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Correspondence between de Saint-Venant and Boussinesq 5: Viscosity and hydraulic resistance
Comptes Rendus. Mécanique, Volume 349 (2021) no. 1, pp. 145-166.

Fluid viscosity is a main feature of fluids; an inviscid fluid does not exist even though a large number of theories has been advanced for flows of such fluids. The velocity of fluid flow may considerably be reduced due to the presence of fluid viscosity both for laminar and turbulent flows. The letters exchanged between de Saint-Venant (dSV) and Boussinesq mainly refer to laminar flow. The most peculiar statement of dSV is that, if the flow in a typical lowland river would be laminar, its velocity would be larger than the speed of sound. It is evident that this statement is wrong because laminar flow has stringent limitations typically expressed by the Reynolds number.

The hydraulic resistance is another peculiar feature in hydrodynamics, given that several theories have been advanced which do not at all reflect the everyday experience. One of these is the d’Alembert paradox, stating that the resistance is equal to zero for a steady flow of an inviscid and incompressible fluid. Typically, a body suspended in a large pipe is considered. Applying the momentum equation in the axial direction sufficiently up- and downstream of the body, the resistance would indeed become zero, as occurs for potential flows. Again, this is far away from everybody’s experience, such as for a swimmer or a walker under heavy wind.

This paper mainly explores the letters exchanged between the two late 19th century scientists dSV and Boussinesq. Their considerations indirectly advanced the turbulence theory later developed by Boussinesq, based on the Boussinesq turbulent exchange coefficient. It also rectifies the role of dSV in improving the fundamental equations currently referred to as the Navier–Stokes equations. Both aspects of turbulence and hydraulic resistance are by now still under considerable research activity.

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DOI : 10.5802/crmeca.71
Mots clés : Académie des Sciences, History of hydraulics, Hydrodynamics, Institut de France, Resistance, Viscosity
Willi H. Hager 1 ; Kolumban Hutter 2 ; Oscar Castro-Orgaz 3

1 VAW, ETH Zurich, CH-8093 Zürich, Switzerland
2 c/o VAW, ETH Zurich, CH-8093 Zürich, Switzerland
3 University of Cordoba, Campus Rabanales, E-14071 Cordoba, Spain
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Willi H. Hager; Kolumban Hutter; Oscar Castro-Orgaz. Correspondence between de Saint-Venant and Boussinesq 5: Viscosity and hydraulic resistance. Comptes Rendus. Mécanique, Volume 349 (2021) no. 1, pp. 145-166. doi : 10.5802/crmeca.71. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.71/

[1] J. R. d’Alembert Essai d’une nouvelle théorie de la résistance des fluides (Essay of A New Theory of Fluid Resistance), David, Paris, 1752 (in French)

[2] A. J. C. B. de Saint-Venant Mémoire sur la théorie de la résistance des fluides: Solution du paradoxe proposé à ce sujet par d’Alembert aux géomètres, comparaison de la théorie aux expériences (Memoir on the theory of fluid resistance: Solution of the paradox proposed on this topic by d’Alembert, comparison of the theory with experiments), C. R. Acad. Sci., Volume 24 (1847) no. 7, pp. 243-246 (in French)

[3] G. G. Stokes On the effect of internal friction of fluids on the motion of pendulums, Trans. Camb. Phil. Soc., Volume 9 (1851), pp. 8-106

[4] L. Prandtl Über Flüssigkeitsbewegung bei sehr kleiner Reibung (On fluid flow under extremely small friction)., Verhandl III, Intern. Math. Kongr. Heidelberg, Volume 2 (1904), pp. 484-491 (in German)

[5] A. J. C. B de Saint-Venant Considérations historiques et pratiques relatives au problème de l’action dynamique mutuelle d’un fluide et d’un solide, spécialement dans l’état de permanence supposé acquis par leurs mouvements (Fluid resistance. Historical and practical considerations relative to the problem of the mutual dynamic action of a fluid and a solid, especially under steady conditions supposed obtained by its movement), Résistance des fluides, Firmin-Didot, Paris, 1887 also printed in Mémoires de l’Académie des sciences de l’Institut de France, Paris, 44, no. 5, p. 1-192 (in French)

[7] J. Boussinesq Théorie des expériences de M. Poiseuille sur l’écoulement des liquides dans les tubes capillaires (Theory on the experiments of Mr. Poiseuille on the flow of liquids in capillary tubes), C. R. Acad. Sci., Volume 65 (1867) no. 2, pp. 46-48 (in French)

[8] J.-V. Poncelet Introduction à la mécanique industrielle (Introduction to Industrial Mechanics), Carilian-Goeury, Paris, 1841 (in French)

[9] W. H. Hager; O. Castro-Orgaz; K. Hutter Correspondence between de Saint-Venant and Boussinesq. 1: Birth of the shallow-water equations, C. R. Mec., Volume 347 (2019) no. 9, pp. 632-662 | DOI

[10] J. L. Poiseuille Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres (Experimental research on liquid flow in pipes of very small diameters) I. Influence de la pression sur la quantité de liquide qui traverse les tubes de très petits diamètres, C. R. Acad. Sci., Volume 11 (1840) no. 24, pp. 961-967

[11] J. L. Poiseuille Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres (Experimental research on liquid flow in pipes of very small diameters) II. Influence du diamètre sur la quantité de liquide qui traverse les tubes de très petits diamètres, C. R. Acad. Sci., Volume 11 (1840) no. 26, pp. 1041-1048

[12] J. L. Poiseuille Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres (Experimental research on liquid flow in pipes of very small diameters) IV. Influence de la température sur la quantité de liquide qui traverse les tubes de très petits diamètres, C. R. Acad. Sci., Volume 12 (1841) no. 2, pp. 112-115

[13] C. Bresse Cours de mécanique appliquée 2: Hydraulique (Course on Applied Mechanics 2: Hydraulics), Mallet-Bachelier, Paris, 1860 (in French)

[14] J. Boussinesq Essai sur la théorie les eaux courantes (Essay on the theory of flowing water), Mémoires, Volume 23 (1877), pp. 1-660 (présentées par divers savants à l’Académie des Sciences, Paris) | Zbl

[15] J. Boussinesq Additions et éclaircissements au mémoire intitulé : Essai sur la théorie des eaux courantes (Additions and spécifications on the memoir entitled: Essay on the theory of flowing waters), Mémoires, Volume 24 (1877), pp. 1-62 (présentées par divers savants à l’Académie des Sciences, Paris)

[16] W. H. Hager; K. Hutter; O. Castro-Orgaz Correspondence between de Saint-Venant and Boussinesq. 3: de Saint-Venant’s professional career and private life, C. R. Mec., Volume 348 (2020) no. 4, pp. 245-273

[17] W. H. Hager; K. Hutter; O. Castro-Orgaz Correspondence between de Saint-Venant and Boussinesq. 2: Boussinesq’s professional and private life up to 1886, C. R. Mec., Volume 348 (2020) no. 2, pp. 77-111

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