Fluid film lubrication naturally divides into two categories. Thin-film lubrication is usually met with in counter-formal contacts, principally in rolling bearings and in gears. The thickness of the film in these contacts is of order of 1µxm or less, and the conditions are such that the pressure dependence of viscosity and the elastic deformation of the bounding surfaces must both be taken into account.

Thick-film lubrication is encountered in externally pressurized bearings, also called hydrostatic bearings, and in self-acting bearings, called hydrodynamic bearings. Of the latter, there are two kinds: journal bearings and thrust bearings. The film thickness in these conformal-contact bearings is at least an order of magnitude larger than in counter-formal bearings. In consequence, the prevailing pressures are orders of magnitude smaller, so that neither the pressure dependence of viscosity nor the elastic deformation of the surfaces plays an important roles. If, in addition, the lubricant is linearly viscous and the reduced Reynolds number is small, the classical Reynolds theory, as derived in the previous chapter, will apply.

This chapter discusses isothermal processes only. It should be realized, however, that bearings never operate under truly isothermal conditions, and under near isothermal conditions only in exceptional cases. Viscous dissipation and consequent heating of the lubricant are always present, and the change in viscosity must be accounted for when analyzing thick-film lubrication problems. In restricted cases, where design and operating conditions are such as to suggest “uniform” temperature rise of the lubricant, the “effective viscosity” approach of Chapter 9 might be employed.

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 fluid mechanics. This discussion will not be limited only to concepts necessary to understand 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. Cauchy's equations of motion will be obtained by substituting the rate of change of linear momentum of a fluid particle and the forces acting on it, into Newton's second law of motion. This will yield three equations, one in each of the three coordinate directions. But these three equations will contain twelve unknowns: three velocity components, (u, v, w) and nine stress components (Txx, Txy,…, Tzz). To render the problem well posed, i.e., to have the number of equations agree with the number of unknowns so that solutions might be obtained, 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 matrix. However, on specifying incompressibility of the fluid (incompressible lubricants are the only type treated in this chapter) a tenth unknown, the fluid pressure, makes its debut.

Elastohydrodynamic lubrication (EHL) is the name given to hydrodynamic lubrication when it is applied to solid surfaces of low geometric conformity that are capable of, and are subject to, elastic deformation. In bearings relying on EHL principles, the pressure and film thickness are of order 1 GP and 1 µm, respectively – under such conditions, conventional lubricants exhibit material behavior distinctly different from their bulk properties at normal pressure. In fact, without taking into account the viscosity-pressure characteristics of the liquid lubricant and the elastic deformation of the bounding solids, hydrodynamic theory is incapable of explaining the existence of continuous lubricant films in highly loaded gears and rolling-contact bearings. This is illustrated in the next section, by applying isoviscous lubrication theory to a rigid cylinder rolling on a plane.

When two convex, elastic bodies come into contact under zero load, they touch along a line (e.g., a cylinder and a plane or two parallel cylinders) or in a point (e.g., two spheres or two crossed cylinders). On increasing the normal contact load from zero, the bodies deform in the neighborhood of their initial contact and yield small, though finite, areas of contact; this deformation ensures that the surface stresses remain finite. For a nominal line contact the shape of the finite contact zone is an infinite strip, for a nominal point contact it is an ellipse. Nominal line contacts possess only one spatial dimension and are, therefore, easier to characterize than the two-dimensional nominal point contacts.

It follows from the previous chapters that one of the key elements in the analysis of flow separation from a solid body surface is the investigation of the flow behavior in the region of boundary-layer interaction with the external inviscid flow. Although the interaction region is normally very small, it plays a major role in the separation phenomenon because of the mutual influence of the near-wall viscous flow and external inviscid flow in this region, with a sharp pressure rise prior to the separation point leading to very rapid deceleration of fluid particles near the wall and ultimately to the appearance of the reverse flow downstream of the separation. The complexity of the physical processes in the interaction region is accompanied, as might be expected, by mathematical difficulties in solving the equations that describe the flow in this region. While everywhere outside the interaction region the solution may often be obtained in analytical or at least self-similar form, the analysis of the interaction region requires that special numerical methods be used.

