The primary goal of this chapter is to convert verbal notions that embody the basic axioms of nonrelativistic continuum mechanics into usable mathematical expressions. They will have generality beyond fluid mechanics in that they apply to any continuum material, for example solids. First, we must list those axioms. These axioms will speak to the evolution in time of mass, linear momenta, angular momenta, energy, and entropy.

Here we give an introduction to topics in geometry that will be relevant to the mechanics of fluids. More specifically, we will consider elementary aspects of differential geometry. Geometry can be defined as the study of shape, and differential geometry connotes that methods of calculus will be used to study shape. It is well known that fluids in motion may transform location and shape, such as shown in Fig. 2.1.

In this chapter, we consider a variety of topics related to the governing equations as a system. We briefly discuss boundary and interface conditions, necessary for a complete system, summarize the partial differential equations in various forms, present some special cases of the governing equations, present the equations in a dimensionless form, and consider a few cases for which the linear momenta equation can be integrated once.

In this chapter we will consider the kinematics and dynamics of fluid elements rotating about their centers of mass. Such an element is often described as a vortex, and is a commonly seen in fluids. However, a precise definition of a vortex is difficult to formulate. Rotating fluids may be observed, among other places, in weather patterns and airfoil wakes.

Here we consider some basic problems in one-dimensional viscous flow. Application areas range from ordinary pipe flow to microscale fluid mechanics, such as found in micro-electronic or biological systems. A typical scenario is shown in Fig. 10.1. We will select this and various problems that illustrate the effects of advection, diffusion, and unsteady effects.

In this chapter, we briefly introduce how to apply methods from the discipline of nonlinear dynamical systems to fluid flow. The governing equations for a fluid are a nonlinear system of partial differential equations with space and time as independent variables. Here we will adopt methods to rationally reduce the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations.

Here we advance from geometry to consider kinematics, the study of motion in space. We will not yet consider forces that cause the motion. If we knew the position of every fluid particle as a function of time, we could also describe the velocity and acceleration of each particle. We could also make statements about how groups of particles translate, rotate, and deform. This is the essence of kinematics, the tool to describe the motion of an infinitesimally small fluid particle, as well as a continuum of such particles.

In this chapter, we turn to the problem of completing the set of equations presented in Section 4.8 by introducing specific constitutive equations. They are material-specific and thus depend upon the constitution of the material.

We close this book with a brief discussion of one of the most important and challenging unsolved problems in the mechanics of fluids: turbulence. As it remains as much descriptive art as predictive science, it is appropriate to call upon visual and poetic sources for inspiration to examine this daunting subject. In the visual realm, the subject has been illustrated with a well-known sketch from da Vinci, seen in Fig. 14.1.

Here we consider some standard problems in multi-dimensional viscous flow. As for one-dimensional viscous flow, application areas are widespread and can include ordinary pipe flows as well as microscale fluid mechanics. We will restrict attention to problems that are steady and laminar. Most of the problems will be incompressible, except for one dealing with a problem in natural convection, Section 11.2.6, and another in compressible boundary layers, Section 11.2.7.

In this chapter, we introduce the complex numbers system – an extension of the well-known real numbers. Complex numbers arise naturally in many problems in mathematics and science and allow us to study polynomial equations that may not have real solutions (such as x2 + 5 = 0). As we will see, many familiar algebraic properties remain valid in the complex number system. In particular, we show that the complex numbers form a field and that the quadratic formula and the triangle inequality can still be used in this new number system.

Linear Algebra is a branch of mathematics that deals with linear equation, systems of linear equations, and their representation as functions between algebraic structures called vector spaces. Linear Algebra is an essential tool in many disciplines such as engineering, statistics, and computer science and is also central within mathematics, in areas such as analysis and geometry. Much like the definition of a field, a vector space is defined through a list of axioms, motivated by concrete observations in familiar spaces, such as the standard two-dimensional plane and the three-dimensional space. We begin this chapter by taking a closer look at real n-dimensional spaces and vectors and then move on to discussing abstract vector spaces and linear maps. Our experience with sets and functions, developed in previous chapters, as well as certain proof techniques, will prove to be useful in our discussion.

We begin our journey by taking a closer look at some familiar notions, such as quadratic equations and inequalities. And, rather than using mechanical computations and algorithms, we focus on more fundamental questions: Where does the quadratic formula come from and how can we prove it? What are the rules that can be used with inequalities, and how can we justify them? These questions will lead us to looking at a few proofs and mathematical arguments. We highlight some of the main features of a mathematical proof, and discuss the process of constructing mathematical proofs. We also review informally the types of numbers often used in mathematics and introduce relevant terminology.