Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl , Olivier Darrigol , Oxford U. Press, New York, 2005. $74.50 paper (356 pp.). ISBN 0-19-856843-6
Olivier Darrigol is a distinguished historian of science who had previously published a prizewinning volume, Electrodynamics from Ampère to Einstein (Oxford U. Press, 2000), on another branch of physics. His latest book, Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, will be of most interest to historians, especially those who already have some knowledge of the scientists, engineers, and scientific topics that the author covers.
The text is very mathematical. Darrigol repeats the detailed derivations of the earlier mathematicians, including their sidetracks and mistakes, but with modern vector notation and methods. His book could also attract a wider audience: By skating over the details and gaining a broad view of the achievements and interactions of the pioneers, scientists who use hydrodynamics will find Darrigol’s treatment interesting and entertaining. He covers more than a century of hydrodynamics in detail, from the 18th century to the early 20th century, when physicists began to tackle boundary effects in nonidealized flows.
In the 18th and 19th centuries there was a clear distinction between hydrodynamics and hydraulics, and the fields evolved independently. Hydrodynamics was the application of advanced mathematics to idealized flows rarely encountered by engineers, who used empirical formulas to make progress with practical hydraulics problems. Only in the 20th century were the challenges of real flows properly dealt with by physically based theories, and those developments, too, are well described by Darrigol. The early contributors to the tortuous development of the equations of motion of inviscid fluids include the Swiss mathematicians Leonhard Euler, Johann Bernoulli, and Johann’s son Daniel; and the Frenchmen Joseph Louis de Lagrange and Jean le Rond d’Alembert. Although these names are now attached to their respective mathematical results, a great deal of overlap and interaction exists between each formulation.
One problem that the early theorists could solve was the motion of waves on a free surface. French mathematicians provided strong input, but most water-wave phenomena were well known in navigation contexts, both at sea and in canals, before they could be explained. British physicists paid more attention to such practical questions than did their continental peers; they provided useful solutions to those questions and added to fundamental theory. A striking example of the different approach by British hydraulic engineers was John Scott Russell’s accidental discovery of what are now known as solitary waves. A canal boat, towed at high speed and then suddenly stopped, produced a large wave of permanent form that continued to progress independently along the canal. Russell’s observations and the deductions from them were brilliant; his theoretical insights, however, were not very sound, and he was strongly criticized by his contemporaries—in particular in the work of George Biddell Airy, George Gabriel Stokes, and William Thomson (later known as Lord Kelvin).
Much more firmly based contributions to other aspects of wave theory were made by British mathematicians. Among them were Airy, who wrote an influential review, Tides and Waves , published in 1845, and Stokes, who studied finite oscillatory waves. Thomson made notable contributions to the theory of capillary waves, the understanding of three-dimensional ship-wave patterns, and the concept of group velocity.
The French civil engineer Claude Louis Navier was the first to break away from the concept of a perfect fluid by obtaining a hydrodynamic equation for viscous flow in 1821. Two other French mathematicians, Augustin Cauchy and Siméon Poisson, rediscovered Navier’s equation before Stokes independently derived it in 1845, but it was many years before the Navier–Stokes equation became a standard tool in hydrodynamics. Stokes used the equation in the linear approximation to derive the formula for the velocity of a sphere sinking through a viscous fluid; but this method only helped for laminar flow.
In the 1860s, Hermann von Helmholtz invented another approach to the problem of fluid friction; his approach was based on vortex-like solutions of Euler’s equations. He introduced the concept of a vortex sheet and first applied the theory to sound generation in organ pipes and then to vortex rings. His ideas were enthusiastically received and were followed up by his colleague and friend Thomson and other British mathematicians. The gap between small- and large-scale physics began to narrow. What is now called the Kelvin–Helmholtz instability was observed both in the laboratory and in the atmosphere, and a photograph of the experiment is presented and described in Worlds of Flow.
Stokes and Thomson studied other instabilities, but the two friends had different views about the roles of viscosity and surfaces of discontinuity. Lord Rayleigh studied inviscid flow between fixed walls and concluded that parallel flow without an inflexion point in the velocity profile is stable. The engineer Osborne Reynolds showed that for turbulence to occur in pipe flow, a small viscosity is essential. He demonstrated the concept experimentally and introduced the dimensionless ratio later named in his honor. Factors that impede ship motion, including the effects of large eddies, surface waves, and skin friction, were tackled by many mathematicians in the late 19th century. The decisive step to understanding skin friction was taken by the German mathematician Ludwig Prandtl, who introduced the boundary-layer approach. Prandtl showed that viscosity is only significant in a thin layer near a solid surface and that flow outside the layer can be treated as inviscid. He went on to study the formation of turbulent boundary layers and the criteria for boundary-layer separation. Wing theory, the understanding of lift and drag, and the development of the whole of modern aeronautics have depended on his ideas.
Despite its extensive treatment of the history of hydrodynamics, I have a personal disappointment with Worlds of Flow : It is a pity that the author chose to end the story with Prandtl, with only a passing reference to Geoffrey Taylor, a towering figure in British fluid mechanics whose life and contributions overlapped with Prandtl’s. Nonetheless, by presenting in detail the interactions between many mathematicians and engineers, and by emphasizing the different styles characteristic of scientists in different countries, Darrigol has provided a fascinating insight into the development of hydrodynamics.
