On the Models of the Fluid-Polymer Systems

Abstract

The purpose of this work is to study fluid-polymer systems. A fluid-polymer system is a system consisting of solvent fluids and polymers, either suspended in the bulk (polymeric fluid systems) or attached on the boundaries. Mathematically, they are coupled multi-scale systems of partial differential equations, consisting of a fluid portion modeled by the Navier-Stokes equation, and a polymer portion modeled by the Fokker-Planck equation. Key difficulties lie in the coupling of two equations.

We propose a new approach to show the well-posedness of a certain class of polymeric fluid systems. In this approach, we use ``moments" to translate a multi-scale system to a fully macroscopic system (consisting of infinitely many equations), solve the macroscopic system, and recover the solution of the original multi-scale system. As an application, we obtain the large data global well-posedness of a certain class of polymeric fluid systems.

We also show the local well-posedness when a polymeric fluid system is written in Lagrangian coordinates. This approach allows us to show the uniqueness in lower regularity space and the Lipschitz dependence on initial data.

Finally, we propose a new boundary condition which describes the situation where polymers are attached on the fluid-wall interface. Using kinetic theory, we derive a dynamic boundary condition which can be interpreted as a ``history-dependent slip" boundary condition, and we prove global well-posedness in 2D case. Also, we show that the inviscid limit holds for an incompressible Navier-Stokes system with this boundary condition.