Numerical solution of the interaction problem serves not only to provide meaningful physical information on the development of events in the interaction region, but in many cases also appears to provide the only way of being sure that the solution for this problem really exists, and hence that the entire asymptotic structure of the flow, anticipated in the course of the asymptotic analysis of the Navier–Stokes equations, is self-consistent.

The major successes achieved since the late 1960s in the development of the theory of separated flows of liquids and gases at high Reynolds numbers are in many respects associated with the use of asymptotic methods of investigation. The most fruitful of these has proved to be the method of matched asymptotic expansions, which has also become widely used in many other fields of mechanics and mathematical physics (Van Dyke, 1964, 1975; Cole, 1968; Lagerstrom and Casten, 1972). By means of this method, important problems have been solved concerning boundary-layer interaction and separation in subsonic and supersonic flows, the nature of these phenomena has been made clear, and basic laws and controlling parameters have been ascertained. The number of original papers devoted to different problems in the asymptotic theory of separated flows now amounts to many dozens. We can confidently speak of the appearance of a new and very productive direction in the development of theoretical hydrodynamics. However, it has not yet received an adequate systematic account, and the objective of this book is to make an attempt to partially fill this gap.

Of necessity, the authors have restricted themselves to consideration of a range of problems most familiar to them, associated with two-dimensional separated flows of an incompressible fluid, assumed to be laminar. Thus, the book excludes the results of many investigations of separated gas flows at transonic, supersonic, and hypersonic speeds, although chronologically the problem of boundary-layer separation in supersonic flow was solved earlier than the others (see Chapter 1).

Here we leave completely untouched the study of turbulent boundary-layer interaction and separation, as well as the large class of problems of internal flows in channels and tubes.

The problem of studying flow separation from the surface of a solid body, and the flow that is developed as a result of this separation, is among the fundamental and most difficult problems of theoretical hydrodynamics. The first attempts at describing separated flow past blunt bodies are known to have been made by Helmholtz (1868) and Kirchhoff (1869) as early as the middle of the nineteenth century, within the framework of the classical theory of ideal fluid flows. Although in the work of Helmholtz one can find some indication of the viscous nature of mixing layers, which are represented in the limit as tangential discontinuities, nonetheless before Prandtl (1904) developed the boundary-layer theory at the beginning of the twentieth century there were no adequate methods for investigating such flows. Prandtl was the first to explain the physical nature of flow separation at high Reynolds number as separation of the boundary layer. In essence, Prandtl's boundary-layer theory proved to be the foundation for all further studies of the asymptotic behavior of either liquid or gas flows with extremely low viscosity, i.e., flows of media similar to air or water, with which one is most commonly concerned in nature and engineering. Therefore we begin with a brief description of the main ideas of this theory.

Boundary-Layer Theory

The Navier–Stokes equations with corresponding boundary and initial conditions are usually used as the basic system of relations describing a viscous fluid flow. These equations reflect sufficiently well the behavior of real liquids and gases.

The Analogy between Unsteady Separation and Separation from a Moving Surface

For the steady fluid motions considered in the preceding chapters, flow separation leads to a change in the flow structure as a whole. The limiting state when the Reynolds number tends to infinity is defined by the Helmholtz–Kirchhoff theory for ideal fluid flows with free streamlines, and the location of the separation point, in accordance with the criterion of Prandtl, coincides with the point where the shear stress at the surface of the body vanishes (see Chapter 1).

The situation is somewhat different if the flow is unsteady. To illustrate this fact we consider the example of flow past a circular cylinder that is set into motion instantaneously from rest. The starting motion of a body in a viscous fluid can be likened to the introduction of the no-slip condition at the surface of a body as it moves through a fluid that has no internal friction. Therefore, at the first instant the flow is potential and is described by the well-known solution for unseparated steady flow past a cylinder with zero circulation. At the body surface, where the no-slip condition is imposed, there arises the process of vorticity diffusion and convection, which leads to the formation of a boundary layer.

