.0 m/s, what is the average drag on the piston caused by the lubricating oil? 15. The speed of a sailboat is limited by the drag of water on the hull. At low speeds, you can use a laminar flow model to estimate drag. The water touching, or wetting the hull moves with the boat. Successive layers of water slide against each other, with each layer moving at a lower speed than previous layers. At a distance of approximately 0.3 mm, water does not slide in layers. You can use this distance as the thickness of the fluid layer in the viscosity equation. (a) When moving at constant velocity, a sailboat has a wetted hull surface area of 21.5 m 2. The sails apply a force of 223 N in the direction the boat is moving. What is the speed of the boat? (Assume the water s temperature is 20 C.) (b) If the sailboat has a winged keel that reduces the wetted surface by 15%, what is the speed of the boat? SECTION 4.2 RESISTANCE IN FLUID SYSTEMS 199