Here, we elaborate in more detail on a few technical notions that we have used throughout this book. Among other things, the subjects include the implicit function theorem, the notion of a ridge, classic Lyapunov exponents, differentiable manifolds, Hamiltonian systems, the AVISO data set, surfaces normal to a vector field, calculus of variations, Beltrami flows and the Reynolds transport theorem.

If we accept that transport barriers should be material features for experimental verifiability, we must also remember a fundamental axiom of mechanics: material response of any moving continuum, including fluids, must be frame-indifferent.This means that the conclusions of different observers regarding material behavior must transform into each other by exactly the same rigid-body transformation that transforms the frames of the observers into each other. This requirement of the frame-indifference of material response is called objectivity in classical continuum mechanics. Its significance in fluid mechanics is often overlooked or forgotten, which prompts us to devote a whole chapter to this important physical axiom. We clarify some common misunderstandings of the principle of objectivity in fluid mechanics and discuss in detail the mathematical requirements imposed by objectivity on scalars, vectors and tensors to be used in describing transport barriers.

This chapter discusses two different approaches to describing fluid flow: a Lagrangian approach (following a fluid element as it moves) and a Eulerian approach (watching fluid pass through a fixed volume in space). Understanding each of these descriptions of flow is needed to fully understand the dynamics of fluids. This chapter is devoted to diving into the differences between the two descriptions of fluid motion. Understanding this chapter will help tremendously in the understanding of the upcoming chapters when the Navier–Stokes equations and energy equation are discussed. This chapter will introduce the material derivative. It is extremely important to understand this derivative before the Navier–Stokes equations themselves are tackled.

In the previous chapter, the forces acting on a moving fluid element were exhaustively studied. Using Newtons second law of motion, the Navier–Stokes equations for both compressible and incompressible flows were obtained. This chapter uses an alternative approach to developing the Navier–Stokes equations. Namely, by starting from a Eulerian description (as opposed to a Lagrangian description), the integral form and conservation form of the Navier–Stokes equations are developed. The continuity and Navier–Stokes equations in its various forms are tabulated and reviewed in this chapter. This chapter ends by solving some very simple, yet common, problems involving the incompressible Navier–Stokes equations.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

In this chapter, a concept known as scaling is introduced. Scaling (also known as nondimensionalization) is essentially a form of dimensional analysis. Dimensional analysis is a general term used to describe a means of analyzing a system based off the units of the problem (e.g. kilogram for mass, kelvin for temperature, meter for length, coulomb for electric change, etc.). The concepts of this chapter, while not entirely about the fluid equations per se, is arguably the most useful in understanding the various concepts of fluid mechanics. In addition, the concepts discussed within this chapter can be extended to other areas of physics, particularly areas that are heavily reliant on differential equations (which is most of physics and engineering).

In addition to the continuity equation, there is another very important equation that is often employed alongside the Navier–Stokes equations: the energy equation. The energy equation is required to fully describe compressible flows. This chapter guides the student through the development of the energy equation, which can be an intimidating equation. A discussion on diffusion and its interplay with advection is also included, leading to the idea of a boundary layer. The chapter ends with the addition of the energy equation in shear-driven and pressure-driven flows.