Even a cursory look at the table of contents of this book will reveal that the material is arranged in an unusual way, starting immediately with the application of techniques (Part I) which are justified and explained in greater detail in later chapters (Parts II and III). Indeed, the book is not meant to be read sequentially and I have not attempted to force the material into a sequential exposition. Rather, I have organized it as a series of modules through which each reader can fashion an individual pathway according to background, needs and interests.

The widespread availability of computers and symbolic manipulation packages has made mathematics relevant to many more people than in the past. A serious problem, however, remains the high “cost of entry” of mathematics. While some might argue that in a perfect world people would spend the time required to learn all the mathematics they might need before applying it, this is a utopia certainly, and probably also a misguided ideal as any user of mathematics – having gone through many cycles of learning and applying – can testify.

Hopefully, the modular structure of the book will serve a variety of users addressing different needs ranging from a quick impression of a mathematical tool to a fairly deep understanding of its basis and “inner workings.” The many cross-references, detailed index and table of contents will render possible a non-systematic “navigation” through the material.

For the successful operation of mechanical devices, from spinning computer disks to automobiles to large electric generators of nuclear power stations, it is essential that the components that are destined to move relative to one another do so with low friction and rate of wear. This is made possible through appropriate design and utilization of fluid film bearings. Traditionally, fluid film lubrication was a purely mechanical subject, but with the emergence of bioengineering, the technology also finds application in lubricating artificial joints, contact lenses, and mechanical heart pumps, to name a few. In this second edition, I have tried to give a flavor of some of these advances. The contents of the first edition remain valid by and large, as they deal with the fundamentals that have changed little. Thus, this edition represents addition, rather than revision, of material. Chapter 2 is rewritten, however, to align it with a more complete discussion of constitutive theory. Chapter 3, the chapter covering thick-film lubrication, features a section on surface texturing; another section treats surface roughness in a more thorough manner. The chapter on turbulence includes the handling of flow with significant inertia. In the treatment of elastohydrodynamic lubrication, covered in Chapter 8, I tried to convey basic ideas of the multigrid method and touched on multilevel multi integration. The chapter on lubrication with non-Newtonian fluids discusses the “qualitative” EHL, and contains a discourse on lubrication with piezoviscous fluids, relative to the Reynolds equation. This chapter also comprises a thorough discussion of blood as lubricant, with a view to the application of lubrication theory to artificial organs. Chapters 12 and 13 are new. In the first of these, I concentrate on ultra-thin films, both liquid and gaseous. The chapter discusses and classifies recent research results and, particularly for gas films but also for liquid films, outlines design principles. The chapter closes with the listing of 92 up-to-date references. The last chapter, Biotribology, is devoted to lubrication of the hip joint. Its two principal sections discuss lubrication of natural joints and artificial joints. The first of these presents the various theories of natural joint lubrication including microelastohydrodynamic lubrication, biphasic models, and boundary lubrication. The second section lists the various existing constructs of total hip replacement and their relative performance. This last chapter surveys 60 references in all.

Fluid Mechanics

The equations employed to describe the flow of lubricants in bearings result from simplifications of the governing equations of fluid mechanics. It is appropriate, therefore, to devote a chapter to summarizing pertinent results from that subject. This summary will not be limited to concepts necessary to appreciate the classical theory of lubrication. A more than elementary discussion of fluid behavior is called for here, as various nonlinear effects will be studied in later chapters.

Our discussion begins with the mathematical description of motion, followed by the definition of stress. We will then derive Cauchy's equations of motion by substituting the rate of change of linear momentum of a fluid body and the forces acting on it into Newton's second law. This will yield three equations, one in each of the three coordinate directions. For an incompressible fluid these three equations will contain twelve unknowns: three velocity components (u, v, w) and nine stress components (Txx, Txy, …, Tzz). For compressible fluids there is an additional unknown, the density; for incompressible fluids the density is known a priori. To render the problem well posed, i.e., to have the number of equations agree with the number of unknowns so that a unique solution might exist, we will need to find additional equations. A fourth equation is easy to come by, by way of the principle of conservation of mass. The situation further improves on recognizing that only six of the nine stress components are independent, due to symmetry of the stress tensor. However, on specifying incompressibility of the fluid, a tenth unknown, the fluid pressure makes its debut. In contrast, for a compressible fluid both the pressure and the density are variables, but as they are connected through the equation of state, we count them as representing a single unknown. In summary, for both compressible and incompressible fluid we end up having four equations and ten unknowns. Six additional equations are required.