Blasius (1908) formulated this problem and gave an approximate solution. It turns out that at a certain instant the point of zero skin friction starts moving upstream along the body surface from the rear stagnation point. At the same time a region of reverse flow appears within the boundary layer (Figure 5.1).

The authors are honored by the publication of the English edition by Cambridge University Press, edited by two colleagues who are well known for their work in fluid mechanics – Professor A. F. Messiter of the University of Michigan and Professor M. D. Van Dyke of Stanford University. The authors are deeply indebted to them for their interest in this book, the initiative for the translation into English, and for their considerable effort in the editing, which has no doubt improved the original text. Our work rests heavily on the present state of development of the asymptotic analysis of problems in fluid mechanics, which in many respects is due to Professor Van Dyke and Professor Messiter.

Since the publication of the Russian edition of this book in 1987, there have appeared numerous papers devoted to the asymptotic theory of separated flows. Therefore, in the preparation of the English edition the authors have attempted to bring the book more up to date. In particular, Chapters 2 and 6 have been expanded by the addition of two new sections, (Section 4 and Section 6, respectively) and Chapter 7 has been completely revised. Elsewhere we have confined ourselves to improvements and short additions in the text and to expanding the list of references.

Experimental Observations

Boundary-layer separation at the leading edge of a thin airfoil is the principal factor that limits the lift force acting on an airfoil in a fluid stream. Jones (1934) was the first to describe this kind of separation. Since that time, many researchers have turned to the experimental study of flow around the leading edge of an airfoil. In addition to a great number of original studies, several surveys have been devoted to this theme. The reviews by Tani (1964) and Ward (1963) can be regarded as the most complete.

Experiments show that as the angle of attack increases the picture of the flow around the airfoil changes in the following way. When the angle of attack is small, the flow over the profile is attached. Then the pressure has its maximum at the stagnation point O of the flow, where the zero streamline divides into two – one branch lies along the lower surface of the airfoil, and the second one bends around the leading edge of the airfoil and then lies along its upper surface. As we move from the stagnation point along the upper branch, the pressure first falls rapidly, reaching a minimum at some point M (Figure 4.1a), and then starts to increase, so that the boundary layer downstream from point M finds itself under the influence of an adverse pressure gradient. Its magnitude increases with growth of the angle of attack, finally resulting in boundary-layer separation, which occurs earlier for smaller relative airfoil thickness.

When the boundary layer separates, one can observe the appearance of a closed region of recirculating flow on the upper surface of the airfoil (Figure 4.1a).

The Background of the Problem

This chapter will be devoted to the analysis of one of the fundamental problems of hydrodynamics: ascertaining the limiting state for the steady flow behind a body of finite size as the Reynolds number Re → ∞ in cases where unseparated flow over the body is impossible, for example, behind a blunt body such as a circular cylinder or a plate placed normal to the oncoming flow. Although in reality such flows already become unsteady at Reynolds numbers of the order of 101–102, and undergo transition to a turbulent state with further increase in Reynolds number, the solution of this problem is of great interest in principle. Moreover, one might anticipate that such a solution would allow the study of fluid flows at moderate Reynolds numbers when a steady flow regime is still maintained but the methods of the theory of slow motion (Re < 1) are no longer applicable.

There are several points of view about possible ways of solving this problem. According to the first of these, already stated by Prandtl (1931) and then developed by Squire (1934) and Imai (1953, 1957b), the limiting flow configuration as Re → ∞ is the classical Kirchhoff (1869) flow with free streamlines and a stagnation zone that extends to infinity and expands asymptotically according to a parabolic law (Figure 6.1). This picture has been severely criticized both because of poor quantitative agreement with experimental results (for measurements of body drag) and also for some fundamental reasons.