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

Abstract

We analyze a linear 3D/3D fluid–structure interaction problem between a thin layer of a viscous fluid and a thin elastic plate-like structure with the aim of deriving a simplified reduced model. Based on suitable energy dissipation inequalities quantified in terms of two small parameters, thickness of the fluid layer and thickness of the elastic structure, we identify the right relation between the system coefficients and small parameters which eventually provide a reduced model on the vanishing limit. The reduced model is a linear sixth-order thin-film equation describing the out-of-plane displacement of the structure, which is justified in terms of weak convergence results relating its solution to the solutions of the original fluid–structure interaction problem. Furthermore, approximate solutions to the fluid-structure interaction problem are reconstructed from the reduced model and quantitative error estimates are obtained, which provide even strong convergence results.

Appendix A: Griso Decomposition and Korn Inequality on Thin Domain

The following result is directly from [31, Theorem 2.3], tailored to the specific boundary conditions and geometry considered in this paper.

Theorem A.1

Let $h>0$, then every $\varvec{u}^h\in V_S(\Omega _h)$ can be decomposed as

$$\begin{aligned} \varvec{u}^h(x) = \varvec{w}^h(x') + (x_3 - h/2)\varvec{e}_3\times \varvec{r}^h(x') + {{\tilde{\varvec{u}}}}^h(x)\,,\quad (x',x_3)\in \Omega _h\,, \end{aligned}$$

(108)

or written componentwise

$$\begin{aligned} u_1^h(x)&= w_1^h(x') + (x_3-h/2)r_2^h(x') + {\tilde{u}}_1^h(x)\,,\\ u_2^h(x)&= w_2^h(x') - (x_3-h/2)r_1^h(x') + {\tilde{u}}_2^h(x)\,,\\ u_3^h(x)&= w_3^h(x') + {\tilde{u}}_3^h(x)\,, \end{aligned}$$

where

$$\begin{aligned} \varvec{w}^h(x') = \frac{1}{h}\int _0^h \varvec{u}^h(x',x_3){\mathrm {d}}x_3\,,\quad \varvec{r}^h(x') = \frac{3}{h^3}\int _0^h (x_3 - h/2)\varvec{e}_3\times \varvec{u}^h(x',x_3){\mathrm {d}}x_3\,, \end{aligned}$$

and ${{\tilde{\varvec{u}}}}^h\in V_S(\Omega _h)$ is so called warping or residual term. The main part of the decomposition, denoted by $\varvec{u}_E^h = \varvec{w}^h(x') + (x_3 - h/2)\varvec{e}_3\times \varvec{r}^h(x')$, is called the elementary plate displacement. Moreover, the following estimate holds

$$\begin{aligned} \Vert {\text {sym}}\nabla \varvec{u}_E^h\Vert _{L^2(\Omega _h)}^2 + \Vert \nabla {\tilde{\varvec{u}}}^h\Vert _{L^2(\Omega _h)}^2 + \frac{1}{h^2}\Vert {\tilde{\varvec{u}}}^h\Vert _{L^2(\Omega _h)}^2 \le C\Vert {\text {sym}}\nabla {\varvec{u}}^h\Vert _{L^2(\Omega _h)}^2\,, \end{aligned}$$

(109)

where $C>0$ is independent of $\varvec{u}^h$ and h.

Theorem A.2

(Korn inequality on thin domains) Let $\omega \subset {\mathbb R}^2$ be Lipschitz domain and $\gamma \subset \partial \omega$ part of its boundary of positive measure, then there exists a constant $C_K>0$ and $h_0>0$ such that for every $0<h<h_0$

$$\begin{aligned}&\Vert (\psi _1,\psi _2,h\psi _3)\Vert _{H^1(\Omega ;{\mathbb R}^3)}^2 \\&\quad \le C_K\Big (\Vert (\psi _1,\psi _2,h\psi _3)\Vert _{L^2(\Omega ;{\mathbb R}^3)}^2+\Vert {\text {sym}}\nabla _h\varvec{\psi }\Vert _{L^2(\Omega ;{\mathbb R}^9)}^2\Big )\,, \quad \forall \varvec{\psi }\in H^1(\Omega ;{\mathbb R}^3)\,, \end{aligned}$$

where $\Omega = \omega \times (0,1)$. The Korn constant $C_K$ depends only on $\omega$ and $\gamma$.

Proof

The proof follows by the Griso’s decomposition of $\varvec{\psi }\in H_\gamma ^1(\Omega ;{\mathbb R}^3)$ (see [31]) and application of the Korn inequality for functions defined on $\omega$. $\square$