So far, we have concentrated on developing a method for modeling fluids. We established the idea of the continuum to allow us to describe flows with continuous functions, and we introduced the use of the control volume (CV) to free us from having to follow individual fluid particles (see Chapter 3). We described a method of accounting for stresses in fluids (see Chapter 4) and showed how stress and motion are related through the constitutive equation (see Chapter 5). We developed a solution methodology that led to the microscopic balance equations and, finally, to solutions of simple flow problems (see Chapter 6). We have all of the tools necessary to solve flow problems, and we now turn to the task of modeling and understanding the flow behaviors described in Chapter 2.

In this chapter, we concentrate on internal flows, which are flows through closed conduits. Chapter 8 discusses both external flows, in which fluid moves over or around obstacles, and an important class of flows called boundary-layer flows.

In this chapter and in Chapter 8, we address complex, realistic, and practical flow problems with our modeling methods. We begin the analysis of complex problems with a microscopic analysis on an idealized system.

Our task is to learn to model flows. To set up the models, we draw on our intuition of how fluids behave; for example, we often can guess the direction that a flow takes under the influence of particular forces. Intuition also may enable us to identify symmetries in a flow field. Intuition comes from experience, however, and for introductory students, experience may be in short supply.

One solution to a lack of experience is to experiment with fluids. Unfortunately, not all of us have access to pumps, flow meters, and piping systems; therefore, it is worthwhile to take a laboratory course in fluid mechanics, if possible. Another way to build experience with fluid behavior is to view flow-visualization videos. Between 1961 and 1969, a group of experts in fluid mechanics (the National Committee for Fluid Mechanics Films [NCFMF]) produced a series of flow- demonstration films [112] that introduce fluid behavior; the films and film notes are now available on the Internet. There also are books [170] and other media [65] that catalog fluid behavior, as well as Web sites on which researchers have posted flow-visualization videos, including the Gallery of Fluid Motion [133], and elsewhere [182]. These sites bring to life all types of fluid behavior, from the mundane to the esoteric.

The mass- and momentum-balance techniques described in Chapters 3–5 are general and apply to any control volume (CV). We apply those techniques to a general microscopic control volume in Chapter 6 and use the microscopic balances in Chapters 7 and 8. Microscopic-control-volume calculations yield the equations that govern three-dimensional velocity and stress fields. If the equations can be solved, the information that microscopic balances provide is complete. Solving the microscopic balances is difficult, however, because the continuity equation and the Navier-Stokes equation are a set of four nonlinear, coupled, partial differential equations (PDEs).

For many fluids problems, the information sought is relatively large scale and flow details are not very important. For these problems—such as the calculation of the total force on a wall; overall flow rate in a device; and the total work performed by a pump, a turbine, or a mixer-balancing on a larger CV can be a fast and simple way to arrive at quantities of interest. Macroscopic CV balances are mathematically easier to calculate than microscopic CV balances, although they generally require information that must be determined experimentally.

In this chapter, we derive and learn to use the macroscopic mass, momentum, and energy balances, including the mechanical energy balance (MEB), which is discussed in Chapter 1.

Chapter 2 describes fluid behaviors, and Chapters 1 and 2 introduce basic fluids calculations. We turn now to developing a modeling method that allows us to understand fluids behavior in detail.

Fluids move and deform in predictable ways that are governed by the laws of physics. To apply the laws of physics to fluids, we must develop a mathematical picture or model of fluid motion. With an effective model, we can predict fluid patterns and stresses and apply these predictions to engineering calculations.

To build up the fluid model that we use, we begin with a reminder about how to calculate the motions of individual rigid bodies. To apply these methods to fluids, we then introduce the continuum model, a mathematical picture of fluids in which we consider small packets of fluid to be individual bodies. We discuss how we apply the laws of physics to these small fluid packets or particles to deduce velocities and forces for the fluid particles. Finally, we introduce the control volume, a point of view used for fluid modeling that focuses our calculations on a physical region in space rather than on individual bodies in motion. This difference in strategy—that is, considering a control volume rather than individual bodies—is a key difference between the modeling techniques of fluid mechanics and those of solid-body mechanics.

