A two-dimensional time dependent model of an interaction between a thin elastic plate and a Newtonian viscous fluid described by the non-steady Stokes equations is considered. It depends on a small parameter \(\varepsilon \) that is the ratio of the thicknesses of the plate and the fluid layer. The Young’s modulus of the plate and its density may be great or small parameters equal to some powers (positive or negative) of \(\varepsilon \) while the density and the viscosity of the fluid are supposed to be of order one. An asymptotic expansion is constructed and justified for various magnitudes of the rigidity and density of the plate. The limit problems are studied in all these cases. They are Stokes equations with some special coupled or uncoupled boundary conditions modeling the interaction with the plate. The estimates of the difference between the exact solution and a truncated asymptotic expansion are established. These estimates justify the asymptotic approximations.

中文翻译：

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粘性流体-薄弹性板相互作用：关于板的刚度和密度的渐近分析

考虑了由非稳态斯托克斯方程描述的薄弹性板与牛顿粘性流体之间相互作用的二维时变模型。它取决于一个小的参数\（\ varepsilon \），它是板和流体层的厚度之比。板的杨氏模量及其密度可以是大参数，也可以是小参数，它们等于\（\ varepsilon \）的某些幂（正或负）而流体的密度和粘度应为一阶。构造渐近扩展并证明其对于板的各种刚度和密度的合理性。在所有这些情况下都研究极限问题。它们是斯托克斯方程，带有一些特殊的耦合或非耦合边界条件，可模拟与板的相互作用。建立精确解和截断渐近展开之间的差值的估计。这些估计证明了渐近近似是正确的。