The significance of the vorticity field has been stressed at a number of points in the discussion thus far. For example, the Helmholtz decomposition showed that the vorticity can be used with the dilatation to re-create the velocity field. Furthermore, vorticity does often concentrate into small regions, as in a tornado, and so by following the vorticity dynamics, one has an economical and insightful means of capturing the essence of the flow field. Another significant example is in the case of turbulent flows, where in many ways the dynamics of the vorticity field offers the most direct means of understanding such key processes in the flow as momentum transport and the passage of energy between scales.

An equation that governs the physics of the vorticity field in a moving fluid can be obtained by taking a curl of the Navier-Stokes equation (16.2). Under many circumstances, as in the case of incompressible flow, the pressure is removed by the action of taking the curl, so that the resulting vorticity equation offers a route toward determining the flow without directly taking the pressure into account. This may represent an advantage in the analysis of some flows by providing a more straightforward approach toward understanding the physics of the flow than can occur with the coupled velocity and pressure. In particular, the pressure continuously adapts to the evolution of the velocity field so that it may be more difficult to establish the origin and cause of flow behaviors in this case than through examining the vorticity field and its evolution. In flows for which vortical structures dominate the physics, as in turbulent flow, there very well may be a definite advantage to analyzing the flow through numerical schemes that model the vorticity equation. In this case, the terms in the vorticity equation represent the physical processes that are most relevant for understanding the dynamics of the flow field.

What Is a Fluid?

What distinguishes fluids from solids? Because we all have an intuitive understanding of the difference, the question is really about precisely identifying the formal differences between them. In this regard, it is useful to consider the different ways that fluids and solids react to applied forces. For example, in the case of a solid, the material offers resistance if we press down on it. If the applied force is not so large as to shatter the solid, then it is clear that the solid is quite capable of resisting the force so as to reach a state of equilibrium. The solid arrives at a state where it ceases to move or deform.

Consider now specifically the case of a gas, as shown in Fig. 1.1. In the figure, a piston is pushing down on the gas, and although initially the gas might compress because of the applied force, it is also able to eventually reach a point for any given applied force where it does not compress further. In other words, the gas is capable of resisting the downward force in the same way as the solid. We may conclude that resistance to a normal force is not a good candidate for framing the distinguishing properties of fluids and solids.

The situation for fluids and solids is different if we consider an applied shear force, as in the experiment indicated in Fig. 1.2. Following the application of a shear force to the top surface of the solid, as shown in Fig. 1.2(a), an equilibrium is reached in which the body has deformed a fixed amount. Alternatively, if the container holds a fluid, as in Fig. 1.2(b), and a shear force is applied to the top lid, the fluid cannot prevent the lid from sliding to the side. This is true no matter how small the applied force may be. This is not to say that the fluid does not offer resistance – it does – but the resistance it offers cannot be enough to create a stationary equilibrium.

This book is inspired by a graduate-level course in fluid dynamics that I have taught at the University of Maryland for many years. The typical student taking this course, which is the starting point for graduate studies in fluid mechanics, has had one undergraduate course on fluids and a limited exposure to vector and tensor analysis. Consequently, the goal of this book is to provide a background in the physics and mathematics of fluid mechanics necessary for the pursuit of advanced studies and research at the graduate level. It is my experience that an effective route to these objectives is via a synthesis of the best features of two very excellent books, namely, An Introduction to Fluid Dynamics by George Batchelor, which presents the physics of fluid mechanics with exceptional clarity, and An Introduction to Continuum Mechanics by M. E. Gurtin (and now expanded and revised as The Mechanics and Thermodynamics of Continua by Gurtin, Fried, and Anand), which demonstrates the advantages of direct tensor notation in simplifying the expression of physical laws. Thus, to a large extent, this book combines the physics of Batchelor with the mathematics of Gurtin. The hope is that, in this way, an environment is created that helps make the subject of fluid dynamics clear, focused, and readily understandable. As a practical matter, this book should serve as an effective stepping-stone for new graduate students to enhance their accessibility to the books by Batchelor and Gurtin as well as those by many others.

Stylistically, this book follows an arc through the material that builds steadily toward the derivation and then application of the Navier-Stokes equations. The sequence of topics is also chosen so as to provide some significant exposure to examples of fluid flow and problem solving, before a relatively long and unavoidable set of chapters that deal in detail with the derivation of the flow equations.

It was shown in Chapter 7 that inviscid flow theory predicts the absence of drag on the flow past solid bodies. This unphysical result merely focuses attention on the fact that when fluid flows around an object, there are frictional effects originating in the interactions of the molecules of the fluid with the solid surface that cannot be ignored. The presence of viscosity creates a resistance to motion that is the origin of the drag force. A functionally useful boundary condition that reflects this phenomenon is the nonslip condition to the effect that the fluid velocity tangential to a solid boundary must move with the corresponding velocity of the surface.

The Navier-Stokes equation accommodates the frictional force by inclusion of momentum diffusion at a rate determined by the viscosity in product with the velocity derivatives. If an object is moving swiftly through fluid and the Reynolds number is large, the convective terms in the Navier-Stokes equation are expected to be dominant over the viscous terms. For the viscous effect not to be irrelevant, the velocity gradients that underlie viscous momentum diffusion must be very large, and this can only happen in a relatively thin region near the body, where the velocity rapidly changes from that of the object to that of the free stream. Such regions are referred to as boundary layers, and their study is of considerable importance because many flows are at high speed.

The simplest example of a boundary layer is that forming on a flat plate such as that shown in Fig. 20.1. The oncoming fluid is assumed to be at a uniform velocity U, and a zero velocity condition develops on the plate surface, leading to the reduction of momentum in a thin layer near the boundary. In essence, momentum diffusing to the surface is lost, giving the impression of a spreading boundary layer. This boundary layer is equivalent to that formed by a plate moving at speed U into a quiescent fluid and is known as a zero-pressure-gradient boundary layer because pressure variation along the plate is not present to drive the flow.

Energy and mass transfer within flowing fluids is ubiquitous and of essential importance in a wide range of technologies. This chapter provides an introduction to some of the ideas involved in analyzing the effects of convection on the development of the energy or mass field. It also develops an opportunity to further explore the properties of the many flows that have been discussed previously by seeing their influence on the dissemination of contaminant fields.

The dynamics of the velocity field can be either coupled or decoupled from the transport of heat or mass if they are present. For example, in non-isothermal flows with temperature-dependent viscosity, compressibility, combustion, and other effects, it is likely that the velocity field can only be determined simultaneously with the thermal field. Conversely, this may not be the case for a plume of sufficiently dilute contaminant species or of thermal energy released into a flow field. In this chapter, our focus is confined to the latter situation, in which the heat or mass added to the flow acts passively, so determination of the velocity field can be done without regard to the energy or mass concentration field.

The first scenario of interest concerns a generalization of the boundary layer analysis of the previous chapter to include the presence of simultaneous fluid and thermal boundary layers. Following this is a presentation of a numerical scheme known as the Monte Carlo method, which, with a small expenditure of programming effort, can be adapted to the solution of a wide range of convection diffusion problems that cannot be solved analytically. The effectiveness of the Monte Carlo method is demonstrated for the same thermal boundary layer considered in the next section. Some additional applications of the Monte Carlo scheme to flows considered in this book are suggested in the problem section at the end of the chapter.