Editing Boundary layer (section)

===Prandtl's transposition theorem===
[[Ludwig Prandtl|Prandtl]] observed that from any solution <math>u(x,y,t),\ v(x,y,t)</math> which satisfies the boundary layer equations, further solution <math>u^*(x,y,t),\ v^*(x,y,t) </math>, which is also satisfying the boundary layer equations, can be constructed by writing<ref>{{Cite journal|doi = 10.1002/zamm.19380180111|title = Zur Berechnung der Grenzschichten|year = 1938|last1 = Prandtl|first1 = L.|journal = Zeitschrift für Angewandte Mathematik und Mechanik|volume = 18|issue = 1|pages = 77–82|bibcode = 1938ZaMM...18...77P}}</ref>

:<math>u^*(x,y,t) = u(x,y+f(x),t), \quad v^*(x,y,t) = v(x,y+f(x),t) - f'(x) u(x,y+f(x),t)</math>

where <math>f(x)</math> is arbitrary. Since the solution is not unique from mathematical perspective,<ref>Van Dyke, Milton. Perturbation methods in fluid mechanics. Parabolic Press, Incorporated, 1975.</ref> to the solution can be added any one of an infinite set of eigenfunctions as shown by [[Keith Stewartson|Stewartson]]<ref>{{Cite journal|doi=10.1002/sapm1957361173|title=On Asymptotic Expansions in the Theory of Boundary Layers|year=1957|last1=Stewartson|first1=K.|journal=Journal of Mathematics and Physics|volume=36|issue=1–4|pages=173–191}}</ref> and [[Paul A. Libby]].<ref>{{Cite journal|doi=10.1017/S0022112063001439|title=Some perturbation solutions in laminar boundary-layer theory|year=1963|last1=Libby|first1=Paul A.|last2=Fox|first2=Herbert|journal=Journal of Fluid Mechanics|volume=17|issue=3|page=433|doi-broken-date=14 January 2025 |s2cid=123824364 }}</ref><ref>{{Cite journal|doi=10.1017/S0022112064000830|title=Some perturbation solutions in laminar boundary layer theory Part 2. The energy equation|year=1964|last1=Fox|first1=Herbert|last2=Libby|first2=Paul A.|journal=Journal of Fluid Mechanics|volume=19|issue=3|pages=433–451|bibcode=1964JFM....19..433F|s2cid=120911442 }}</ref>