Comptes Rendus
no. 10
Mixed boundary value problem in Potential Theory: Application to the hydrodynamic impact (Wagner) problem
Yves-Marie Scolan1  ; Alexander A. Korobkin2
1 ENSTA-Bretagne, LBMS/DFMS, 2, rue François-Verny, 29806 Brest cedex 9, France
2 University of East Anglia, School of Mathematics, Norwich, United Kingdom
Comptes Rendus. Mécanique, Volume 340 (2012) no. 10, pp. 702-705.

A three-dimensional solution of the mixed boundary value problem posed in Potential Theory is proposed. The support of the Neumann condition is conformally mapped onto a unit disk. On that disk, the solution is broken down as Fourier series of azimuthal angle and linear combinations of known functions of the radial coordinate. It is shown that the whole problem reduces highly nonlinear equations for the coefficients of the mapping function. The present method of solution is to be applied to hydrodynamic impact problem.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2012.09.006
Mots clés : Dynamical systems, Potential Theory, Mixed Neumann–Dirichlet problem
Yves-Marie Scolan 1 ; Alexander A. Korobkin 2

1 ENSTA-Bretagne, LBMS/DFMS, 2, rue François-Verny, 29806 Brest cedex 9, France
2 University of East Anglia, School of Mathematics, Norwich, United Kingdom 
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Yves-Marie Scolan; Alexander A. Korobkin. Mixed boundary value problem in Potential Theory: Application to the hydrodynamic impact (Wagner) problem. Comptes Rendus. Mécanique, Volume 340 (2012) no. 10, pp. 702-705. doi : 10.1016/j.crme.2012.09.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.09.006/

[1] H. Wagner Über Stoss- und Gleitvorgänge an der Oberfläche von Flüssigkeiten, ZAMM, Volume 12 (1932), pp. 193-215

[2] A.A. Korobkin Formulation of penetration problem as a variational inequality, Dinamika Sploshn. Sredy, Volume 58 (1982), pp. 73-79

[3] S.D. Howison; J.R. Ockendon; S.K. Wilson Incompressible water-entry problems at small deadrise angles, J. Fluid Mech., Volume 222 (1991), pp. 215-230

[4] S. Zaremba Sur un problème mixte relatif à lʼéquation de Laplace, Bull. Acad. Sci. Cracovie, Classe Sci. Math. Nat., Sér. A (1910), pp. 313-344

[5] Z. Nehari Conformal Mapping, Dover Publications, New York, 1975

[6] A.A. Korobkin; Y.-M. Scolan Three-dimensional theory of water impact. Part 2. Linearized Wagner problem, J. Fluid Mech., Volume 549 (2006), pp. 343-373

[7] E.P. Stephan Boundary integral equations for mixed boundary value problems in R3, Math. Nachr., Volume 134 (1987), pp. 21-53

[8] Y.-M. Scolan, A.A. Korobkin, Towards a solution of the three-dimensional Wagner problem, in: 23rd International Workshop on Water Waves and Floating Bodies, Korea, April 2008.

[9] I.N. Sneddon Mixed Boundary Value Problems in Potential Theory, J. Wiley & Sons, Inc., 1966

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