Fluid Mechanics with teacher's notes

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$Holt Math Minnesota Test Prep Workbook for Grade 9$

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transformed into internal energy because of the internal friction. Ideal fluids are considered nonviscous, so they lose no kinetic energy due to friction as they flow. Ideal fluids are also characterized by a steady flow. In other words, the velocity, density, and pressure at each point in the fluid are constant. The flow of an ideal fluid is also nonturbulent, which means that there are no eddy currents in the moving fluid. Although no real fluid has all the properties of an ideal fluid, the ideal fluid model does help explain many properties of real fluids, so the model is a useful tool for analysis. Unless otherwise stated, the fluids in the rest of our discussion of fluid flow will be treated as ideal fluids. PRINCIPLES OF FLUID FLOW Fluid behavior is often very complex. A detailed analysis of the forces acting on a fluid may be so complicated that even a supercomputer cannot create an accurate model. However, several general principles describing the flow of fluids can be derived relatively easily from basic physical laws. The continuity equation results from mass conservation Imagine that an ideal fluid flows into one end of a pipe and out the other end, as shown in Figure 9-11. The diameter of the pipe is different on each end. How does the speed of fluid flow change as the fluid passes through the pipe? Because mass is conserved and because the fluid is incompressible, we know that the mass flowing into the bottom of the pipe, m 1, must equal the mass flowing out of the top of the pipe, m 2, during any given time interval: m1 = m2 This simple equation can be expanded by recalling that m = rV and by using the formula for the volume of a cylinder, V = A∆x. r1V1 = r2V2 r1A1∆x1 = r2A2∆x2 The length of the cylinder, ∆x, is also the distance the fluid travels, which is equal to the speed of flow multiplied by the time interval (∆x = v∆t). r 1A 1v 1∆t = r 2A 2v 2∆t For an ideal fluid, both the time interval and the density are the same on each side of the equation, so they cancel each other out. The resulting equation is called the continuity equation: CONTINUITY EQUATION A 1v 1 = A 2v 2 area speed in region 1 area speed in region 2 Copyright © by Holt, Rinehart and Winston. All rights reserved. Motor-oil labels include a viscosity index based on standards set by the Society of Automotive Engineers (SAE). High-viscosity oils (high SAE numbers) are meant for use in hot climates and in high-speed driving. Low-viscosity oils (low SAE numbers) are meant for use in severe winter climates, when the engine is cold for much of the time. A1 v1 ∆x1 Fluid Mechanics v2 ∆x2 A2 Figure 9-11 The mass flowing into the pipe must equal the mass flowing out of the pipe in the same time interval. 333 SECTION 9-3 Teaching Tip The viscosity index on motor oil cans consists of an SAE number (typically from 5 to 50) followed by the letter W. The higher the SAE number, the more viscous the oil is. Low-viscosity oils are meant for use in severe winter climates because low-viscosity oils flow more easily in cold temperatures. For hot or high-speed driving conditions, a high-viscosity oil can be used because the excessive heat effectively thins the oil. Visual Strategy Figure 9-11 Point out that the volumes of the two shaded sections of the pipe must be equal in order for the continuity equation to apply. Assume A1 is smaller than Q pictured in Figure 9-11. Would ∆x1 need to be longer or shorter for the mass of liquid in each section to still be equal? A longer 333
SECTION 9-3 Misconception STOP Alert One might think that the water coming out of the hose is at a higher pressure than the water in the hose, but the opposite is true. The pressure outside the hose is atmospheric pressure, while the pressure inside the hose is higher than atmospheric pressure. In fact, the water flows out of the hose because the greater pressure in the hose pushes it out. Teaching Tip The tilt of an airplane wing also adds to the lift on the plane. The front of the wing is tilted upward so that air striking the bottom of the wing is deflected downward. At the same time, the deflected air produces an upward reaction force on the wing, contributing substantially to the lift. 334 Figure 9-12 Placing your thumb over the end of a garden hose reduces the area of the opening and increases the speed of the water. 