Significant Learning Outcom ... http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... C H A P T E R Introduction 1 SIG(cid:20)IFICA(cid:20)T LEAR(cid:20)I(cid:20)G OUTCOMES Fluid mechanics applies concepts related to force and energy to practical problems such as the design of gliders. (Photo courtesy of DG Flugzeugbau GmbH.) Conceptual Knowledge · Describe fluid mechanics. · Contrast gases and liquids by describing similarities and differences. · Explain the continuum assumption. Procedural Knowledge · Use primary dimensions to check equations for dimensional homogeneity. · Apply the grid method to carry and cancel units in calculations. · Explain the steps in the “Structured Approach for Engineering Analysis” (see Table 1.4). Prior to fluid mechanics, students take courses such as physics, statics, and dynamics, which involve solid mechanics. Mechanics is the field of science focused on the motion of material bodies. Mechanics involves force, energy, motion, deformation, and material properties. When mechanics applies to material bodies in the solid phase, the discipline is called solid mechanics. When the material body is in the gas or liquid phase, the discipline is called fluid mechanics. In contrast to a 1 of 2 1/14/2009 9:57 PM Significant Learning Outcom ... http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... solid, a fluid is a substance whose molecules move freely past each other. More specifically, a fluid is a substance that will continuously deform—that is, flow under the action of a shear stress. Alternatively, a solid will deform under the action of a shear stress but will not flow like a fluid. Both liquids and gases are classified as fluids. This chapter introduces fluid mechanics by describing gases, liquids, and the continuum assumption. This chapter also presents (a) a description of resources available in the appendices of this text, (b) an approach for using units and primary dimensions in fluid mechanics calculations, and (c) a systematic approach for problem solving. Copyright © 2009 John Wiley & Sons, Inc. All rights reserved. 2 of 2 1/14/2009 9:57 PM Liquids and Gases http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... 1.1 Liquids and Gases This section describes liquids and gases, emphasizing behavior of the molecules. This knowledge is useful for understanding the observable characteristics of fluids. Liquids and gases differ because of forces between the molecules. As shown in the first row of Table 1.1, a liquid will take the shape of a container whereas a gas will expand to fill a closed container. The behavior of the liquid is produced by strong attractive force between the molecules. This strong attractive force also explains why the density of a liquid is much higher than the density of gas (see the fourth row). The attributes in Table 1.1 can be generalized by defining a gas and liquid based on the differences in the attractive forces between molecules. A gas is a phase of material in which molecules are widely spaced, molecules move about freely, and forces between molecules are minuscule, except during collisions. Alternatively, a liquid is a phase of material in which molecules are closely spaced, molecules move about freely, and there are strong attractive forces between molecules. Table 1.1 COMPARISO(cid:25) OF SOLIDS, LIQUIDS, A(cid:25)D GASES Attribute Solid Liquid Gas Typical Visualization Macroscopic Solids hold their shape; Liquids take the shape of Gases expand to fill a closed Description no need for a container the container and will container stay in open container Mobility of Molecules have low Liquids typically flow Molecules move around Molecules mobility because they are easily even though there freely with little interaction bound in a structure by are strong intermolecular except during collisions; this strong intermolecular forces between molecules is why gases expand to fill forces their container Typical Density Often high; e.g., density Medium; e.g., density of Small; e.g., density of air at 3 3 3 of steel is 7700 kg/m water is 1000 kg/m sea level is 1.2 kg/m Molecular Small—molecules are Small—molecules are Large—on average, Spacing close together held close together by molecules are far apart intermolecular forces Effect of Shear Produces deformation Produces flow Produces flow Stress Effect of Produces deformation Produces deformation Produces deformation (cid:25)ormal Stress that may associate with associated with volume associated with volume volume change; can cause change change failure Viscosity NA High; decreases as Low; increases as temperature increases temperature increases Compressibility Difficult to compress; Difficult to compress; Easy to compress; bulk bulk modulus of steel is bulk modulus of liquid modulus of a gas at room 160 × 109 Pa water is 2.2 × 109 Pa conditions is about 1 of 2 1/14/2009 10:00 PM Liquids and Gases http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... Attribute Solid Liquid Gas 5 1.0 × 10 Pa Copyright © 2009 John Wiley & Sons, Inc. All rights reserved. 2 of 2 1/14/2009 10:00 PM The Continuum Assumption http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... 