On some probability density function based moment closure approximations of micro-macro models for viscoelastic polymeric fluids

In this paper we will discuss several issues related to the moment-closure approximation of mul- tiscale models for viscoelastic polymeric fluids. These moment-closure approaches are based on special ansatz for the probability density function (PDF) in the finite extensible nonlinear elastic (FENE) dumbbell micro-macro models which consist of the coupled incompressible Navier-Stokes equations and the Fokker-Planck equations. We present the exact energy law of the resulting closure systems and introduce a post-modification scheme to preserve the positivity of PDF. The scheme not only reduces the region of negative PDF values but also preserves the structure of the induced stress tensor resulting from the molecular behaviors such as stretching and rotation. Numerical verifications are provided for the moment-closure system with some standard external flows. We also explore the relation of the maximum entropy principle (MEP) and the moment-closure approach.