The European Physical Journal H

, Volume 40, Issue 3, pp 375–384 | Cite as

Euler’s friction of fluids theory and the estimation of fountain jet heights

  • Sylvio R. Bistafa


In 1761, Leonhard Euler (1707–1783) published a treatise with the title “Attempt at a Theory of the Friction of Fluids”, in which he assumed that, as is the case for solid friction, fluid friction is proportional to pressure. Several experiments were proposed by Euler to derive a friction factor, which were intended to experimentally confirm his equations. Detailed developments of five different problems of discharge were presented in his treatise, taking into account the loss of head in the conduits. In the Appendix, an example is given of the calculation of the jet heights of a particular fountain, fed with conduits of different cross-sectional areas. Application of the current method for the calculation of head losses in pipes reveals that Euler grossly overestimated the fountain jet heights.


Friction Factor Head Loss Solid Friction Pipe Roughness Water Fountain 
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  1. Brown, G. 2002. The History of the Darcy-Weisbach Equation for Pipe Flow Resistance. Environmental and Water Resources History, American Society of Civil Engineers, 34-43.Google Scholar
  2. Darrigol, O. 2005. Worlds of Flow: A history of hydrodynamics from the Bernoullis to Prandtl. Oxford University Press, Oxford. Google Scholar
  3. Eckert, M. 2002. Euler and the Fountains of Sanssouci. Arch. Hist. Exact Sci. 56: 451–468.zbMATHMathSciNetCrossRefADSGoogle Scholar
  4. Euler, L. 1761. Tentamen theoriae de frictione fluidorum [Attempt at a theory of the friction of fluids]. [Eneström index] E 260. Originally published in Novi Commentarii academiae scientiarum Petropolitanae 6, 1761, 338-388. Also published in Opera Omnia: Series II, Volume 12, 169–214. Both publications and a Portuguese translation by the author can be found in ‘The Euler Archive’ (
  5. Fellmann, E.A. 2007. Leonhard Euler. (Gautschi, E. & Gautschi, W.,Trans.). Birkhäuser Verlag: Basel. Google Scholar
  6. Knobloch, E. 2008. Euler, the historical perspective. Physica D 237: 1887–1893. zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. Swamee, P.K., Jain, A.K. 1976. Explicit equation for pipe flow problems., J. Hydr. Div., ASCE 102: 657–664. Google Scholar
  8. Truesdell, C.A. 1955. Rational Fluid Mechanics, 1687–1765. Opera Omnia. (Series II, Volume 12), Lausanne.Google Scholar
  9. Weisbach, J. 1845. Lehrbuch der Ingenieur-und Maschinen-Mechanik, Vol. 1. Theoretische Mechanik, Vieweg und Sohn, Braunschweig. 535 pages (in German). Retrieved from
  10. White, F.M. 1979. Fluid Mechanics. New York, NY: McGraw-Hill.Google Scholar

Copyright information

© EDP Sciences and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Polytechnic SchoolUniversity of São PauloSão PauloBrazil

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