Comptes Rendus
Exact solutions of the problem of free-boundary unsteady flows
[Solutions exactes du problème des écoulements instationnaires à surface libre]
Comptes Rendus. Mécanique, Volume 341 (2013) no. 6, pp. 533-537.

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.

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.

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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