Mathematical methods in fluid mechanics: Regular and singular perturbations

Syllabus: Asymptotic series and expansions; applications of the regular perturbation method in some flow problems; failure of the Poincare expansions; method of strained coordinates; renormalization technique; method of matched asymptotic expansions; flows around a sphere or a cylinder with small Reynolds numbers; method of multiple scales; turning point problems.

Learning Outcomes:On successful completion of this module, students should be able to:1. explain and apply the regular perturbation method for solving differential equations, specially flow problems, by means of parameter or coordinate perturbation;2. recognize the limitations of the regular perturbation method; 3. choose and apply alternative suitable singular perturbation methods if the regular perturbation method fails for given differential equations;4. recognize relations and distinctions of different singular perturbation methods, e.g. methods of strained coordinates, renormalization, multiple scales.

Prerequisites:

Basic knowledge of ordinary and partial differential equations and the corresponding solution methods; basic knowledge of fluid mechanics. Knowledge of lecture “Mathematical methods in fluid mechanics: Exact and symmetry methods” is not required.

Regular cycles: Each Winter Semester

See Tucan

By arrangement with Prof. Dr.-Ing. Yongqi Wang or Dipl.-Math. Dominik Plümacher

Oral exams by arrangement with Prof. Dr.-Ing. Yongqi Wang

Prof. Dr.-Ing. Yongqi Wang | Dipl.-Math. Dominik Plümacher

lecture notes;

Nayfeh, A.H.: Perturbation Methods, John Wiley & Sons, 1975;

Van Dyke, M.: Pertubation Methods in Fluid Mechanics, Parabolic Press, 1975.