Mathematical methods in fluid mechanics: Exact and symmetry methods

General Informations

Syllabus: Basic equations of incompressible fluid flow; examples of exact solutions of the Navier-Stokes equations; introduction into the mathematical concept of symmetry; the theory of Lie Groups; Lies 1. and 2. fundamental theorem; dimensional analysis; invariance of differential equations; the Lie algorithm for determining symmetries; invariant solutions of non-linear partial differential equations; direct construction method of conservation laws in divergence form.

Learning Outcomes: On successful completion of this module, students should be able to:1. simplify the complexity of the Navier-Stokes equations for various simple flow problems and reach their exact solutions 2. apply the analytic theory, based on Lie symmetries, for solving ordinary and partial differential equations, specially for flow problems 3. analyze the symmetries and invariances of given differential equations by means of the theory of Lie groups4. development of potential local conservation laws of differential equations with the aid of the direct construction method.


1) Basic knowledge of mathematics

2) Fundamentals of Fluid Mechanics

Regular cycles: Each Summer Semester


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Office hours

By arrangement with Prof. Dr.-Ing. Martin Oberlack

Lecture Materials

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Oral exams by arrangement


Prof. Dr.-Ing. Martin Oberlack | Dominik Plümacher, M.Sc.


  • Lecture Notes
  • Bluman, Kumei: Symmetries and Differential equations, Springer Verlag, 1996
  • Stephani: Differentialgleichungen, Symmetrien und Lösungsmethoden, Spektrum Akademischer Verlag, 1994
  • Cantwell: Introduction to Symmetrie Analysis, Cambridge University Press, 2002
  • Bluman, G.W., Cheviakov, A.F., and Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Applied Mathematical Sciences Vol. 168. Springer 2010.