Chapter 1

Introduction

Abstract

Fluids, materials that deform continuously under an applied shear stress, are omnipresent in the world around us, and beyond. Fluid mechanics is the branch of science concerned with moving and stationary fluids. Here fluids are treated as being continuous even though their substance is discrete at the molecular level. At the macroscopic level, the molecular character of fluids is manifested as diffusive transport of species, heat, and momentum. With the continuum approximation, the dependent field variables of velocity, pressure, density, temperature, etc. are considered to be well defined at every point in space and classical thermodynamics is applied even when the fluid is not perfectly in equilibrium. In static situations, gravity and thermodynamic gradients in the fluid determine whether or not the situation is stable to small perturbations. When fluids move, they obey Newton’s second law but there is no restriction of the system of units used to describe the relevant forces and accelerations. This fact and the requirement for dimensional homogeneity in physically meaningful equations allows potentially useful scaling laws to be developed from considerations of the dimensions of relevant parameters, fluid properties, and fundamental constants.

Keywords

Definition of a fluid; Continuum hypothesis; Fluid particle; Molecular transport; Fluid statics; Thermodynamics; Stratified fluids; Static stability; Dimensional analysis

Outline

1.1 Fluid Mechanics 2

1.2 Units of Measurement 3

1.3 Solids, Liquids, and Gases 4

1.4 Continuum Hypothesis 5

1.5 Molecular Transport Phenomena 7

1.6 Surface Tension 11

1.7 Fluid Statics 13

1.8 Classical Thermodynamics 15

1.9 Perfect Gas 20

1.10 Stability of Stratified Fluid Media 21

1.11 Dimensional Analysis 26

Exercises 35

Literature Cited 48

Supplemental Reading 48

Chapter Objectives

• To properly introduce the subject of fluid mechanics and its importance

• To state the assumptions upon which the subject is based

• To review the basic background science of liquids and gases

• To present the relevant features of fluid statics

• To establish dimensional analysis as an intellectual tool for use in the remainder of the text

1.1. Fluid Mechanics

Fluid mechanics is the branch of science concerned with moving and stationary fluids. Given that the vast majority of the observable mass in the universe exists in a fluid state, that life as we know it is not possible without fluids, and that the atmosphere and oceans covering this planet are fluids, fluid mechanics has unquestioned scientific and practical importance. Its allure crosses disciplinary boundaries, in part because it is described by a nonlinear field theory and also because fluid phenomena are readily observed. Mathematicians, physicists, biologists, geologists, oceanographers, atmospheric scientists, engineers of many types, and even artists have been drawn to study, harness, and exploit fluid mechanics to develop and test formal and computational techniques, to better understand the natural world, and to attempt to improve the human condition. The importance of fluid mechanics cannot be overstated for applications involving transportation, power generation and conversion, materials processing and manufacturing, food production, and civil infrastructure. For example, in the twentieth century, life expectancy in the United States approximately doubled. About half of this increase can be traced to advances in medical practice, particularly antibiotic therapies. The other half largely resulted from a steep decline in childhood mortality from water-borne diseases, a decline that occurred because of widespread delivery of clean water to nearly the entire population – a fluids-engineering and public-works achievement. Yet, the pursuits of mathematicians, scientists, and engineers are interconnected: engineers need to understand natural phenomena to be successful, scientists strive to provide this understanding, and mathematicians pursue the formal and computational tools that support these efforts.

Advances in fluid mechanics, like any other branch of physical science, may arise from mathematical analyses, computer simulations, or experiments. Analytical approaches are often successful for finding solutions to idealized and simplified problems and such solutions can be of immense value for developing insight and understanding, and for comparisons with numerical and experimental results. Thus, some fluency in mathematics, especially multivariable calculus, is helpful in the study of fluid mechanics. In practice, drastic simplifications are frequently necessary to find analytical solutions because of the complexity of real fluid flow phenomena. Furthermore, it is probably fair to say that some of the greatest theoretical contributions have come from people who depended rather strongly on their physical intuition. Ludwig Prandtl, one of the founders of modern fluid mechanics, first conceived the idea of a boundary layer based solely on physical intuition. His knowledge of mathematics was rather limited, as his famous student Theodore von Karman (1954, page 50) testifies. Interestingly, the boundary layer concept has since been expanded into a general method in applied mathematics.

As in other scientific fields, mankind’s mathematical abilities are often too limited to tackle the full complexity of real fluid flows. Therefore, whether we are primarily interested in understanding flow physics or in developing fluid-flow applications, we often must depend on observations, computer simulations, or experimental measurements to test hypotheses and analyses, and to develop insights into the phenomena under study. This book is an introduction to fluid mechanics that should appeal to anyone pursuing fluid mechanical inquiry. Its emphasis is on fully presenting fundamental concepts and illustrating them with examples drawn from various scientific and engineering fields. Given its finite size, this book provides – at best – an incomplete description of the subject. However, the purpose of this book will have been fulfilled if the reader becomes more curious and interested in fluid mechanics as a result of its perusal.