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.

The object of the appendix It is clear that elastoplasticity problems are not easily amenable to analytical methods, but for a few exceptions as in the case of the spherical envelope in Chapter 6. In particular, the elastoplastic borderline separating the region where the material still behaves elastically and the already plasticized region is an unknown in such problems. For complex geometries then a numerical implementation seems necessary (Chapter 11). However, the few cases that admit analytical solutions are typical of a methodology of which any student and practitioner of elastoplasticity must be aware. We have thus selected four examples, the first in plane strain (the wedge problem), the second in torsion, the third exhibiting a complex loading and the fourth accounting for anisotropy in a composite material.

Elastoplastic loading of a wedge

General equations

A wedge of angle β < π/2 is made of an isotropic elastoplastic material, satisfying Hooke's law in the elastic regime and Tresca's criterion without hardening at the yield limit. On its upper face it is subjected to a pressure p which increases with time (Fig. A3.1). We look first for the fully elastic solution and then for the elastoplastic solution in which the plasticized zone progresses until the whole wedge has become plastic. The solution of this problem in the elastoplastic framework is due to Naghdi (1957) – see also Murch and Naghdi (1958) and Calcotte (1968, pp. 158–64).

The object of the chapter In the absence of plastic strain, the problem of brittle fracture by extension of cracks can be presented in a thermodynamic framework, analogous to that of elastoplasticity. This means that the fracture criterion (or the criterion of crack propagation) replaces the plasticity criterion. One important notion is the notion of mechanical field singularity (displacement, stresses).

Introduction and elementary notions

We are interested in the problem of fracture, a phenomenon that occurs, more or less violently, under monotonic loading (whereas fatigue concerns cyclical loading). More specifically, we are interested in the problem of cracking, that is, the progagation of macroscopic cracks (of size of the order of one millimetre), whereas the beginning of cracking belongs to the microscopic and to the metal analyses which will not be examined here. (Microscopic cracks are one cause of damage – see Chapter 10.) The aim of this study is to arrive at a formulation of the crack-propagation laws, based upon fracture criteria and the definition of the conditions that may insure resistance to this fracture. We are certainly aware of the interest that such a subject implies for industry; it suffices to think about aeronautical engines and nuclear installations. Actually, our main interest is brittle fracture, that is, the kind that occurs without considerable plastic strain (i.e. the separation mechanism of crystallographic facets through cleavage), whereas ductile rupture is produced by different mechanisms accompanied by great plastic strains).

The object of the chapter This chapter provides a short introduction to the notion of homogenization (i.e., determining the parameters of a unique fictitious material that ‘best’ represents the real heterogeneous material or composite) and then, at some length, its application to the case where all or some of the constitutive components have an elastoplastic behaviour. The essential notions are those of representative volume element, procedure of localization, and the representation of some microscopic effects by means of internal variables. Composites with unidirectional fibres, polycrystals and cracked media provide examples of application.

Notion of homogenization

Homogenization is the modelling of a heterogeneous medium by means of a unique continuous medium. A heterogeneous medium is a medium of which material properties (e.g., elasticity coefficients) vary pointwise in a continuous or discontinuous manner, in a periodic or nonperiodic way, deterministically or randomly. While, obviously, homogenization is a modelling technique that applies to all fields of macroscopic physics governed by nice partial differential equations, we focus more particularly on the mechanics of deformable bodies with a special emphasis on composite materials (as used in aeronautics) and polycrystals (representing many alloys.) Most of the composite materials developed during the past three decades present a brittle, rather than ductile behaviour. As emphasized in Chapter 7, the elastic behaviour then prevails and there is no need to consider the homogenization of an dastoplastic behaviour.