Rigorous Derivation of a Linear Sixth-Order Thin-Film Equation as a Reduced Model for Thin Fluid–Thin Structure Interaction Problems

Abstract

Starting from a nonlinear 2D/1D fluid-structure interaction problem between a thin layer of a viscous fluid and a thin elastic structure, on the vanishing limit of the relative fluid thickness, we rigorously derive a sixth-order thin-film equation describing the dynamics of vertical displacements of the structure. The procedure is essentially based on quantitative energy estimates, quantified in terms of the relative fluid thickness, and a uniform no-contact result between the structure and the solid substrate. The sixth-order thin-film equation is justified in the sense of strong convergence of rescaled structure displacements to the unique positive classical solution of the thin-film equation. Moreover, the limit fluid velocity and the pressure can be expressed solely in terms of the solution to the thin-film equation.

A review on rigorous derivation of reduced models for fluid-structure interaction systems

“…Since equation (7) implies that the pressure is constant in the vertical direction, equation ( 6) is a Poisson equation with respect to z with Dirichlet boundary condition. Its explicit solution is then given by,…”

“…although, all obtained results will also hold for nontrivial initial conditions under some additional smallness assumptions (cf. for instance [6,Appendix]). A nontrivial volume force on the structure could also be involved, again under certain scaling assumptions (cf.…”

“…A nontrivial volume force on the structure could also be involved, again under certain scaling assumptions (cf. again [6]), but the trivial one is in fact motivated by applications in microfluidics [47]. The previously settled framework also incorporates physically more relevant problem which involves prescribed pressure drop between inlet and outlet of the channel, instead of the periodic boundary conditions.…”

“…Moreover, these quantitative a priori estimates provide the necessary scaling assumption on model parameters which ensures the nontrivial reduced model. The same ideas have been recently employed in [6], where the authors analyzed a linear 3D/3D FSI problem, in which the simultaneous dimension reduction in the fluid and the structure has been performed and again a linear sixth-order thin-film equation has been derived as the reduced model under particular scaling assumptions. We briefly report on this problem and results in Section 3.2.…”

“…Unfortunately, the wellposedness results for such problems are very scarce, hence, for the beginning we restrict our analysis to a 2D/1D FSI problem for which the existence of global in time strong solutions is available from [24]. Rigorous derivation of the nonlinear sixth-order thin-film equation, which is known in the engineering literature [29,26,34], as the reduced model for this FSI problem is still work in progress [7], and only main ideas are outlined in Section 3.3.…”