Preamble

For us, the material presented in Chapters 1 through 6 of this book is a preamble: the material allows us to solve problems that arise in the analysis of physical problems, which so often end in partial differential equations. It is true that the analysis of Fourier transforms, for example, or eigenfunction expansions are, in themselves, interesting mathematical pursuits. However, our reason for studying what has come before this chapter, and the motivation for most applied mathematicians, is pragmatic: We are thereby enabled to approach solutions to those very physical problems we wish to solve.

With the advent and now exploding use of tools, such as RANS, DNS, and LES for solving the Navier–Stokes equations numerically, one might presume that the methodologies of this book are out of date. However, it has been our experience in our years of fluid dynamics research that the cross–fertilization of numerics and analysis, functioning synergistically alongside each other, provides insights into physical problems that are not available from either one standing alone. So, this chapter and Chapter 8 present some relatively simple, but real–world problems, that use more than one method from the previous chapters.

Lee Waves

We now turn to a problem that is important in atmospheric flows, namely the standing gravity waves downstream of mountains, known as Lee waves. Early work on Lee waves may be found in (Janowitz) and (Miles), for example; an excellent, recent summary of the topic is in (Wurtele, et al).

In this book, it is our intent to equip graduate students in applied mathematics and engineering with a range of classical analytical methods for the solution of partial differential equations. In our research specialties, numerical methods, on the one hand, and perturbation and variational methods, on the other, constitute contemporary tools that are explicitly not covered in this book, since there are significant books that are devoted to those topics specifically.

This book grew not from the authors' desire to write a textbook but rather from many years for each of us in compiling notes to be distributed to our graduate students in courses devoted to the solution of partial differential equations. One of us taught mostly engineers (MRF at The Ohio State University), and the other (IH at Howard University and later at Rensselaer Polytechnic Institute) mostly mathematicians. It surprised us to learn, on becoming re–acquainted at RPI, how similar are our perspectives about this material, and in particular the level of rigor with which it ought to be presented. Further, both of us wanted to create a book that would include many of the techniques that we have learned one way or another but are quite simply not in books.

The topics chosen for the book are those that we have found to be of considerable use in our own research careers. These are topics that are applicable in many areas, such as aeronautics and astronautics; biomechanics; chemical, civil, and mechanical engineering fluid mechanics; and geophysical flows.

As noted in §1.5.1, a reference frame consists of a spatial coordinate system and clock. The principle of material frame indifference requires constitutive equations to be invariant or unaffected by an arbitrary time-dependent translation, rotation, and reflection of the coordinate axes and by an arbitrary translation of the time axis. In contrast, constitutive equations expressed in tensor form are invariant under time-independent transformations of the coordinate axes, but they are not necessarily invariant under time-dependent transformations.

The basis for frame indifference is the intuitive idea that the response of a material should be independent of the motion of the observer (Oldroyd, 1950; Noll, 1955, p. 45; Noll, 1959). The principle cannot be proved, but simple examples have been given (Truesdell and Noll, 1965; Hunter, 1983, p. 123) where its validity appears plausible. An example taken from Hunter (1983, p. 123) is discussed below.

A pilot bails out of an aircraft and opens his parachute. Suppose that the force P exerted by the parachute on the pilot is measured by the extension of a spring attached to the harness of the parachute. Consider two reference frames, one fixed to the pilot and the other to an observer standing on the ground. Suppose that the spring is visible in both the frames. When relativistic effects are ignored, it is usually implicitly assumed that at any time t, the distance between two points in space has the same value regardless of the motion of the reference frame (see, e.g., Resnick, 1968, p. 5). This assumption implies that the pilot and the observer on the ground will record the same extension for the spring.

A granular material is a collection of solid particles or grains, such that most of the particles are in contact with at least some of their neighboring particles. The terms “granular materials,” “bulk solids,” “particulate solids,” and “powders” are often used interchangeably in the literature. Common examples of granular materials are sand, gravel, food grains, seeds, sugar, coal, and cement. Figure 1.1 shows the typical size ranges for some of these materials.

Granular materials are commonly encountered in nature and in various industries. For example, with reference to the chemical industry, Ennis et al. (1994) note that about 40% of the value added is linked to particle technology. Similarly, Bates (2006) notes that more than 50% of all products sold are either granular in form or involve granular materials in their production. In spite of the importance of granular materials, their mechanics is not well understood at present. Nevertheless, some progress has been made during the past few decades. The goal of this book is to describe some of the experimental observations and models related to the mechanical behavior of flowing granular materials. As studies in this area are increasing rapidly, our account is necessarily incomplete. However, it is hoped that the book will provide a useful starting point for the beginning student or researcher.

A material is called a dry granular material if the fluid in the interstices or voids between the grains is a gas, which is usually air. On the other hand, if the voids are completely filled with a liquid such as water, the material is called a saturated granular material.