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On the return to equilibrium problem for axisymmetric floating structures in shallow water. (English) Zbl 1435.74020

Summary: In this paper we address the return to equilibrium problem for an axisymmetric floating structure in shallow water. First we show that the equation for the solid motion can be reduced to a delay differential equation involving an extension-trace operator whose role is to describe the influence of the fluid equations on the solid motion. It turns out that the compatibility conditions on the initial data for the return to equilibrium configuration are not satisfied, so we cannot use the result from our work [SIAM J. Math. Anal. 52, No. 1, 306–339 (2020; Zbl 1428.76036)] for the nonlinear problem. Hence, assuming small amplitude waves, we linearize the equations in the exterior domain and we keep the nonlinear equations in the interior domain. For such configurations, the extension-trace operator can be computed explicitly and the delay term in the differential equation can be put in convolution form. The solid motion is therefore governed by a nonlinear second order integro-differential equation, whose linearization is the well-known Cummins equation. We show global in time existence and uniqueness of the solution using the conservation of the total fluid-structure energy.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
45J05 Integro-ordinary differential equations
35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Citations:

Zbl 1428.76036

Software:

DLMF; ddesd
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References:

[1] Cummins W 1962 The Impulse Response Function and Ship Motions ((Cambridge, MA: MIT Libraries): (Navy Department, David Taylor Model Basin)
[2] John F 1949 On the motion of floating bodies. I Commun. Pure Appl. Math.2 13-57 · Zbl 0033.03104 · doi:10.1002/cpa.3160020102
[3] Ursell F 1964 The decay of the free motion of a floating body J. Fluid Mech.19 305-19 · Zbl 0137.45202 · doi:10.1017/s0022112064000738
[4] Maskell S J and Ursell F 1970 The transient motion of a floating body J. Fluid Mech.44 303-13 · Zbl 0215.29003 · doi:10.1017/s0022112070001842
[5] Wehausen J and Laitone E 2002 Surface Waves (Berlin: Springer)
[6] Lannes D 2017 On the dynamics of floating structures Ann. PDE3 11 · Zbl 1403.35239 · doi:10.1007/s40818-017-0029-5
[7] Wahl F 2018 Modeling and analysis of interactions between free surface flows and floating structures PhD Thesis
[8] Bocchi E 2020 Floating structures in shallow water: local well-posedness in the axisymmetric case SIAM J. Math. Anal.52 306-39 · Zbl 1428.76036 · doi:10.1137/18M1174180
[9] Iguchi T and Lannes D 2020 Hyperbolic free boundary problems and applications to wave-structure interactions Indiana Univ. J. Mathematics (arXiv:1806.07704)
[10] Maity D, San Martín J, Takahashi T and Tucsnak M 2019 Analysis of a simplified model of rigid structure floating in a viscous fluid J. Nonlinear Sci.29 1975-2020 · Zbl 1428.35383 · doi:10.1007/s00332-019-09536-5
[11] Schochet S 1986 The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit Commun. Math. Phys.104 49-75 · Zbl 0612.76082 · doi:10.1007/bf01210792
[12] Métivier G 1991 Ondes soniques J. Math. Pures Appl.70 197-268 · Zbl 0728.35068
[13] Abramowitz M and Stegun I A (ed) (eds) 1992 Handbook of Mathematical Functions with Formulae, Graphs, and Mathematical Tables (New York: Dover Publications)
[14] NIST Digital Library of Mathematical Functions, Release 1.0.25 of 2019-12-15 (eds) F W J Olver et alhttp://dlmf.nist.gov/
[15] Nikolski N K 2002 Operators, Functions, and Systems: An Easy Reading(Mathematical Surveys and Monographs 92) vol 1 (Providence, RI: AMS) · Zbl 1007.47001
[16] Harper Z 2010 Laplace transform representations and Paley-Wiener theorems for functions on vertical strips Doc. Math.15 235-54 · Zbl 1203.30040
[17] Yosida K 1995 Functional Analysis(Classics in Mathematics) (Berlin: Springer) · Zbl 0830.46001 · doi:10.1007/978-3-642-61859-8
[18] Rognlie D M 1969 Generalized integral trasnform PhD Thesis (Iowa State University, Department of Mathematics)
[19] Liu Z and Magal P 2019 Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions Discrete & Continuous Dyn. Syst. B 25 2271-92 · Zbl 1443.34082 · doi:10.3934/dcdsb.2019227
[20] Shampine L F 2005 Solving ODEs and DDEs with residual control Appl. Numer. Math.52 113-27 · Zbl 1063.65061 · doi:10.1016/j.apnum.2004.07.003
[21] Armesto J, Guanche R, Jesus F D, Iturrioz A and Losada I 2015 Comparative analysis of the methods to compute the radiation term in cummins’ equation J. Ocean Eng. Mar. Energy1 377-93 · doi:10.1007/s40722-015-0027-1
[22] Watson G N 1995 A Treatise on the Theory of Bessel Functions(Cambridge Mathematical Library) (Cambridge: Cambridge University Press) · Zbl 0849.33001
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