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Trapped flexural waves supported by a pair of identical cylinders in a two-layer fluid. SN APPLIED SCIENCES 2020. [DOI: 10.1007/s42452-020-03229-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022] Open
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Motygin OV, McIver P. On uniqueness in the problem of gravity–capillary water waves above submerged bodies. Proc Math Phys Eng Sci 2009. [DOI: 10.1098/rspa.2009.0012] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In this paper, we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; examples of numerical computation of trapped modes are given. We also prove a uniqueness theorem that provides simple bounds for the possible non-uniqueness parameters.
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Affiliation(s)
- Oleg V. Motygin
- Institute of Problems in Mechanical Engineering, Russian Academy of SciencesV. O. Bol'shoy prospect 61, St Petersburg 199178, Russia
| | - Philip McIver
- School of Mathematics, Loughborough UniversityLoughborough, Leicestershire LE11 3TU, UK
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Harter R, Abrahams ID, Simon MJ. The effect of surface tension on trapped modes in water-wave problems. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.0063] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In this paper the effect of surface tension is considered on two two-dimensional water-wave problems involving pairs of immersed bodies. Both models, having fluid of infinite depth, support localized oscillations, or trapped modes, when capillary effects are excluded. The first pair of bodies is surface-piercing whereas the second pair is fully submerged. In the former case it is shown that the qualitative nature of the streamline shape is unaffected by the addition of surface tension in the free surface condition, no matter how large this parameter becomes. The main objective of this paper, however, is to study the submerged body problem. For this case it is found, by contrast, that there exists a critical value of the surface tension above which it is no longer possible to produce a completely submerged pair of bodies which support trapped modes. This critical value varies as a function of the separation of the two bodies. It can be inferred from this that surface tension does not always play a qualitatively irrelevant role in the linear water-wave problem.
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Affiliation(s)
- Robert Harter
- School of Mathematics, The University of ManchesterOxford Road, Manchester M13 9PL, UK
| | - I. David Abrahams
- School of Mathematics, The University of ManchesterOxford Road, Manchester M13 9PL, UK
| | - Michael J Simon
- School of Mathematics, The University of ManchesterOxford Road, Manchester M13 9PL, UK
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Porter R. Surface wave scattering by submerged cylinders of arbitrary cross–section. Proc Math Phys Eng Sci 2002. [DOI: 10.1098/rspa.2001.0885] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- R. Porter
- School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
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