This textbook is intended to be read by undergraduate students enrolled in engineering or engineering science curricula who wish to study fluid mechanics on an intermediate level. No previous knowledge of fluid mechanics is assumed, but the students who use this book are expected to have had two years of engineering education which include mathematics, physics, engineering mechanics and thermodynamics.

This book has been written with the intention to direct the readers to think in clear and correct terms of fluid mechanics, to make them understand the basic principles of the subject, to induce them to develop some intuitive grasp of flow phenomena and to show them some of the beauty of the subject.

Care has been taken in this book to strictly observe the chain of logic. When a concept evolves from a previous one, the connection is shown, and when a new start is made, the reason for it is clearly stated, e.g., as where the turbulent boundary layer equations are derived anew. As related later, there may be programs of study in which certain parts of this book may be skipped. However, from a pedagogical point of view it is important that these skipped parts are there and that the student feels that he or she has a complete presentation even though it is not required to read some particular section.

It is believed that an important objective of engineering education is to induce the students to think in clear and exact terms.

The nonlinear terms on the left-hand side of the Navier–Stokes equations result from the acceleration of the fluid. We already know that these terms contribute much to the difficulties one encounters in the solution of the equations. Indeed, we have noted the relative ease with which exact solution have been obtained in Chapter 6 for fully developed flows, where those nonlinear terms vanish. As we now look for approximations, obtained by the solutions of approximate forms of the Navier–Stokes equations, the first idea that comes to mind is to remove the terms which cause the greatest difficulty, i.e., the nonlinear acceleration terms.

Our subject here is fluid mechanics, and in this context we must ask if there exist real flows in which the acceleration terms are negligible; and if they exist, how do we identify them? The answer to this enquiry is that there are at least two such families of flows: flows in narrow gaps and creeping flows.

Flow in Narrow Gaps

Consider the two-dimensional flow of an incompressible fluid in the narrow gap between the two plates shown in Fig. 9.1. For simplicity we assume the plates to be flat. The results obtained in this analysis may be extended to cases where the plates are curved, as long as the gap is much smaller than the radius of curvature. Another possible extension is to three-dimensional flows, where a w-component of the velocity also exists. Here we present the simplest examples of gap flows and limit the analysis to two-dimensional flows between flat plates.

The Field of Fluid Mechanics

Fluid mechanics extends the ideas developed in mechanics and thermodynamics to the study of motion and equilibrium of fluids, namely of liquids and gases.

The beginner in the study of fluid mechanics may have some intuitive notion as to the nature of a fluid, a notion that centers around the idea of a fluid not having a fixed shape. This idea indicates at once that the field of fluid mechanics is more complex than that of solid mechanics. Fluid mechanics has to deal with the mechanics of bodies that continuously change their shape, or deform. Similarly, the ideas developed in classical equilibrium thermodynamics have to be extended to allow for the additional complexity of properties which vary continuously with space and time, normally encountered in fluid mechanics.

Fluid mechanics bases its description of a fluid on the concept of a continuum, with properties which have to be understood in a certain manner. So far, neither the idea of a continuum nor that of a fluid have been properly defined. We, therefore, begin with the explanation of what a continuum is and how local properties are defined. We then describe the various forces that act in a continuum leading to the definition of the concept of stress at a point. The concepts of stress and continuum are then used to define a fluid.

Moving fluids are subject to the same laws of physics as are moving rigid bodies or fluids at rest. The problem of identification of a fluid particle in a moving fluid is, however, much more difficult than the identification of a solid body or of a fluid particle in a static fluid. When the fluid properties, e.g., its velocity, also vary from point to point in the field, the analysis of these properties may become quite elaborate.

There are cases where such an identification of a particle is made, and then this particle is followed and the change of its properties is investigated. This is known as the Lagrangian approach. In most cases one tries to avoid the need for such an identification and presents the phenomena in some field equations, i.e., equations that describe what takes place at each point at all times. This differential analysis, which is called the Eulerian approach, is the subject of the next chapter.

There exist, however, quite a few cases where there are sufficient restrictions on the flow field, such that while the individual fluid particles still elude identification, whole chunks of fluid can be identified. When this is the case, some useful engineering results may be obtained by integrating the relevant properties over the appropriate chunk. This approach is known as integral analysis.

In most cases where integral representation is possible, there are walls impermeable to the fluid.

Most flows encountered in nature and in engineering practice are turbulent. It is therefore important to understand the fundamental mechanisms at work in such flows. Turbulent flows are unsteady and contain fluctuations that are random in space and time. An important characteristic is the richness of scales of eddy motion present in such flows: In a fully developed turbulent flow all scales appear to be fully occupied or saturated in a sense, from the largest ones that can fit within the size of the flow region down to the smallest scale allowed by dissipative processes. Turbulent flows are also highly vortical, a consequence of vortex stretching and tilting by larger random vorticity fields.

The reason turbulence is so prevalent in fluids of low viscosity is that steady laminar flows tend to become unstable at high Reynolds or Rayleigh numbers and therefore cannot be maintained indefinitely as steady laminar flows. Instability to small disturbances is an initial step in the process whereby a laminar flow goes through transition to turbulence. In investigations of instability of flows that are homogeneous in one or more spatial dimensions, one usually formulates a linear problem of the evolution of an infinite train of small-amplitude waves so as to find whether such waves will grow or decay with time. More general disturbances may be analyzed by Fourier superposition.

A complete description of the transition process requires one to consider the development of disturbances of finite amplitudes. This is generally a difficult theoretical task since it leads to nonlinear problems. A few simplified model problems, giving some insight into the nature of the transition process, are tractable, however, such as the evolution of a finiteamplitude wave train in a parallel shear flow, finite-amplitude density interface waves, weak nonlinear interaction of several wave trains, and the influence of distortion by large-scale motion on smaller-scale wave trains. The treatment of non-wave-like amplitude disturbances lacking spatial and temporal periodicity is more difficult.

Because of the random nature of fully developed turbulent flow fields, statistical methods are usually employed for their description. However, in the statistical averages much of the information that may be relevant to the understanding of the turbulent mechanisms may be lost, especially phase relationships. This may not seem too serious for flows in which the motion appears to be completely disorganized, such as in nearly isotropic or homogeneous turbulent flows.