# Hydrodynamics of Ideal Liquids

• Kolumban Hutter
• Yongqi Wang
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

## Abstract

This chapter starts with kinematic concepts such as ‘motion’, ‘velocity’, ‘Eulerian and Lagrangean descriptions’, and then proceeds to describe ‘streamlines’, ‘trajectories’ and ‘streak lines’, illuminating these with illustrative examples. Next, the balance laws of mass and linear momentum are discussed both in global and local form and specialized to Eulerian fluids. The Bernoulli equation, defined as the path-integration of the scalar product of the momentum equation with the velocity field is given extensive space; it is discussed both when referred to non-inertial and inertial frames and when the integration is conducted along any path or along streamlines. Ample space is devoted to applications of the Bernoulli equation to typical examples, e.g., venturi pipes, Prandtl pipes, Torricelli flow out of a vessel, including clepsydra clocks. Global formulations of the momentum equation are equally touched and applied to the problem of Borda’s exit flows, impact of a jet on a wall, mixing processes of non-uniform velocities in plane conduits, hydraulic jumps and flow of a density preserving fluid through a periodic grid of wings. Aerodynamics is given a first glimpse by studying plane flow around infinitely long wings, specifically by deriving the Kutta-Joukowski condition of smooth flow off the wing’s trailing edge, which fixes the circulation around the wing. The chapter closes with a presentation of the balance of moment of momentum and its application to the Segner water wheel and Euler’s turbine equation.

## Keywords

Kinematics Streamlines Trajectories Streaklines Balances of mass and linear momentum Bernoulli equation Plane wing theory Balance of moment of momentum

## References

1. 1.
Becker, E.: Technische Strömungslehre. Teubner, Stuttgart (1985)Google Scholar
2. 2.
Bresse, C.: Cours de Mécanique Appliquiée, 3rd edn, vol. 2. Hydraulique. Mallet-Bachelier, Paris (1879)Google Scholar
3. 3.
Carmody, T., Kobus, H.: Translation of Hydrodynamica (Daniel Bernoulli) and Hydraulica (Johann Bernoulli), 456 pp. Dover Publ. Inc., New York (1968)Google Scholar
4. 4.
Chadwick, P.: Continuum Mechanics: Concise Theory and Problems. Dover Publications Inc., Mineola, New York (2012)Google Scholar
5. 5.
Gersten, K.: Einführung in die Strömungsmechanik, 6. Vieweg, Aufl (1991)Google Scholar
6. 6.
Hager, W.H.: Hydraulicians in Europe, 1800–2000. IAHR Monograph (2003)Google Scholar
7. 7.
Hunt, B.W.: Numerical solution of an integral equation for flow from a circular orifice. J. Fluid Mech. 31 (1968)Google Scholar
8. 8.
Hutter, K., Joehnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin, etc. (2004)Google Scholar
9. 9.
Kleinert, A.: Johann Andreas (von) Segner (1704–1777). Martin-Luther Universität, Halle-Wittenberg, FB Mathematik und Informatik, Reports on Didactics and History of Mathematics, vol. 19, pp. 15–20 (2002)Google Scholar
10. 10.
Kozeny, J.: Hydraulik. Ihre Grundlagen und praktische Anwendung. Springer, Wien (1953)Google Scholar
11. 11.
Kuhlmann, H.: Strömungsmechanik. Pearson (2007)Google Scholar
12. 12.
Kundu, P.K., Kohen, I.M., Dowling, D.R.: Fluid Mechanics, 5th edn. Elservier (2012)Google Scholar
13. 13.
McNown, J.S.: When time flowed—The Story of the Clepsydra. La Houille Blanche 5 (1976)Google Scholar
14. 14.
Mills, A.A.: Newton’s water clocks and the fluid mechanics of clepsydrae. Notes and Records of the Royal Society of London, vol. 37, pp. 35–61 (1982–1983). Errata, vol. 38 (1983)Google Scholar
15. 15.
Panton, R.L.: Incompressible Flow. Wiley (1984)Google Scholar
16. 16.
17. 17.
18. 18.
Siegloch, W.: Technische Fluidmechanik. VDI (1996)Google Scholar
19. 19.
Spencer, A.J.M.: Continuum Mechanics. Courier Corporation, Mineola, New York (2004)Google Scholar
20. 20.
Spurk, J.H., Aksel, N.: Fluid Mechanics, 2nd edN. Springer, Berlin etc. (2008). [Also in German: Strömungslehre, Einführung in die Theorie der Strömungen, 3.Aufl. Springer, Berlin etc. (2007)]Google Scholar
21. 21.
Tritton, D.J.: Physical Fluid Mechanics. Clarendon Press (1988)Google Scholar
22. 22.
von Mises, R.: Elemente der Technischen Hydromechanik. Teil I. Teubner Verlag, Leipzig und Berlin (1914)Google Scholar
23. 23.
von Mises, R.: Berchnung von Ausfluß- und Überganszahlen. Zeitschrift Verein Deutscher Ingenieure, Band, vol. 61 (1917)Google Scholar
24. 24.
von Mises, R.: Mathematical Theory of Compressible Fluid Flow. Dover, Completed by H. Geiringer and G. S. S. Ludford (2004). ISBN: 13-978-0486439419 (1958)Google Scholar
25. 25.
White, F.M.: Fluid Mechanics, 7th edn. McGraw-Hill (2011)Google Scholar
26. 26.
Wilcox, D.C.: Basic Fluid Mechanics. DCW Industries (1998)Google Scholar
27. 27.
Zierep, J., Bühler, K.: Strömungsmechank. Springer, Berlin, etc. (1991)Google Scholar