The term biotribology, to cover “all aspects of tribology related to biological systems,” was coined only four decades ago (Dowson and Wright, 1973); however, investigations into friction, lubrication and wear of biological tissues date back much further. As early as the nineteenth century, Young (1809) and Poiseuille (1840) studied the flow properties of blood, essential today for the design of artificial organs. Reynolds likened lubrication of articulated joints to lubrication of machine elements. In the last sentence of his classical paper, Reynolds remarked that hydrodynamic lubrication “is as fundamental to animal mechanics as the lubricating action of the journal is to mechanical contrivances” (Reynolds, 1886). Jin and Dowson (2005) listed six areas of biotribology in addition to lubrication of joints and of red blood cells in capillaries, such as wear of dentures and the tribology of contact lenses. Nevertheless, in this chapter, we restrict ourselves to one topic only, the lubrication of articular joints.

Lubrication of Articular Joints

The loading cycle to which an articular joint is subjected is complex and the demands this places on the joint are numerous (Paul, 1967). For example, across the hip the cycle displays two force maxima, at heal strike (HS) and at toe-off (TO), as illustrated in Figure 13.1, the latter reaching four times the body weight. As additional complications, the forces that operate within the joint are three dimensional, time dependent and vary with speed and length of stride, and, of course, vary from person to person.

Fluid film bearings are machine elements which should be studied within the broader context of tribology, “the science and technology of interactive surfaces in relative motion and of the practices related thereto.” The three subfields of tribology -- friction, lubrication, and wear -- are strongly interrelated. Fluid film bearings provide but one aspect of lubrication. If a bearing is not well designed, or is operated under other than the design conditions, other modes of lubrication, such as boundary lubrication, might result, and frictional hearting and wear would also have to be considered.

Chapter 1 defines fluid film bearings within the context of the general field of tribology, and is intended as an introduction; numerous references are included, however, should a more detailed background be required. Chapters 2, 3, and 4 outline classical lubrication theory, which is based on isothermal, laminar operation between rigid bearing surfaces. These chapters can be used for an advanced undergraduate or first-year graduate course. They should, however, be augmented with selections from Chapter 8, to introduce the students to the all-important rolling bearings, and from Chapter 9, to make the student realize that no bearing operation is truly isothermal. Otherwise, the book will be useful to the industrial practitioner and the researcher alike. Sections in small print may be omitted on first reading -- they are intended for further amplification of topics. In writing this book, my intent was to put essential information into a rational framework for easier understanding. So the objective was to teach, rather than to compile all available information into a handbook. I have also included thought-provoking topics; for example, lubrication with emulsions, the treatment of which has not yet reached maturity. I expect significant advances in this area as it impacts on the environment.

Although the lubrication approximation has been derived for thin films, there is, nevertheless, a thin film limit to its validity. When the characteristic dimensions of the fluid-containing device approach the mean free path (for gases) or the dimension of the molecules (for liquids) the continuum assumption, one of the basic assumptions of the approximation, breaks down. In such cases the Reynolds equation must be amended or replaced by other mathematical systems.

We have two distinct models at our disposal for representing fluids, continuum and particle. While the latter is valid under the whole range of conditions, though its use is limited by practical considerations, the continuum model applies only with restrictions. The equations that are available for fluid characterization, and how they relate to the two models, are shown in Table 12.1 (Gad-el-Hak, 1999).

The term tribology, meaning the science and technology of friction, lubrication, and wear, is of recent origin (Lubrication Engineering Working Group, 1966), but its practical aspects reach back to prehistoric times. The importance of tribology has greatly increased during its long history, and modern civilization is surprisingly dependent on sound tribological practices.

The field of tribology affects the performance and life of all mechanical systems and provides for reliability, accuracy, and precision of many. Tribology is frequently the pacing item in the design of new mechanical systems. Energy loss through friction in tribo-elements is a major factor in limits on energy efficiency. Strategic materials are used in many tribo-elements to obtain the required performance.