Comptes Rendus
no. 6
Exact solutions of the problem of free-boundary unsteady flows
[Solutions exactes du problème des écoulements instationnaires à surface libre]

Présenté par : Gérard Iooss

Evgenii Karabut1
1 Lavrentyev Institute of Hydrodynamics, Lavrentyev av. 15, 630090 Novosibirsk, Russia
Comptes Rendus. Mécanique, Volume 341 (2013) no. 6, pp. 533-537.

Une approche permettant de déterminer des fonctions analytiques dans un secteur angulaire et respectant certaines conditions aux limites est proposée. Si le rapport de lʼangle du secteur au nombre π est rationnel, le problème aux limites est réduit à un système fini dʼéquations différentielles ordinaires. Une telle approche, appliquée au problème du mouvement inertiel dʼun secteur liquide, a permis de sommer la série comportant des petits dénominateurs intervenant dans le problème et de trouver quatre exemples exacts dʼécoulements auto-semblables comportant une surface libre.

Some approach to the solution of boundary value problems for finding functions that are analytical in a wedge is proposed. If the ratio of the angle at the wedge vertex to the number π is rational, then the boundary value problem is reduced to the finite system of ordinary differential equations. Such approach, applied to the problem of inertial motion of a liquid wedge, made it possible to sum the series with small denominators arising in the problem and find four exact examples of self-similar flows with a free boundary.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2013.02.009
Keywords: Free boundary flows, Small denominators, Power series, Conformal mapping
Mots-clés : Écoulements limites libres, Petits dénominateurs, Série de pouvoir, Cartographie de conformal

Evgenii Karabut 1

1 Lavrentyev Institute of Hydrodynamics, Lavrentyev av. 15, 630090 Novosibirsk, Russia 
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Evgenii Karabut. Exact solutions of the problem of free-boundary unsteady flows. Comptes Rendus. Mécanique, Volume 341 (2013) no. 6, pp. 533-537. doi : 10.1016/j.crme.2013.02.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.02.009/

[1] L.V. Ovsyannikov General equations and examples, The Problem of the Unstable Flow with a Free Boundary, Nauka, Novosibirsk, 1967, pp. 5-75 (in Russian)

[2] M.S. Longuet-Higgins A class of exact, time-dependent, free-surface flows, J. Fluid Mech., Volume 55 (1972) no. 3, pp. 529-543

[3] E.A. Karabut Small perturbations in the problem of the jet impingement: Constitutive equations, J. Appl. Mech. Tech. Phys., Volume 50 (2009) no. 6, pp. 944-958

  • E. A. KARABUT; A. G. PETROV; E. N. ZHURAVLEVA Semi-analytical study of the Voinovs problem, European Journal of Applied Mathematics, Volume 30 (2019) no. 2, p. 298 | DOI:10.1017/s0956792518000098
  • E. A. Karabut; E. N. Zhuravleva Reproduction of solutions in the plane problem on motion of a free-boundary fluid, Doklady Physics, Volume 61 (2016) no. 7, p. 347 | DOI:10.1134/s1028335816070107
  • E. A. Karabut; E. N. Zhuravleva Unsteady flows with a zero acceleration on the free boundary, Doklady Physics, Volume 59 (2014) no. 10, p. 480 | DOI:10.1134/s1028335814100139
  • E. A. KARABUT; A. A. KUZHUGET Conformal mapping, Padé approximants, and an example of flow with a significant deformation of the free boundary, European Journal of Applied Mathematics, Volume 25 (2014) no. 6, p. 729 | DOI:10.1017/s0956792514000242
  • E. A. Karabut; E. N. Zhuravleva Unsteady flows with a zero acceleration on the free boundary, Journal of Fluid Mechanics, Volume 754 (2014), p. 308 | DOI:10.1017/jfm.2014.401

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