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Gravity–capillary flows over obstacles for the fifth-order forced Korteweg–de Vries equation

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Abstract

The aim of this work is to investigate gravity–capillary waves resonantly excited by two topographic obstacles in a shallow water channel. By considering the weakly non-linear regime the forced fifth-order Korteweg–de Vries equation arises as a model for the free-surface displacement. The water surface is initially taken at rest over a uniform flow and the initial value problem for this equation is computed numerically using a pseudospectral method. We study nearly-resonant flows with intermediate capillary effects. Details of the wave interactions are analysed for obstacles with different sizes. Our numerical results indicate that the flow is not necessarily governed by the larger obstacle.

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Acknowledgements

The authors are grateful to IMPA-National Institute of Pure and Applied Mathematics for the research support provided during the Summer Program of 2020 and to the unknown reviewers for their constructive comments and suggestions which enriched the manuscript. M.F. is grateful to Federal University of Paraná for the visit to the Department of Mathematics. R.R.-Jr is grateful to University of Bath for the extended visit to the Department of Mathematical Sciences.

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Authors and Affiliations

  1. UFRPE/Rural Federal University of Pernambuco, UACSA/Unidade Acadêmica do Cabo de Santo Agostinho, BR 101 Sul, 5225, Ponte dos Carvalhos, Cabo de Santo Agostinho, Pernambuco, 54503-900, Brazil

    Marcelo V. Flamarion

  2. Departamento de Matemática, Centro Politécnico, UFPR/Federal University of Paraná, Jardim das Américas, Caixa Postal 19081, Curitiba, PR, 81531-980, Brazil

    Roberto Ribeiro-Jr

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  1. Marcelo V. Flamarion
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  2. Roberto Ribeiro-Jr
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Correspondence to Marcelo V. Flamarion.

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Flamarion, M.V., Ribeiro-Jr, R. Gravity–capillary flows over obstacles for the fifth-order forced Korteweg–de Vries equation. J Eng Math 129, 17 (2021). https://doi.org/10.1007/s10665-021-10153-z

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  • DOI: https://doi.org/10.1007/s10665-021-10153-z

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