We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

我们研究了相互耦合的介观-宏观-壳-壳系统方程，该系统对稀的不可压缩聚合物流体进行建模，该流体正在演化并与Koiter型柔性壳相互作用。聚合物构成溶剂-溶质混合物，其中溶剂通过不可压缩的Navier-Stokes方程在宏观尺度上建模，而溶质通过Fokker-Planck方程（Kolmogorov正向方程）在介观尺度上建模，用于概率密度函数。珠-弹簧聚合物链构型。该混合物与非线性弹性壳相互作用，该非线性弹性壳用作聚合物流体的物理空间域的运动边界。我们使用Koiter的经典模型来描述壳运动，从而产生完全非线性的四阶双曲方程。