In Chapter 7, we applied analysis methods to flows inside pipes and other closed conduits. We started with a practical challenge of estimating the extent of a home flood and developed our solution method by thinking about that problem from various angles (Figure 8.1). We first decided on the goal of our analysis; then, star ting with the simplest models, we systematically investigated flows of increasing complexity until we found a solution to the burst-pipe problem through dimensional analysis and data correlations. This protocol is general, and it can be applied to other flows, as demonstrated in this chapter.

We turn now to external flows. External flow is a term used to describe flows over or around obstacles. The wind blowing on a skyscraper is an example of an external flow (see Example 2.5), as is an electric fan cooling a printed circuit board in a computer or a cleaning jet directed past the fender of a freshly painted automobile. Objects moving through fluids also create external flows (see Figure 2.11). Ships on the ocean, mixing blades in viscous liquids, and skydivers (Figure 8.2) are all operating in external flows. External flows are not unidirectional, steady flows; thus, both inertia and viscosity affect flow behavior.

In this text, our goal is to explain flow; this chapter surveys how far we have come. With the completion of nine chapters of study, we find that we can make sense of much of the fluid behavior we observe. A reexamination of those behaviors helps consolidate our knowledge.

Section 10.1 is an integrated summary of the concepts of viscosity, drag, and boundary layers. Section 10.2 provides guidance on how numerical tools are used to pursue advanced flow-field models. In Section 10.3, we turn to turbulent flow, which until now was addressed only through data correlations for friction factor and drag coefficient. Sophisticated applications involving turbulent flow (e.g., airplane flight, mixing, and reactor design) require more detailed understanding of turbulent flow structure than discussed so far. We introduce the statistical study of turbulence in Section 10.3. Lift-briefly introduced in Chapter 8—is studied most effectively with advanced tools such as vorticity and circulation (Section 10.4). Section 10.5 continues with vorticity to show how this tool improves our understanding of curvy flow. Compressible fluid flow is discussed in Section 10.6. The flow behaviors not addressed in this text are accessible through advanced study based on the introductory methods in Chapters 1–9.

This book forms the basis of a one-semester introductory course in fluid mechanics for engineers and scientists. Students working with this text are expected to have a background in multivariable calculus, linear algebra, and differential equations; review of these topics as applied to fluid mechanics is provided in Chapter 1. Problem solving is taught by example throughout the text. We include numerous solved examples and end-of-chapter problems, and a complete solution manual is available for instructors.

Fluid mechanics can be a difficult subject. Nonlinear physics governs flow, and thus we often resort to a variety of simplifications to obtain solutions. Different simplifications are used under different conditions, making fluid mechanics intimidating, at least to a beginner. An Introduction to Fluid Mechanics presents the topic through a discovery process, as described in this preface, that mimics engineering practice. The process used seeks solutions by answering the following questions:

1. What is the problem?

2. What do we need to know, and do, to address the problem?

3. What is the solution to the problem?

4. What other problems/opportunities may be addressed now that we have solved this problem?

This organizational choice builds critical thinking skills by emphasizing the thought processes that lead to model development. The book is divided into four parts that answer these four questions for the study of fluid mechanics.

Getting motivated

Flows are beautiful and complex. A swollen creek tumbles over rocks and through crevasses, swirling and foaming. A child plays with sticky taffy, stretching and reshaping the candy as she pulls and twists it in various ways. Both the water and the taffy are fluids, and their motions are governed by the laws of nature. Our goal is to introduce readers to the analysis of flows using the laws of physics and the language of mathematics. On mastering this material, readers can harness flow to practical ends or create beauty through fluid design.

In this text we delve into the mathematical analysis of flows; however, before beginning, it is reasonable to ask if it is necessary to make this significant mathematical effort. After all, we can appreciate a flowing stream without understanding why it behaves as it does. We also can operate machines that rely on fluid behavior—drive a car, for example—without understanding the fluid dynamics of the engine. We can even repair and maintain engines, piping networks, and other complex systems without having studied the mathematics of flow. What is the purpose, then, of learning to mathematically describe fluid behavior?

The answer is quite practical: Knowing the patterns that fluids form and why they are for med, and knowing the stresses that fluids generate and why they are generated, is essential to designing and optimizing modern systems and devices.