Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl

AWorlds of FlowHistory of Hydrodynamics fromthe Bernoullis to PrandtlOLIVIER DARRIGOLOXFORDUNIVERSITY PRESSThisbook has been printed digitally and produced in a standard specificationin order to ensure its continuing availabilityOXFORDUNIVERSITY PRESSGreat Clarendon Street, Oxford OX2 6DPOxford University Press is a department of the University of Oxford.It furthers the University's objective of excellence in research, scholarship,and education by publishing worldwide inOxford New YorkAuckland Cape Town Dar es Salaam Hong Kong KarachiKuala Lumpur Madrid Melbourne Mexico City NairobiNew Delhi Shanghai Taipei TorontoWith offices inArgentina Austria Brazil Chile Czech Republic France GreeceGuatemala Hungary Italy Japan South Korea Poland PortugalSingapore Switzerland Thailand Turkey Ukraine VietnamOxford is a registered trade mark of Oxford University Pressin the UK and in certain other countriesPublished in the United Statesby Oxford University Press Inc., New York© Oxford University Press 2005Not to be reprinted without permissionThe moral rights of the author have been assertedDatabase right Oxford University Press (maker)Reprinted 2008All rights reserved.

No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,without the prior permission in writing of Oxford University Press,or as expressly permitted by law, or under terms agreed with the appropriatereprographics rights organization. Enquiries concerning reproductionoutside the scope of the above should be sent to the Rights Department,Oxford University Press, at the address aboveYou must not circulate this book in any other binding or coverAnd you must impose this same condition on any acquirerISBN 978-0-19-856843-8Cover image: The maelstrom in Saltstraumen close to Beda along the Norwegiancoast.

(These whirlpools are created by an instability of the flow due to the verystrong tides encountering a large rock.) By Courtesy of Jer6me Cuny.Printed and bound by CPI Antony Rowe, EastbournePREFACENow I think hydrodynamics is to be the root of all physical science, and is atpresent second to none in the beauty of its mathematics.' (William Thomson,December 1857)In Victorian Britain, William Thomson wanted hydrodynamics to reign over all thephysical sciences, from the intimate constitution of ether and matter to the beautifulpattern of ship waves.

Although today's scientists tend to ascribe a less dominant role tothis theory, it still inspires many human endeavors and it challenges the skills of a largecommunity of experts. Many of the problems of modem hydrodynamics have ropts inThomson's times. Despite tremendous improvements in methods, the basic equations andthe basic questions have been more stable in this field than in any other domain of physics.These circumstances probably explain the interest that students of hydrodynamics oftenmanifest in the history of their subject.The present book aims at satisfying this curiosity, in a way that only requires anelementary knowledge of modem fluid mechanics.

Through its focus on the concrete,worldly circumstances of theoretical progress, it should appeal to a great variety of users ofhydrodynamics. By attending to the technicalities of concept formation and evolution, itmay help budding specialists understand the foundations of their own field. By exploringdeep genetic connections with other domains of knowledge, it should also interest histor­ians of science. In addition, this history answers a few questions recently raised in thephilosophy of science, regarding the failures of purely deductive conceptions of physicaltheory and the role of asymptotic theory and singular approximations.Hydrodynamics may be defined as the art of subjecting flow to the general principles ofdynamics. This book tells how, in the eighteenth century, a small elite of Swiss and Frenchgeometers inaugurated this theory; how nineteenth-century engineers, physicists, andmathematicians widely expanded its concepts, methods, and purposes; and how its con­crete applications nevertheless failed until, in the first third of the twentieth century, itevolved into a reliable guide for the engineers of flow.There were many natural and artificial worlds of flow to which hydrodynamics wasmeant to be relevant: the hydraulics of conduits, rivers, and canals; the physiology ofblood and sap circulation; aspects of navigiltion including tides, waves, and ship resist­ance; acoustic phenomena; the damping of the seconds pendulum in air; atmosphericmotion; bird flight, ballooning, and aviation; British theories of the ether as a perfectliquid; and hydrodynamic analogies for electromagnetic phenomena.

