All the solutions considered so far have had very regular streamlines or particle paths but a glance at the weather or a smoke plume shows that spatial and temporal complexity arises in many common situations. In this chapter we will look at some situations where, in spite of this complexity, a mathematical model can be made of at least some aspects of the physical problem. We begin with the problem of flow in a porous medium where the spatial complexity can be averaged to give a smoothly varying macroscopic model.

Flow in a porous medium

Surprisingly, the Hele-Shaw model of Chapter 4 can be used as a model for the geometrically complicated problem of flow of a viscous fluid through a porous medium. Such a model is relevant in many practical situations such as oil recovery, hydrology, soil mechanics, filter design, and fluidized beds as well as numerous other phenomena in the earth sciences. The most basic model assumes that, besides there being an obvious microscopic flow scale defined by the ‘pore’ size and illustrated in figure 5.1, there is a much larger macroscopic scale over which the problem is to be studied; this macroscopic scale may be the overall dimensions of the industrial device or the oil field. We can make progress by working on an intermediate scale which is small compared to the macroscopic scale yet still contains enough pores for an averaged velocity u and pressure p to be defined.

In this chapter we consider relatively low Reynolds number flow of a thin film. Such a film may exist between two rigid walls, as in a bearing, or in a droplet, e.g. paint, spreading under gravity on a rigid surface. In either case the geometry of the problem allows us to simplify equation (3.2) in a way that is similar to the technique used to derive boundary layer theory in Chapter 2. The differences are that the order of magnitude of the width of the thin layer is dictated by the data of the problem and, since the layer is confined geometrically, there is no need to match with an outer flow.

Lubrication theory for slider bearings

The simple observation that a sheet of paper can slide across a smooth floor shows that a thin layer of fluid can support a relatively large normal load while offering very little resistance to tangential motion. More important mechanical examples occur in the lubrication of machinery and this motivates the study of slider bearings. A slider bearing consists of a thin layer of viscous fluid confined between nearly parallel walls that are in relative tangential motion.

A two-dimensional bearing is shown in figure 4.1 in which the plane y = 0 moves with constant velocity U in the x-direction and the top of the bearing (the slider) is fixed.

These lecture notes are intended to provide third-year mathematics undergraduates who are already familiar with inviscid fluid dynamics with some of the basic facts about the modelling and analysis of viscous flows. Writing the notes has been an interesting task because so many of the phenomena to be described are not only associated with vitally important mechanisms in everyday life but they are also readily observable without any need for instrumentation. More sophisticated realisations are also readily available, for instance in the very valuable collection of photographs “An Album of Fluid Motion”, edited by Van Dyke (2). Thus it is all the more stimulating when the mathematics that emerges when these phenomena are modelled is novel and suggestive of new methodologies.

The notes are strictly not self-contained and should be read in conjunction with standard texts which are referenced. We have concentrated on trying to present some of the salient physical ideas and mathematical ramifications as starkly as possible and, to this end, many of the exercises have been designed to be worked as an integral part of the notes; they are only put at the end of chapters for convenience. The starred exercises cover more advanced material and can be omitted at a first reading.

Experience has shown us that, in the twentieth century, theoretical mechanics generally has been one of the best vehicles for learning about physical applied mathematics.

The beginning

In the spring of 1984 Klaus Hasselmann invited wave modellers to Hamburg to discuss possible joint work. The WAM (Wave Modelling) group emerged. I dutifully reported on progress (Komen, 1985a,b, 1986, 1987a,b, 1990, 1991a,b). Now, ten years later, the WAM group has achieved what it wanted to achieve: a third generation computer model has been developed, which is able to predict the wave conditions at sea; it is used for global and regional applications; the model is a useful tool for interpreting satellite observations of the ocean and increased our understanding of the role of waves in air/sea interaction and the coupling between the atmosphere and the ocean. The results obtained are presented in this book.

WAM was one of a number of international collaborations: JONSWAP, SWAMP, SWIM, WAM, … Observations during the Joint North Sea Wave Project (Hasselmann et al, 1973) and also by Mitsuyasu and collaborators (Mitsuyasu, 1966, 1969, Mitsuyasu et al, 1971) had established the importance of the nonlinear transfer in governing the shape and evolution of a wind sea spectrum. Based on this finding ‘parametric’ wave models had been developed based on the approximation of a self-similar shape of the wind-sea spectrum.

Summary and outlook

The WAM group has realized its objectives: a third generation wave model has been developed; it runs in global and regional modes; it extended our understanding of the underlying physics; and data assimilation schemes have been developed and tested. This book is testimony of what has been achieved. However, careful reading will also reveal open ends, which range from small inconsistencies to major open issues. In this chapter we give an outlook on expected developments.

Compared to the so-called second generation models considerable progress has been made regarding the formulation of the evolution equation for the wave spectrum. The present WAM model is based on an explicit formulation of the physics of generation of waves by wind, nonlinear wave-wave interactions and dissipation due to whitecapping and bottom processes, rather than on the approach of ad hoc modelling which was commonplace with second generation models. The latter approach was shown to be inadequate under extreme circumstances such as hurricanes (SWAMP, 1985), while the WAM model gives for rapidly varying wind fields very satisfactory results (see chapter IV). Nevertheless, under ‘normal’ circumstances both approaches give similar results for the wave height. The reason for this is that although second generation models have inadequate physics they have been tuned to a considerable extent. Thus, the benefits of a third generation model are mainly related to a better representation of the spectrum itself and to a more explicit formulation of the underlying physics of wave evolution.

Despite the progress, we still are not able to make wave predictions that always fall within the error bands of the observations.