FLUID MECHANICS

the study of the effects of forces and energy on liquids and gases. Like other branches of classical mechanics, the subject subdivides into statics (often called hydrostatics) and dynamics (fluid dynamics, hydrodynamics, or aerodynamics). Hydrostatics is a comparatively elementary subject with a few classical results of importance but little scope for further development. Fluid dynamics, in contrast, is a highly developed branch of science that has been the subject of continuous and expanding research activity since about 1840. The development of fluid dynamics has been strongly influenced by its numerous applications. Some of the fields of application to engineering, the environmental sciences, and the biological sciences are evident: aeronautical engineering, marine engineering, meteorology, oceanography, and the study of blood flow, the dynamics of swimming, and the flight of creatures. There are also many less immediately obvious applications. Fluid dynamics is studied both theoretically and experimentally, and the results are described both mathematically and physically. The phenomena of fluid motion are governed by known laws of physicsconservation of mass, the laws of classical mechanics (Newton's laws of motion), and the laws of thermodynamics. These can be formulated as a set of nonlinear partial differential equations, and in principle one might hope to infer all the phenomena from these. In practice, this has not been possible; the mathematical theory is often difficult, and sometimes the equations have more than one solution, so that subtle considerations arise in deciding which one will actually apply. As a result, observations of fluid motion both in the laboratory and in nature are also essential for understanding the motion of fluids. Liquids and gases are classified together as fluids because, over a wide range of situations, they have identical equations of motion and thus exhibit the same flow phenomena. Scaling analysis makes it possible to infer when two geometrically similar situationsof perhaps quite different size and involving different fluids (either both liquids, both gases, or one of each)will give rise to the same type of flow. It leads to the formulation of various nondimensional parameters, with names like Reynolds number, Mach number, Froude number, in terms of which fluid-dynamical results are usually presented. Flow configurations equally applicable to liquids and gases include flow through pipes, flow due to relative motion between a body and ambient fluid, and thermal convectiongravitationally driven flow due to temperature differences. Sometimes the effect of rotation of the whole system (of particular significance in meteorology and oceanography) is included. A common feature of all these flows is their tendency to undergo a spontaneous transition from one type of motion to another. The best-known type of transition is that from laminar flow (a smooth, regular type of flow) to turbulent flow (in which rapid, irregular fluctuations arise). Instability can also lead to a complicated flow with a highly regular structure (such as an orderly array of vortices or of convection cells). Much current research is concerned with gaining an understanding of these various transitions and, in particular, of how a deterministic set of equations can account for the chaotic behaviour of turbulent fluids. During flow at speeds comparable to the speed of sound, the density of fluids changes significantly. This phenomenon is of practical importance only for gases, in which shock waves may occur. These waves involve an almost discontinuous change in the velocity, temperature, pressure, and density of the fluid. The main phenomena of importance for liquids but not for gases are those associated with free surfaces, such as the upper boundary of a liquid in a partly filled vessel. The fact that the speed of water waves varies with wavelength and with amplitude leads to a wide variety of effects. These include the hydraulic jump (or bore)a sudden change in water level, analogous to a shock waveand the solitona single large-amplitude pulse that propagates without change of form. science concerned with the response of fluids to forces exerted upon them. It is a branch of classical physics with applications of great importance in hydraulic and aeronautical engineering, chemical engineering, meteorology, and zoology. The most familiar fluid is of course water, and an encyclopaedia of the 19th century probably would have dealt with the subject under the separate headings of hydrostatics, the science of water at rest, and hydrodynamics, the science of water in motion. Archimedes founded hydrostatics in about 250 BC when, according to legend, he leapt out of his bath and ran naked through the streets of Syracuse crying Eureka!; it has undergone rather little development since. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as Leonhard Euler and Daniel Bernoulli began to explore the consequences, for a virtually continuous medium like water, of the dynamic principles that Newton had enunciated for systems composed of discrete particles. Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G.G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a drag force upon it; the theory of potential flow, which worked so well in other contexts, yielded results that at relatively high flow rates were grossly at variance with experiment. This problem was not properly understood until 1904, when the German physicist Ludwig Prandtl introduced the concept of the boundary layer (see below Hydrodynamics: Boundary layers and separation). Prandtl's career continued into the period in which the first manned aircraft were developed. Since that time, the flow of air has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid dynamics. The term fluid mechanics, as used here, embraces both fluid dynamics and the subject still generally referred to as hydrostatics. One other representative of the 20th century who deserves mention here besides Prandtl is Geoffrey Taylor of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and quantum mechanics, and he made several unexpected and important discoveries in the field of fluid mechanics. The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlineari.e., one that involves the fluid velocity twice over. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in chaos that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century. (For a discussion of the concept of chaos, see physical science, principles of.) Fluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section. For further details, see thermodynamics and liquid. Additional reading A classic text which enshrines all the results of 19th-century fluid dynamics is Horace Lamb, Hydrodynamics, 6th ed. (1932, reissued 1945). This remains useful, but many later books, besides being more up-to-date, provide a better-balanced perspective of the subject and have better illustrations. N. Curle and H.J. Davies, Modern Fluid Dynamics, vol. 1, Incompressible Flow (1968); and G.K. Batchelor, Introduction to Fluid Dynamics (1967, reissued 1973), can both be recommended to serious students who are not put off by mathematics. D.J. Tritton, Physical Fluid Dynamics, 2nd ed. (1988), adopts a somewhat different approach and contains interesting material on turbulence and convective instabilities. Readers who are interested in the practical aspects of the subject and who want information concerning hydrostatics as well as fluid dynamics should consult one of the many good texts intended for engineers; among these, B.S. Massey, Mechanics of Fluids, 5th ed. (1983), is excellent. The development of the subject as a practical science is traced in Hunter Rouse and Simon Ince, History of Hydraulics (1957, reprinted 1980). Thomas E. Faber