# 1.1: Preface

- Page ID
- 18050

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This book began as lecture notes for an Oregon State University course in fluid mechanics, designed for beginning graduate students in physical oceanography. Because of its fundamental nature, this course is often taken by students outside physical oceanography, e.g., atmospheric science, civil engineering, physics and mathematics.

In later courses, the student will discover esoteric fluid phenomena such as internal waves that propagate through the sky, water phase changes that govern clouds, and planetary rotation effects that control large-scale winds and ocean currents. In contrast, this course concerns phenomena that we have all been familiar with since childhood: flows you see in sinks and bathtubs, in rivers, and at the beach. In this context, we develop the mathematical techniques and scientific reasoning skills needed for higher-level courses and professional research.

Prerequisites are few: basic linear algebra, differential and integral calculus and Newton’s laws of motion. As we go along we discover the need for the more advanced tools of tensor analysis.

The science of fluid mechanics is vast. Most books on the topic are concerned with technological applications, e.g., flow through pipes and machinery, that have little relevance in nature. But even among naturally occurring flows we cannot, and should not, try to cover everything. What I have done here is to identify three canonical flow structures that are common in nature (Figure \(\PageIndex{1}\)):

- vortices
- waves
- hydraulic jumps

The inner workings of these three phenomena involve all of the basic flow processes, and their study demands a thorough understanding of the theory. The goal, then, is to learn what we need to know to thoroughly understand vortices, waves and hydraulic jumps. Master this, and you will be well prepared to study the much broader range of fluid phenomena found in nature.

The discussion is in three, roughly equal parts:

- Chapters 2-4 are mainly mathematical; we review some advanced aspects of linear algebra and then develop the tools of tensor analysis.
- Chapters 5 and 6 are the crux. The mathematical tools from part 1 are used to develop a theoretical description of flow. The development is thorough, rigorous and (I hope) intuitive.
- In chapters 7-9, we apply the theory to our three common phenomena. Besides exercising the analytical skills we have developed, these examples allow us to test the assumptions that underlie the theory by comparing the results with our everyday experience. In some cases we find that the theory is inadequate, laying the groundwork for further study.

Homework exercises are included (chapter 12) and are integral to the course. The main text is designed to be covered in 40 hours of lectures. Appendices give auxiliary information and additional topics that can be covered or assigned. It is expected that students will devote an addition 80 hours to homework and independent study. Instructors are invited to contact the author (smythw@oregonstate.edu) for additional materials such as suggested assignments, solutions and possible exam questions.

This book is also intended for self-study, with detailed explanations and frequent exercises to confirm your understanding. If you take this route, feel free to email me with any questions that may arise.

We will not shy away from proofs of the mathematical results we encounter; indeed, I expect the student to demand them. Professionals with graduate degrees are expected not only to know facts, but to understand why they are true and how they came to be known. Be skeptical. To believe something just because a professor said it is to invite error. As a young student, I was taught that the continents do not move and that the planet Mercury always keeps the same face toward the Sun, two statements that we now know are untrue. I absolutely guarantee that, at some point in each student’s education, and perhaps in this book, a “fact” will be learned that will turn out to be total hogwash. Be on the lookout.

Smaller errors in logic or mathematics turn up all the time. I have frequently had a math error corrected, or learned a clearer way to explain a difficult concept, thanks to an alert student. That is true in every course but perhaps more so in this one, because every student arrives with an intuitive feel for the fluid phenomena that motivate the analysis. If you have taken this course, thank you. I have learned from you.

There is yet room for improvement - if you spot something wrong or unclear, please let me know and I will fix it.

Bill Smyth, December, 2019