Stability and instability in nineteenth-century fluid mechanics
[Stabilité et instabilité en mécanique des fluides au XIXe siècle]
Revue d'histoire des mathématiques, Tome 8 (2002) no. 1, pp. 5-65.

Au dix-neuvième siècle, la stabilité ou l'instabilité de quelques écoulements simples fut l'objet de conjectures, de débats et parfois de preuves mathématiques. Les motivations pour ce type de recherche variaient considérablement, de la turbulence observée d'écoulements réels à la permanence attendue dans les théories hydrodynamiques de la matière. Les mathématiques contemporaines étaient rarement en mesure de fournir des réponses rigoureuses et les intuitions des uns et des autres conduisirent parfois à des résultats faux. Néanmoins, quelques grands de la physique mathématique britannique et allemande - Stokes, Kelvin, Helmholtz et Rayleigh - développèrent certaines idées et méthodes de base de la théorie moderne des instabilités hydrodynamiques. Ils y parvinrent en réfléchissant à des problèmes spécifiques concrets, avec une étonnante diversité de styles.

The stability or instability of a few basic flows was conjectured, debated, and sometimes proved in the nineteenth century. Motivations varied from turbulence observed in real flows to permanence expected in hydrodynamic theories of matter. Contemporary mathematics often failed to provide rigorous answers, and personal intuitions sometimes gave wrong results. Yet some of the basic ideas and methods of the modern theory of hydrodynamic instability occurred to the elite of British and German mathematical physics, including Stokes, Kelvin, Helmholtz, and Rayleigh. This usually happened by reflecting on concrete specific problems, with a striking variety of investigative styles.

Mots clés : hydrodynamic instability, Stokes, Helmholtz, Kelvin, Rayleigh, Reynolds
@article{RHM_2002__8_1_5_0,
     author = {Darrigol, Olivier},
     title = {Stability and instability in nineteenth-century fluid mechanics},
     journal = {Revue d'histoire des math\'ematiques},
     pages = {5--65},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {8},
     number = {1},
     year = {2002},
     zbl = {1119.76301},
     language = {en},
     url = {http://www.numdam.org/item/RHM_2002__8_1_5_0/}
}
TY  - JOUR
AU  - Darrigol, Olivier
TI  - Stability and instability in nineteenth-century fluid mechanics
JO  - Revue d'histoire des mathématiques
PY  - 2002
SP  - 5
EP  - 65
VL  - 8
IS  - 1
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/RHM_2002__8_1_5_0/
LA  - en
ID  - RHM_2002__8_1_5_0
ER  - 
%0 Journal Article
%A Darrigol, Olivier
%T Stability and instability in nineteenth-century fluid mechanics
%J Revue d'histoire des mathématiques
%D 2002
%P 5-65
%V 8
%N 1
%I Société mathématique de France
%U http://www.numdam.org/item/RHM_2002__8_1_5_0/
%G en
%F RHM_2002__8_1_5_0
Darrigol, Olivier. Stability and instability in nineteenth-century fluid mechanics. Revue d'histoire des mathématiques, Tome 8 (2002) no. 1, pp. 5-65. http://www.numdam.org/item/RHM_2002__8_1_5_0/

[1] Batchelor (George K.) [1967] An Introduction to Fluid Dynamics, Cambridge, 1967. | Zbl

[2] Birkhoff (Garrett) [1950] Hydrodynamics: A Study in Logic, Fact, and Similitude, Princeton, 1950. | MR | Zbl

[3] Brillouin (Marcel) [1911] Les surfaces de glissement de Helmholtz et la résistance des fluides, Annales de chimie et de physique, 23 (1911), pp.145-230. | JFM

[4] Caflisch (Russel) [1990] Analysis for the evolution of vortex sheets, UCLA Computational and Applied Mathematics, Report 91-05 (Nov 1990). | MR | Zbl

[5] Darrigol (Olivier) [1998] From organ pipes to atmospheric motions: Helmholtz on fluid mechanics, HSPS 29 (1998), pp.1-51.

[6] Darrigol (Olivier) [2000] Electrodynamics from Ampère to Einstein, Oxford, 2000. | Zbl

[7] Darrigol (Olivier) [2002a] Between hydrodynamics and elasticity theory: The first five births of the Navier-Stokes equation, Archive for History of Exact Sciences, 56 (2002), pp.95-150. | MR | Zbl

[8] Darrigol (Olivier) [2002b] Turbulence in nineteenth-century hydrodynamics, HSPS, 32 (2002), pp.207-262. | MR

[9] Drazin (P.G.) & Reid (W.H.) [1981] Hydrodynamic Stability, Cambridge, 1981. | MR

[10] Hadamard (Jacques) [1903] Leçons sur la propagation des ondes et sur les équations de l'hydrodynamique, Paris, 1903. | JFM

[11] Helmholtz (Hermann) [1858] Über die Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, JRAM, also in HWA 1, pp.101-104. | Zbl

[12] Helmholtz (Hermann) [1868] Über discontinuirliche Flüssigkeitsbewegungen, Königliche Adademie der Wissenschaften zu Berlin, Monatsberichte, also in HWA 1, pp.146-157.