334 P 1 v 1 Chapter 9 P 2 v 2 Figure 9-13 As a leaf passes into a constriction in a drainage pipe, the leaf speeds up. Pressure gauges show that the water pressure on the right is less than the pressure on the left. F Figure 9-14 As air flows around an airplane wing, the air above the wing moves faster than the air below, producing lift. a The speed of fluid flow depends on cross-sectional area Note in the continuity equation that A 1 and A 2 can represent any two different cross-sectional areas of the pipe, not just the ends. This equation implies that the fluid speed is faster where the pipe is narrow and slower where the pipe is wide. The product Av, which has units of volume per unit time, is called the flow rate. The flow rate is constant throughout the pipe. The continuity equation explains an effect you may have experienced when placing your thumb over the end of a garden hose, as shown in Figure 9-12. Because your thumb blocks some of the area through which the water can exit the hose, the water exits at a higher speed than it would otherwise. The continuity equation also explains why a river tends to flow more rapidly in places where the river is shallow or narrow than in places where the river is deep and wide. The pressure in a fluid is related to the speed of flow Suppose there is a water-logged leaf carried along by the water in a drainage pipe, as shown in Figure 9-13. The continuity equation shows that the water moves faster through the narrow part of the tube than through the wider part of the tube. Therefore, as the water carries the leaf into the constriction, the leaf speeds up. If the water and the leaf are accelerating as they enter the constriction, an unbalanced force must be causing the acceleration, according to Newton’s second law. This unbalanced force is a result of the fact that the water pressure in front of the leaf is less than the water pressure behind the leaf. The pressure difference causes the leaf and the water around it to accelerate as it enters the narrow part of the tube. This behavior illustrates a general principle, known as Bernoulli’s principle, which can be stated as follows: BERNOULLI’S PRINCIPLE The pressure in a fluid decreases as the fluid’s velocity increases. The lift on an airplane wing can be explained, in part, with Bernoulli’s principle. As an airplane flies, air flows around the wings and body of the plane, as shown in Figure 9-14. Airplane wings are designed to direct the flow of air so that the air speed above the wing is greater than the air speed below the wing. Therefore, the air pressure above the wing is less than the pressure below, and there is a net upward force on the wing, called lift. Bernoulli’s equation relates pressure to energy in a moving fluid Imagine a fluid moving through a pipe of varying cross-section and elevation, as shown in Figure 9-15. As the fluid flows into regions of different cross-sectional area, the pressure and speed of the fluid along a given streamline in the pipe can change. However, if the speed of a fluid changes, then the fluid’s kinetic energy also changes. The change in kinetic energy may be compensated for by a change in gravitational potential energy or by a change in pressure so energy is still conserved. Copyright © by Holt, Rinehart and Winston. All rights reserved.
Page 1 and 2: Chapter 9 Overview Section 9-1 defi
Page 3 and 4: Section 9-1 Demonstration 1 Volume
Page 5 and 6: SECTION 9-1 The Language of Physics
Page 7 and 8: SECTION 9-1 Demonstration 4 Lifting
Page 9 and 10: SECTION 9-1 PRACTICE GUIDE 9A Solvi
Page 11 and 12: SECTION 9-2 STOP 326 Misconception
Page 13 and 14: SECTION 9-2 Visual Strategy Figure
Page 15 and 16: SECTION 9-2 Classroom Practice The
Page 17: Section 9-3 Demonstration 7 Fluid f
Page 23 and 24: Section 9-4 Demonstration 10 Temper
Page 27 and 28: Chapter 9 Summary Teaching Tip Ask
Page 29 and 30: 9 REVIEW & ASSESS 12. because the p
Page 31 and 32: 9 REVIEW & ASSESS 42. 7.1 m 43. 17
Page 33 and 34: 9 REVIEW & ASSESS 63. 60.9 m/s 2 64
Page 35 and 36: Chapter 9 Laboratory Exercise NOTE
Page 37 and 38: CHAPTER 9 LAB Boyle’s Law Apparat

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