Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy

Abstract

The long-time existence of a weak solution is proved for a nonlinear, fluid-structure interaction (FSI) problem between an incompressible, viscous fluid and a semilinear cylindrical Koiter membrane shell with inertia. No axial symmetry is assumed in the problem. The fluid flow is driven by the timedependent dynamic pressure data prescribed at the inlet and outlet boundaries of the 3D cylindrical fluid domain. The fluid and the elastic structure are fully coupled via continuity of velocity and continuity of normal stresses. Global existence of a weak solution is proved as long as the lateral walls of the cylinder do not touch each other. The main novelty of the work is the nonlinearity in the structure model: the model accounts for the fully nonlinear Koiter membrane energy, supplemented with a small linear fourth-order derivative term modeling the bending rigidity of shells. The existence proof is constructive, and it is based on an operator splitting scheme. A version of this scheme can be implemented for the numerical simulation of the underlying FSI problem by extending the FSI solver, developed by the authors in [5], to include the nonlinearity in the structure model discussed in this manuscript.

Compressible Fluids Interacting with a Linear-Elastic Shell

“…The results from [32] have been extended to some incompressible non-Newtonian cases in [31]; see also [24]. Results for incompressible fluids in cylindrical domains have been shown in [38,39] and [6]. The paper [38] deals with a cylindrical linear elastic/viscoelastic Koiter shell in two dimensions (the shell is prescribed by a one-dimensional curve).…”

“…The paper [38] deals with a cylindrical linear elastic/viscoelastic Koiter shell in two dimensions (the shell is prescribed by a one-dimensional curve). The papers [6,39] extend this to cylindrical three-dimensional fluid flows. Note that in [39] even nonlinear elastic behavior of the shell is allowed.…”

“…The papers [6,39] extend this to cylindrical three-dimensional fluid flows. Note that in [39] even nonlinear elastic behavior of the shell is allowed.…”

The Viscous Surface Wave Problem with Generalized Surface Energies

“…Cheng-Coutand-Shkoller [CCS07] proved a local existence result for a viscous fluid coupled to a nonlinear elastic biofluid shell, and Cheng-Shkoller [CS10] proved local existence for a model with a Koiter shell. Local existence results for similar models related to hemodynamics were proved by Muha-Ĉanić [MC15,MC16]. We refer to the work of Bonito-Nochetto-Pauletti [BNP11] and Barett-Garcke-Nürnberg [BGN17] and the references contained therein for a discussion of the numerical analysis of such models.…”

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

“…Omitting ∆t, k in the subscript and simplifying the notation ∇ n := ∇ η n , corresponding to (25)- (27), we now define the following weak form of the fluid sub-problem (FSP):…”