This book offersglimpses into each of these worlds, with an emphasis on aspects of fluid mechanics inwhich compressibility and thermal effects play no role (hydrodynamics in the narrowsense).1Thomson to Stokes, 20 Dec. 1857, ST.viPREFACEThere were also many socio-professional worlds of flow, which may be arranged intotwo categories: one comprising the academic elite of mathematics, natural philosophy, andastronomy; and another including the practical men of hydraulics, naval architecture,construction engineering, meteorology, and instrument making. Although the founders ofhydrodynamics belonged to the first category, they hoped to serve the world of engineersas part of their pursuit of the enlightenment ideal of a rational unification of knowledge.The new hydrodynamics largely frustrated this desire. It is indeed a cliche of the history offluid mechanics that until the twentieth century at least two separate disciplines of flowexisted-hydrodynamics and hydraulics-which implied utterly different methods andprofessional identities, and which evolved independently of each other.

Whereas hydro­dynamicists applied advanced mathematics to flows rarely encountered by engineers,hydraulicians used simple empirical or semi-empirical formulas that defied deeper theory.2D'Alembert's paradox of 1768, according to which the steady motion of a perfect liquidexerts no force on a fully-immersed solid, quickly became the emblem of the split betweenthe ideal and the practical worlds of flow.

As may be retrospectively judged, the cause ofthis misfit was not any error in the application of the laws of mechauics. The correctequations of fluid motion, namely Euler's and Navier's, were indeed known at early stagesof the history of hydrodynamics. The real difficulty lay in analyzing the consequences ofthese equations. The results and solutions that could be derived from them contradictedcommon observation-indeed they forbade the soaring of birds and made water rush atunreal velocities in channels or conduits. In1786,the prominent hydraulician CharlesBossut recorded this impotence of the new fluid mechauics:These great geometers [d'Aiembert and Euler] seem to have exhausted the resourcesthat can be drawn from analysis to determine the motion of fluids: their formulas areso complex, by the nature of things, that we may only regard them as geometrictruths, and not as symbols fit to paint the sensible image of the actual and physicalmotion of a fluid.More than a century later, the prominent hydrodynamicist Wilhelm Wien similarlylamented:3In hydrodynamics ...

the real processes differ so much from the theoretical conclu­sions that engineers have had to develop their own approach to hydrodynamicsquestions, usually called hydraulics. In this approach, however, both foundationsand conclusions lack rigor to such an extent that most results remain confined Joempirical formulas of very limited validity.For a long time, knowledge of the fundamental equations of the theory proved utterlyinsufficient in practice-a philosophically interesting situation on which more will be saidin the conclusions in Chapter8.Yet there was no lack of attempts, all through thenineteenth century, to make hydrodynamics more relevant to the practical problems offlow.

Despite their frequent failure, these attempts were the source of almost everyconceptual innovation in the science of flow of those times. For this reason this book islargely a history of the transgression of the borders between ideal and practical worlds of2In this book, I indulge in tbe convenient neologisms 'hydrodynamicist' and 'hydraulician'.3Bossut [1786], vol. I p. xv; Wien [1900] p.

V.PREFACEviiflow, and of the several subcultures that favored them. Polytechnique-trained Frenchengineers such as Navier and Saint-Venant formed one of these subcultures. In VictorianBritain, natural philosophers such as Stokes, Thomson, and Rayleigh, and theory­oriented engineers or engineering professors such as Russell, Rankine, Froude, andReynolds formed two others. In Germany, the quintessential polymath Hermann Helm­holtz by himself defmed still another subculture of mediation between the ideal and thepractical.By elevating themselves above and yet struggling to act in the many practical worlds offlow, investigators like Saint-Venant, Stokes, or Helmholtz blurred the borders betweenthese worlds. In their view, the retardation of water in hydraulic conduits, the friction ofwater on ship hulls, the subduction of winds near the ground, or the steering of airships, allbelonged to the same category of phenomena.

Any insight into one of them was transfer­able to the others. They trusted that ultimately these phenomena would be covered by thesame mathematical theory. This synthetic perspective and the belief in theoretical transferand cross-fertilization contrasted with the cultural isolation of the average practitioner.Whereas, for example, a Rankine or a Froude found it natural to apply pipe retardationformulas to the skin resistance of ships, most naval architects and hydraulicians ignoredone another's fields and thus sometimes arrived at similar Jaws without knowing it.The opposition between ideal and real worlds of flow, the fertility of attempts toreconcile them, and the bridging of different practical worlds, are the organizing themesof the following narrative. Chapter I recounts the eighteenth-century emergence of anideal world of flow under the rule of partial differential equations by tracing the efforts ofDaniel and Johann Bemoulli, Jean le Rond d'Alembert, Leonhard Euler, and JosephLouis Lagrange.