[Développements asymptotiques uniformément valables pour des problèmes de perturbations singulières et principe de raccordement]
After a brief reminder of the notion of asymptotic expansion, a counter-example of the Van Dyke matching principle is solved thanks to a modified form of this principle. This leads to a composite approximation to a given order. The proposed method of successive complementary expansions reverses the analysis by starting with a supposed form of the uniformly valid approximation. This method does not require any matching principle which, in fact, is a by-product. The method is illustrated with the very often studied one-dimensional model of Stokes–Oseen for the circular cylinder.
Après un bref rappel de la notion de développement asymptotique, un contre-exemple du principe de raccordement de Van Dyke est levé grâce à une forme modifiée de ce principe. Ceci conduit à une approximation composite à un ordre donné. La méthode des développements successifs complémentaires proposée renverse l'analyse en partant d'une forme supposée d'une approximation uniformément valable. Cette méthode ne fait appel à aucun principe de raccordement. En fait, un tel principe apparaı̂t comme un résultat. La méthode est illustrée avec le modèle unidimensionnel souvent étudié de Stokes–Oseen pour le cylindre circulaire.
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Mots-clés : mécanique des fluides, couche limite, équations différentielles, théorie asymptotique, perturbations singulières
Jacques Mauss 1 ; Jean Cousteix 2, 3
@article{CRMECA_2002__330_10_697_0, author = {Jacques Mauss and Jean Cousteix}, title = {Uniformly valid approximation for singular perturbation problems and matching principle}, journal = {Comptes Rendus. M\'ecanique}, pages = {697--702}, publisher = {Elsevier}, volume = {330}, number = {10}, year = {2002}, doi = {10.1016/S1631-0721(02)01522-X}, language = {en}, }
TY - JOUR AU - Jacques Mauss AU - Jean Cousteix TI - Uniformly valid approximation for singular perturbation problems and matching principle JO - Comptes Rendus. Mécanique PY - 2002 SP - 697 EP - 702 VL - 330 IS - 10 PB - Elsevier DO - 10.1016/S1631-0721(02)01522-X LA - en ID - CRMECA_2002__330_10_697_0 ER -
Jacques Mauss; Jean Cousteix. Uniformly valid approximation for singular perturbation problems and matching principle. Comptes Rendus. Mécanique, Volume 330 (2002) no. 10, pp. 697-702. doi : 10.1016/S1631-0721(02)01522-X. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01522-X/
[1] Asymptotic Analysis of Singular Perturbations, Stud. Math. Appl, 9, North-Holland, 1979
[2] Asymptotic expansions of Navier–Stokes solutions for small Reynolds numbers, J. Math. Mech, Volume 6 (1957), pp. 585-593
[3] On first order matching process for singular functions, Spectral Theory and Asymptotics of Differential Equations, Proceedings of the Scheveningen Conference on Differential Equations, Math. Studies, 13, North-Holland, 1974
[4] On matching principles, Lectures Notes in Math, 711, Springer-Verlag, 1979, pp. 1-18
[5] Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964
[6] Perturbation Methods, Pure Appl. Math, Wiley, 1973
[7] Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E, Volume 54 (1996) no. 1, pp. 376-394
[8] Perturbation Methods. Applied Mathematics, Cambridge University Press, 1991
[9] Matched Asymptotic Expansions, Ideas and Techniques, Appl. Math. Sci, 76, Springer-Verlag, 1988
[10] J. Cousteix, J. Mauss, Approximations of Navier–Stokes equations at high Reynolds number, in: S. Wang, N. Fowkes (Eds.), BAIL 2002 Conference, The University of Western Australia, Perth, 8–12 July 2002
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