Flows in the Complex Plane

  • Achim FeldmeierEmail author
Part of the Theoretical and Mathematical Physics book series (TMP)


One of the primary goals in this book is to obtain analytical solutions for the equations of motion of fluids. In order that such solutions are found, fluid motion is often restricted to be stationary and two dimensional (2-D) in a plane. All fluid fields depend then on the x- and y-coordinates only, and the fluid velocity has no z-component, \(\vec {u}= u_x(x,y)\;\hat{x}+ u_y(x,y)\;\hat{y}\). The motion of a fluid parcel is restricted to a planar layer, and the whole body of fluid is made up of layers stacked one upon the other. With the usual identification of the vector space \(\mathbb {R}^2\) and the field \(\mathbb {C}\) of complex numbers, complex function theory becomes available for 2-D fluid dynamics. In this chapter, stationary fluid dynamics in 2-D is framed as a problem of complex function theory, by introducing a complex potential Open image in new window for the complex speed field w. We discuss some rather elementary topics from potential theory (Laplace equation, boundary conditions, and  maximum property). An Appendix gives an overview of theorems from complex function theory on analytic and meromorphic functions without proofs. We discuss the Schwarz–Christoffel theorem  on conformal mappings, which is applied on Chap.  4 to flows with jets, wakes, and cavities. Finally, a brief introduction to Riemann surfaces is given, which will find applications in later chapters.


  1. Ahlfors, L.V. 1966. Complex analysis, 2nd ed. New York: McGraw-Hill.zbMATHGoogle Scholar
  2. Axler, S., P. Bourdon, and W. Ramey. 2001. Harmonic function theory, 2nd ed. New York: Springer.CrossRefGoogle Scholar
  3. Basye, R.E. 1935. Simply connected sets. Transactions of the American Mathematical Society 38: 341.MathSciNetCrossRefGoogle Scholar
  4. Betz, A. 1948. Konforme Abbildung. Berlin: Springer.Google Scholar
  5. Bieberbach, L. 1921. Lehrbuch der Funktionentheorie, Band I, Elemente der Funktionentheorie. Leipzig: B.G. Teubner; Wiesbaden: Springer Fachmedien.Google Scholar
  6. Burkhardt, H., and W.F. Meyer. 1900. Potentialtheorie. In Encyklopädie der mathematischen Wissenschaften, vol. 2-1-1, 464. Leipzig: Teubner (digital at GDZ Göttingen).Google Scholar
  7. Cartan, H.P. 1963. Elementary theory of analytic functions of one or several complex variables. Reading: Addison-Wesley; New York: Dover (1995).Google Scholar
  8. Copson, E.T. 1935. Theory of functions of a complex variable. Oxford: Oxford University Press.zbMATHGoogle Scholar
  9. Courant, R. 1950. Dirichlet’s principle, conformal mapping, and minimal surfaces. New York: Interscience Publishers.zbMATHGoogle Scholar
  10. Gamelin, T.W. 2001. Complex analysis. Berlin: Springer.CrossRefGoogle Scholar
  11. Garding, L. 1977. Encounter with mathematics. New York: Springer.CrossRefGoogle Scholar
  12. Golusin, G.M. 1957. Geometrische Funktionentheorie. Berlin: Deutscher Verlag der Wissenschaften.Google Scholar
  13. Grattan-Guinness, I. (ed.). 1994. Companion encyclopedia of the history and philosophy of the mathematical sciences, vol. 1. London: Routledge.zbMATHGoogle Scholar
  14. Hurwitz, A., and R. Courant. 1964. Funktionentheorie. Berlin: Springer.zbMATHGoogle Scholar
  15. Jackson, J.D. 1998. Classical electrodynamics, 3rd ed. New York: Wiley.zbMATHGoogle Scholar
  16. Jänich, K. 2001. Vector analysis. Berlin: Springer.CrossRefGoogle Scholar
  17. Kellogg, O.D. 1929. Foundations of potential theory. Berlin: Springer.CrossRefGoogle Scholar
  18. Knopp, K. 1996. Theory of functions. Mineola: Dover.zbMATHGoogle Scholar
  19. Lamb, H. 1932. Hydrodynamics. Cambridge: Cambridge University Press; New York: Dover (1945).Google Scholar
  20. Lu, J.-K., S.-G. Zhong, and S.-Q. Liu. 2002. Introduction to the theory of complex functions. Singapore: World Scientific.CrossRefGoogle Scholar
  21. Markushevich, A.I. 1965. Theory of functions of a complex variable, vol. II. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar
  22. Markushevich, A.I. 1967. Theory of functions of a complex variable, vol. III. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar
  23. Miles, J.W. 1977. On Hamilton’s principle for surface waves. Journal of Fluid Mechanics 83: 153.ADSMathSciNetCrossRefGoogle Scholar
  24. Monna, A.F. 1975. Dirichlet’s principle: A mathematical comedy of errors and its influence on the development of analysis. Utrecht: Oosthoek, Scheltema & Holkema.zbMATHGoogle Scholar
  25. Nehari, Z. 1952. Conformal mapping. New York: McGraw-Hill; New York: Dover (1975).Google Scholar
  26. Neumann, C. 1884. Vorlesungen über Riemann’s Theorie der Abel’schen Integrale, 2nd ed. Leipzig: Teubner.zbMATHGoogle Scholar
  27. Nevanlinna, R. 1953. Uniformisierung. Berlin: Springer.CrossRefGoogle Scholar
  28. Osgood, W.F. 1907. Lehrbuch der Funktionentheorie, Erster Band. Leipzig: Teubner.Google Scholar
  29. Rankine, W.J. 1864. On plane water-lines in two dimensions. Philosophical Transactions of the Royal Society of London A 154: 369.CrossRefGoogle Scholar
  30. Remmert, R. 1992. Funktionentheorie 1, 3rd ed. Berlin: Springer.CrossRefGoogle Scholar
  31. Schwarz, H.A. 1869. Ueber einige Abbildungsaufgaben. Journal für die reine und angewandte Mathematik 70: 105; in Gesammelte Mathematische Abhandlungen, vol. 2, 65. Berlin: Springer (1890).Google Scholar
  32. Serrin, J. 1959. Mathematical principles of classical fluid mechanics. In Handbuch der Physik = Encyclopedia of Physics, vol. 3/8-1, Strömungsmechanik 1, ed. S. Flügge, and C.A. Truesdell, 125. Berlin: Springer.Google Scholar
  33. Walker, M. 1964. The Schwarz-Christoffel transformation and its applications – a simple exposition. New York: Dover.zbMATHGoogle Scholar
  34. Weyl, H. 1913. Die Idee der Riemannschen Fläche. Leipzig: Teubner.zbMATHGoogle Scholar
  35. Zermelo, E. 1902. Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche. Erste Mitteilung. Zeitschrift für Mathematik und Physik 47: 201.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für Physik und AstronomieUniversität PotsdamPotsdamGermany

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