[13] Helmholtz (Hermann) [1877] Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik, 4th ed., Braunschweig, 1877. | JFM

[14] Helmholtz (Hermann) [1886] Wolken- und Gewitterbildung, Physikalische Gesellschaft zu Berlin, Verhandlungen, also in HWA 3, pp.287-288.

[15] Helmholtz (Hermann) [1888] Über atmospherische Bewegungen I, BB, also in HWA 3, pp.289-308.

[16] Helmholtz (Hermann) [1889] Über atmospherische Bewegungen. Zweite Mittheilung, BB, also in HWA 3, pp.309-332.

[17] Helmholtz (Hermann) [1890] Die Energie der Wogen und des Windes, BB, also in HWA 3, pp.333-355.

[18] Hunt (Bruce) [1991] The Maxwellians, Ithaca, 1991. | MR

[19] Kirchhoff (Gustav) [1869] Zur Theorie freier Flüssigkeitsstrahlen, JRAM 70 (1869), pp.289-298. | JFM

[20] Klein (Felix) [1910] Über die Bildung von Wirbeln in reibungslosen Flüssigkeiten, Zeitschrift für Mathematik und Physik 58 (1910), pp.259-262. | JFM

[21] Knudsen (Ole) [1971] From Lord Kelvin's notebook: Ether speculations, Centaurus 16 (1971), pp.41-53.

[22] Lamb (Horace) [1932] Hydrodynamics, 6th ed., Cambridge, 1932.

[23] Le Conte (John) [1858] On the influence of musical sounds on the flame of a jet of coal-gas, PM 15 (1858), pp.235-239.

[24] Lin (Chia Chiao) [1966] The Theory of Hydrodynamic Stability, 2nd ed., Cambridge, 1966. | MR | Zbl

[25] Lindsay (R.B.) [1976] Strutt, John William, third baron Rayleigh, DSB 13 (1976), pp.100-107.

[26] Love (Augustus E.H.) [1901] Hydrodynamik: Theoretische Ausführungen, in Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol.4.3, pp.86-149.

[27] Mayer (Alfred) [1878] A note on experiments on floating magnets, Nature 17 (1878), pp.487-488.

[28] Orr (William) [1907] The stability or instability of the steady motions of a perfect liquid and of a viscous fluid. Part II: A viscous fluid, Royal Irish Academy, Proceedings 27 (1907), pp.69-138. | JFM

[29] Prandtl (Ludwig) [1904] Über Flüssigkeitsbewegung mit kleiner Reibung, III. Internationaler Mathematiker-Kongress, Verhandlungen, also in Gesammelte Abhandlungen zur Mechanik, Hydro- und Aerodynamik, 3 vols., Göttingen, 1961.

[30] Prandtl (Ludwig) [1949]: Führer durch die Strömungslehre, Braunschweig, 1949.

[31] Rayleigh (Lord) [1876] On the resistance of fluids, PM, also in RSP 1, pp.287-296.

[32] Rayleigh (Lord) [1879] On the instability of jets, PLMS, also in RSP 1, pp.361-371. | JFM | MR

[33] Rayleigh (Lord) [1880] On the stability, or instability, of certain fluid motions, PLMS, also in RSP1, pp.474-487. | JFM

[34] Rayleigh (Lord) [1883] On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems, PT, also in RSP 2, pp.239-257.

[35] Rayleigh (Lord) [1884a] Acoustical observations, V, PM, also in RSP 2, pp.268-275.

[36] Rayleigh (Lord) [1884b] Presidential address, BAR, also in RSP 2, pp.332-354.

[37] Rayleigh (Lord) [1887] On the stability or instability of certain fluid motions, II, PLMS, also in RSP 3, pp.17-23. | JFM

[38] Rayleigh (Lord) [1892] On the question of the stability of the flow of fluids, PM, also in RSP 3, 575-584. | JFM

[39] Rayleigh (Lord) [1895] On the stability or instability of certain fluid motions, III, PLMS, also in RSP 4, pp.203-208. | JFM

[40] Rayleigh (Lord) [1896] Theory of Sound, 2nd ed., 2 vols., London 1896.

[41] Reynolds (Osborne) [1873] The causes of the racing of screw steamers investigated theoretically and by experiment, Institution of Naval Architects, Transactions, also in ReP, pp.51-58.

[42] Reynolds (Osborne) [1878] On the internal cohesion of liquids and the suspension of a column of mercury to a height more than double that of the barometer, Manchester Literary and Philosophical Society, Memoirs, also in ReP 1, pp.231-243.

[43] Reynolds (Osborne) [1883] An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, PT, also in ReP 2, pp.51-105. | JFM

[44] Reynolds (Osborne) [1894] On the dynamical theory of incompressible viscous fluids and the determination of the criterion, PT (1894), also in ReP, vol. 2, pp.535-577. | JFM

[45] Silliman (Robert H.) [1963] William Thomson: Smoke rings and nineteenth-century atomism, Isis 54 (1963), pp.461-474.

