Existence of a weak solution to the fluid-structure interaction problem in 3D

TL;DR

This paper proves the existence of weak solutions for a complex 3D fluid-structure interaction system involving Navier-Stokes fluid dynamics and nonlinear elastic plate models, using a novel hybrid approximation scheme.

Contribution

It introduces a hybrid approximation method combining time-discretization and operator splitting to handle nonlinearities in 3D fluid-structure interaction problems.

Findings

Established existence of weak solutions under specific discretization conditions.

Developed a hybrid scheme that effectively manages nonlinearities.

Provided conditions linking discretization parameters to system nonlinearities.

Abstract

We study a nonlinear fluid-structure interaction problem in which the fluid is described by the three-dimensional incompressible Navier-Stokes equations, and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of Kirchhoff, von K\'arm\'an and Berger plate models. The fluid and the structure are fully coupled via kinematic and dynamic boundary conditions. The existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system. We combine time-discretization and operator splitting to create two sub-problems, one piece-wise stationary for the fluid and one in the Galerkin basis for the plate. To guarantee the convergence of approximate solutions to a weak solution, a sufficient condition is given on the number of time discretization sub-intervals in every step in a form of dependence with number of the Galerkin basis functions and nonlinearity order of the plate equation.