1.2 The Continuum Assumption This section describes how fluids are conceptualized as a continuous medium. This topic is important for applying the derivative concept to characterize properties of fluids. While a body of fluid is comprised of molecules, most characteristics of fluids are due to average molecular behavior. That is, a fluid often behaves as if it were comprised of continuous matter that is infinitely divisible into smaller and smaller parts. This idea is called the continuum assumption. When the continuum assumption is valid, engineers can apply limit concepts from differential calculus. Recall that a limit concept, for example, involves letting a length, an area, or a volume approach zero. Because of the continuum assumption, fluid parameters such as density and velocity can be considered continuous functions of position with a value at each point in space. To gain insight into the validity of the continuum assumption, consider a hypothetical experiment to find density. Fig. 1.1a shows a container of gas in which a volume has been identified. The idea is to find the mass of the molecules ∆M inside the volume and then to calculate density by The calculated density is plotted in Fig. 1.1b. When the measuring volume is very small (approaching zero), the number of molecules in the volume will vary with time because of the random nature of molecular motion. Thus, the density will vary as shown by the wiggles in the blue line. As volume increases, the variations in calculated density will decrease until the calculated density is independent of the measuring volume. This condition corresponds to the vertical line at . If the volume is too large, as shown by , then the value of density may change due to spatial variations. Figure 1.1 When a measuring volume is large enough for random molecular effects to average out, the continuum assumption is valid In most applications, the continuum assumption is valid. For example, consider the volume needed to contain at least a million (106) molecules. Using Avogadro's number of 6 × 1023 molecules/mole, the limiting volume for -13 3 -4 water is 10 mm , which corresponds to a cube less than 10 mm on a side. Since this dimension is much smaller than the flow dimensions of a typical problem, the continuum assumption is justified. For an ideal gas 6 (1 atm and 20°C) one mole occupies 24.7 liters. The size of a volume with more than 10 molecules would be 10-10 mm3, which corresponds to a cube with sides less than 10-3 mm (or one micrometer). Once again this size is much smaller than typical flow dimensions. Thus, the continuum assumption is usually valid in gas flows. The continuum assumption is invalid for some specific applications. When air is in motion at a very low density, such as when a spacecraft enters the earth's atmosphere, then the spacing between molecules is significant in comparison to the size of the spacecraft. Similarly, when a fluid flows through the tiny passages in nanotechnology devices, then the spacing between molecules is significant compared to the size of these 1 of 2 1/14/2009 10:01 PM Dimensions, Units, and Resources http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771... 1.3 Dimensions, Units, and Resources This section describes the dimensions and units that are used in fluid mechanics. This information is essential for understanding most aspects of fluid mechanics. In addition, this section describes useful resources that are presented in the front and back of this text. Dimensions A dimension is a category that represents a physical quantity such as mass, length, time, momentum, force, acceleration, and energy. To simplify matters, engineers express dimensions using a limited set that are called primary dimensions. Table 1.2 lists one common set of primary dimensions. Table 1.2 PRIMARY DIME(cid:31)SIO(cid:31)S Dimension Symbol Unit (SI) Length L meter (m) Mass M kilogram (kg) Time T second (s) Temperature θ kelvin (K) Electric current i ampere (A) Amount of light C candela (cd) Amount of matter (cid:17) mole (mol) Secondary dimensions such as momentum and energy can be related to primary dimensions by using equations. For example, the secondary dimension “force” is expressed in primary dimensions by using Newton's second 2 law of motion, F = ma. The primary dimensions of acceleration are L/T , so (1.1) In Eq. 1.1, the square brackets mean “dimensions of.” This equation reads “the primary dimensions of force are mass times length divided by time squared.” Note that primary dimensions are not enclosed in brackets. Units While a dimension expresses a specific type of physical quantity, a unit assigns a number so that the dimension can be measured. For example, measurement of volume (a dimension) can be expressed using units of liters. Similarly, measurement of energy (a dimension) can be expressed using units of joules. Most dimensions have multiple units that are used for measurement. For example, the dimension of “force” can be expressed using units of newtons, pounds-force, or dynes. Unit Systems In practice, there are several unit systems in use. The International System of Units (abbreviated SI from the 1 of 2 1/14/2009 11:57 PM