[46] Smith (Crosbie) & Norton (Wise) [1989] Energy and Empire: A Biographical Study of Lord Kelvin, Cambridge, 1989. | MR

[47] Stokes (George Gabriel) [1842] On the steady motion of incompressible fluids, TCPS, also in SMPP 1, pp.1-16.

[48] Stokes (George Gabriel) [1843] On some cases of fluid motion, TCPS, also in SMPP 1, pp.17-68.

[49] Stokes (George Gabriel) [1845] On the theory of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, TCPS (read in 1845, pub. in 1849), also in SMPP 1, pp.75-129.

[50] Stokes (George Gabriel) [1846a] On the constitution of the luminiferous ether, viewed with reference to the phenomenon of the aberration of light, PM, also in SMPP 1, pp.153-156.

[51] Stokes (George Gabriel) [1846b] On the critical values in the sum of periodic series, TCPS, also in SMPP 1, pp.236-313.

[52] Stokes (George Gabriel) [1848] On the constitution of the luminiferous ether, PM, also in SMPP 2, pp.8-13.

[53] Stokes (George Gabriel) [1850] On the effect of the internal friction of fluids on the motion of pendulums [read on 9 Dec. 1850], TCPS, also in SMPP 3, pp.1-141.

[54] Thompson (Silvanus) [1910] The Life of William Thomson, Baron Kelvin of Largs, 2 vols., London, 1910. | JFM

[55] Thomson (William - Lord Kelvin) [1849] Notes on hydrodynamics. On the vis-viva of a liquid in motion, The Cambridge and Dublin Mathematical Journal, also in TMPP 1, pp.107-112.

[56] Thomson (William - Lord Kelvin) [1867] On vortex atoms, PRSE, also in TMPP 4, pp.1-12.

[57] Thomson (William - Lord Kelvin) [1868] On vortex motion, PRSE, also in TMPP 4, pp.13-66.

[58] Thomson (William - Lord Kelvin) [1871a] The influence of wind and capillarity on waves in water supposed frictionless, letter to Tait, 16 Aug. 1871, PM, also in TMPP 4, pp.76-79.

[59] Thomson (William - Lord Kelvin) [1871b] Letter to Tait, 23 Aug. 1871, PM, also in TMPP 4, pp.79-80.

[60] Thomson (William - Lord Kelvin) [1871c] Ripples and waves, Nature, also in TMPP 4, pp.86-92.

[61] Thomson (William - Lord Kelvin) [1871d] Hydrokinetic solutions and observations, PM, also in TMPP 4, pp.69-92 (contains 1871 a and 1871 b).

[62] Thomson (William - Lord Kelvin) [1876] Vortex statics, PRSE, also in TMPP 4, pp.115-128.

[63] Thomson (William - Lord Kelvin) [1878] Floating magnets, Nature, also in TMPP 4, pp.135-140.

[64] Thomson (William - Lord Kelvin) [1880a] Vibrations of a columnar vortex, PRSE, also in TMPP 4, pp.152-165.

[65] Thomson (William - Lord Kelvin) [1880b] Maximum and minimum energy of vortex motion, BAR, also in TMPP 4, pp.172-183.

[66] Thomson (William - Lord Kelvin) [1880c] On a disturbing infinity in Lord Rayleigh's solution for waves in a plane vortex stratum, Nature, also in TMPP 4, pp.186-187.

[67] Thomson (William - Lord Kelvin) [1887a] On the formation of coreless vortices by the motion of a solid through an inviscid incompressible fluid, PRS, also in TMPP 4, pp.149-151. | JFM

[68] Thomson (William - Lord Kelvin) [1887b] On the stability of steady and periodic fluid motion, PM, also in TMPP 4, pp.166-172.

[69] Thomson (William - Lord Kelvin) [1887c] Rectilinear motion of viscous fluid between two parallel planes, PM, also in TMPP 4, pp.321-330.

[70] Thomson (William - Lord Kelvin) [1887d] Broad river flow down an inclined plane bed, PM, also in TMPP 4, pp.330-337.

[71] Thomson (William - Lord Kelvin) [1887e] On the propagation of laminar motion through a turbulently moving inviscid fluid, BAR, also in TMPP 4, pp.308-320. | JFM

[72] Thomson (William - Lord Kelvin) [1889] On the stability and small oscillations of a perfect liquid full of nearly straight coreless vortices, Royal Irish Academy, Proceedings, also in TMPP4, pp.202-204.

[73] Thomson (William - Lord Kelvin) [1894] On the doctrine of discontinuity of fluid motion, in connection with the resistance against a solid moving through a fluid, Nature, also in TMPP 4, pp.215-230. | JFM

[74] Thomson (William - Lord Kelvin) [1904] Deep water ship-waves, PM, also in TMPP 4, pp.368-393.

[75] Thomson (William) & Tait (Peter Guthrie) [1867] Treatise on Natural Philosophy, Oxford, 1867.

[76] Tyndall (John) [1867] On the action of sonorous vibrations on gaseous and liquid jets, PM 33, pp.375-391.

[77] Wilson (David B.) [1987] Kelvin and Stokes: A Comparative Study in Victorian Physics, Bristol, 1987.

[78] Wilson (David B.) [1990] (editor of) The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, 2 vols., Cambridge, 1990.