Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

Shashikanth Banavara N.

doi:10.3934/jgm.2020003

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2020, Volume 12, Issue 1: 25-52. Doi: 10.3934/jgm.2020003

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Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

Shashikanth Banavara N.

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003-8001, USA

Received: December 31, 2018

Published: February 2020

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Abstract

The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system.

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Mathematics Subject Classification: Primary: 37K05, 74F10, 76B07; Secondary: 53D17.

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Figure 1. Schematic perspective of a rigid body $ B $ beneath the free surface $ \Sigma_f $ of water. The bottom flat surface $ \mathcal{S} $ is shown by the dashed rectangle. Both $ \Sigma_f $ and $ \mathcal{S} $ extend to infinity in the $ x $ and $ y $ (horizontal) directions. In the text, the origin of the spatial frame $ xyz $ is located at the center of the disc $ C_R \subset \mathcal{S} $

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Figure 2. A vertical slice of the setup in Figure 1, shown along with the body-fixed frame

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