Here we study flows that possess steady solutions that may not persist in time if subjected to small perturbations. Often the behavior of a fluid with no time-dependency is dramatically different than one with time-dependency. Understanding what type of perturbation induces persistent time-dependency is essential for scientific and practical understanding of fluid behavior. An example that we will consider here as well as in later chapters is that of warm air rising or not rising; see Fig. 12.1.

This book considers the mechanics of a fluid, defined as a material that continuously deforms under the influence of an applied shear stress, as depicted in Fig. 1.1. Here the fluid, initially at rest, lies between a stationary wall and a moving plate. Nearly all common fluids stick to solid surfaces. Thus, at the bottom, the fluid remains at rest; at the top, it moves with the velocity of the plate.

This chapter will focus on one-dimensional flow of a compressible fluid. The emphasis will be on inviscid problems, with one brief excursion into viscous compressible flow that will serve as a transition to a study of viscous flow in following chapters. The compressibility we will study here is that which is induced when the fluid particle velocity is of similar magnitude to the fluid sound speed.

This chapter will expand upon potential flow, introduced in Section 7.7, and will mainly be restricted to steady, two-dimensional planar, incompressible potential flow. Such flows can be characterized by a scalar potential field. An example of such a field along with associated streamlines is given in Fig. 8.1.

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.