Darrigol O. Worlds of flow. A history of hydrodynamics from the Bernoullis to Prandtl (794382), страница 3
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My incursion into early twentieth-century hydrodynamics is even more limited: I have exclusively attended to the Gottingen school, although the Cambridgeschool harvested results of fundamental import to the broader development of hydrodynamics.5Rouse and !nee [1957]; Tokaty [1971].XPREFACEMelucci has written an informative dissertation. On tides, there is the clearly-written bookby the modem expert David Edgar Cartwright. The monumental biography of Kelvin byCrosbie Smith and Norton Wise is a rich source on Victorian science and Kelvin's interestin flow analogies and navigation.
On ship hydrodynamics, there is the excellent (unfortunately unpublished) dissertation by Thomas Wright. On aerodynamics and the theory offlight, John Anderson's recent, easily readable history stands out. In a freshly publishedbook, John Ackroyd, Brian Axcell, and Anatoly Ruban give competently commentedtranslations of a few historical papers. Information on Prandtl's school, Theodore vonKarman, aerodynamics, and applied mathematics in the early twentieth century are foundin two rich books by Karman himself and by Paul Hanle, and in a short insightful study byGiovanni Battimelli.
I have myself published a few articles on some aspects of the historyof hydrodynamics. Much of their content is included in this book, with the kind permission of the University of California Press, the Societe Mathematique de France, andSpringer Verlag. 6Part of the research for this book was done in Paris, within the Rehseis research team ofCNRS and Paris VII. It is a pleasure to thank the director of that team, Karine Chemla,for intellectual stimulation and institutional support, and my friends Martha-CeciliaBustamante, Nadine de Courtenay, and Edward Jurkowitz for fruitful discussions.Long, pleasant stays at the Max Planck Institut fiir Wissenschaftsgeschichte in Berlin, atthe Dibner Institute in Cambridge, MA, at Harvard University's History of Sciencedepartment, and at UC-Berkeley's OHST have greatly eased my compilation of bibliographical and archival materials, as well as my subsequent reflections.
I am very grateful toJiirgen Renn, Peter Galison, Jed Buchwald, and Cathryn Carson, who welcomed me inthese institutions and offered useful suggestions for my work. I am also indebted to a fewhydrodynamicists for their warm support and for helpful criticism: Alex Craik, MarieFarges, Elizabeth Guazzelli, Etienne Guyon, and John Hinch (also Saint-Venant, in one ofmy dreams).It is not easy, and perhaps not desirable, to reduce the pleasant gurgling of a mountaincreek or the great surfs of the Silver Coast to the dry symbolism of a few fundamentalequations.
Neither is it easy to predict tides, pipe retardation, or ship resistance from firstprinciples. Over the two centuries spanned in this book, a few valiant men graduallymastered these more technical problems and even gained some insights into the morepoetical kinds of flow. I hope the story of their efforts will convince my reader that the lifeof hydrodynamics has been-and will be-as beautiful as some of the flows it purports toexplain."1-ruesdell [1954]; Grimberg [1998]; Saint-Venant [1887c], [1888]; Bullough [1988]; Craik [2004]; Belhoste[1994]; Picon [1992]; Melucci [1996]; Cartwright [1999]; Smith and Wise [1989]; Wright [1983]; Anderson [1997];Ackroyd, Axcell, and Ruban [2001] (ofwhich I became aware only after writing Chapter 7); Karman [1954]; Hanle[1982]; Battimelli [1984]; Darrigol [1 998], [2002a], [2002b], [2002c], [2003].CONVENTIONS AND NOTATIONFor the ease of the reader, uniform notation is used throughout this book, which of courseimplies some departure from original notation.
To a large extent, this liberty amounts to apermutation of letters. Less innocuously, Cartesian-component equations are rendered asvector equations (although no hydrodynamicist used the vector notation before Prandtland Sommerfeld), and Daniel Bemoulli's statements of proportions are translated intoequations with dimensional proportionality constants. This is not to deny the importanceof investigating how Johann Bemoulli, Euler, and others gradually reached the modernconcept of a physics equation, why Prandtl promoted for the vector notation, and howthese changes affected theory. Such enquiries would, however, exceed the scope of thepresent book.dl, dS, d,.ghkpru,vVelements of length, surface, and volume, respectivelyacceleration of gravityheightwave vectorpressureposition vectortimex- and y-components of the velocity, respectivelyfluid velocityLaplacian operatoreJ.LVeffective, large-scale viscosity (including eddy viscosity)viscosity parameterkinematic viscosityfluid densityJ.L/pdeformation of a water surface, sometimes pulsationshear stressstress tensorvelocity potential(v = 'Vcp)stream function for two-dimensional, incompressible flow00w'V\7 X V\7.
