A Review on Rigorous Derivation of Reduced Models for Fluid–Structure Interaction Systems

Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid–structure interaction problem

“…Theorem 1.1 provides only strong and weak convergence results in corresponding spaces, but from the application point of view it is important to have a quantitative error estimate. In linear models [4,5] we were able to obtain error estimates for approximate solutions in strong norms, and it is our next aim to extend these results to the nonlinear framework of this paper. This will be the subject of our future work.…”

“…where D(v) = 1 2 (∇v + (∇v) T ) denotes the symmetric part of the gradient, and f denotes the density of the fluid external force. The structure is described by a linear equation of visco-elastic plate (5), where 2 is the Jacobian of the transformation from Eulerian to Lagrangian coordinates, n η is the unit outer normal to the deformed configuration Ω η , e z is the unit vector in z direction, and ρ, δ, β are positive parameters describing the density, visco-elasticity and bending properties of the structure, respectively. Equations for the fluid and the structure are coupled via dynamic and kinematic conditions ( 5) and ( 6) representing the balance of forces in e z direction and continuity of the velocity, respectively.…”

“…It is our aim here to perform this passage from an FSI problem to the thinfilm equation rigorously in the sense of convergence of solutions of the FSI problem to the solution of the corresponding thin-film equation, as stated in our main result in theorem 1.1. Such procedure has been already performed by the authors in passing from fully linear 3D/2D [5] and 3D/3D [4] FSI problems to 2D linear sixth-order evolution equations. Now we extend these ideas from linear to the nonlinear setting, but only in case of 2D/1D FSI problems due to availability of the global well-posedness result (cf theorem 2.3).…”