V(di/J= -V dx+ U dy)'V x vvorticityeither vorticity value or pulsation, according to the contextgradient operatorcurl of the vector fieldvdivergence of the vector fieldvxivCONVENTIONS AND NOTATION••Citations are in the author-date format and refer to one of the two bibliographies(primary and secondary). Page numbers refer to the last-mentioned source in thebibliographical item. Abbreviations are listed at the start of the bibliography.Translations are my own, unless the source is itself a translation.CONTENTSConventions and notation12xiiiThe dynamical equations1.1 Daniel Bemoulli's Hydrodynamica1.2 Johann Bemoulli's Hydraulica1.3 D'Alembert's flnid dynamics1.4 Euler's equations1.5 Lagrange's analysis149112326Water waves2.1 French mathematicians2.2 Scott Russell, the naval engineer2.3 Tides and waves2.4 Finite waves2.5 The principle of interference3132475669853 Viscosity3.1 Mathematicians' versus engineers' fluids3.2 Navier: molecular mechanics of solids and fluids3.3 Cauchy: stress and strain3.4 Poisson: the rigors of discontinuity3.5 Saint-Venant: slides and shears3.6 Stokes: the pendulum3.7 The Hagen-Poiseuille law1011031091191221261351404 Vortices4.1 Sound the organ4.2 Vortex motion4.3 Vortex sheets4.4 Foehn, cyclones, and storms4.5 Trade winds4.6 Wave formation1451461481591661721785 Instability5.1 Divergent flows5.2 Discontinuous flow5.3 Vortex atoms5.4 The Thomson-Stokes debate5.5 Parallel flow183184188190197208xiiCONTENTS6 Turbulence6.1 Hydraulic phenomenology6.2 Saint-Venant on tumultuous waters6.3 Boussinesq on open channels6.4 The turbulent ether6.5 Reynolds's criterion7 Drag and lift7.1 Tentative theories7.2 Ship resistance7.3 Boundary layers7.4 Wing theory8 ConclusionAppendix AModem discussion of d'Alembert's paradoxBibliographyBibliographic abbreviationsBibliography of primary literatureBibliography of secondary literatureIndex2192212292332392432642652732833023233263293293293443501THE DYNAMICAL EQUATIONSAdmittedly, as useful a matter as the motion of fluid and related sciences hasalways been an object of thought.
Yet until this day neither our knowledge ofpure mathematics nor our command of the mathematical principles of naturehave permitted a successful treatment. 1 (Daniel Bernoulli, September I 734)Modern derivations of the fundamental equations for non-viscous fluids have an air ofevidence. The fluid is divided into volume elements, and the acceleration of a volume elementis equated to a force divided by a mass.
The force on the element dT is the sum of an externalaction fd,. ( e.g. gravity) and of the resultant -(\7P)dT of the pressures exerted on the surfaceof the element by the surrounding fluid. Ifvdenotes the velocity of the fluid at the pointand timet, andr (t) is the position of the element at timet, then the acceleration of the elementis the time derivative of Vthat is, a V+ (v. 'ii')v. The mass of the element is theproduct of its density pand its volume dr. Hence Euler's equation follows:[r(t), t],(�;p(r, t)Iot)+ (v·'ii')v =f-'ii'P.r(1.1)The conservation of the mass of a fluid element during its motion further gives the'continuity equation':(1.2)so named because it assumes that the fluid remains continuous and does not burst intodroplets during its motion.A closer look at this derivation shows that it relies on concepts and idealizations that areby no means obvious.
Firstly, it assumes that the action of the surrounding fluid on a givenfluid portion can be represented by a normal pressure on its surface, whereas molecularintuition suggests a more complex distribution of the forces between inner and outermolecules. In fact, in real fluids the pressure is only normal in the case of rest; viscosityimplies a tangential component of the pressure . In order to justify the existence andproperties of internal pressure without appealing to the molecular picture, one couldexamine experience.
Unfortunately, experience only informs us of the pressure exertedon the surface of immersed objects, not of the pressure of the fluid on itself. The latternotion requires an unwarranted idealization.Another difficulty lurks in the application of Newton's second law to fluid elements. Ifthe fluid is thought of as an assembly of molecules, this application does not immediately1D. Bernoulli to Shoepflin, in MercureSuisse (Sept. 1734) pp. 42-50; also in Bernoulli [2002] pp. 87-90.2WORLDS OF FLOWfollow from the validity of the law for individual molecules, because a volume element isnot necessarily made of the same molecules during its evolution in time.
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