On Some Pdf 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 multiscale 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 consists 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. 1. Introduction. In this paper, we consider the hydrodynamical systems of dilute polymeric fluids. The viscoelastic flow of rheological complex fluids can be described by a multiscale (micro-macro) model. The multiscale-multiphysics model includes the coupling between the continuum mechanic theory [5] in macroscopic level and the kinetic theory in microscopic level. This micro-macro model also reflects the interaction between two different scales as the macroscopic flow/deformation will affect the microscopic structure through kinematic transport/deformation relation; while the averaging (coarsening) effects of the microscopic molecular configurations such as stretching and orientation will affect the macroscopic flow field through the induced elastic stresses. In many applications, we are more interested in the macroscopic quantity, such as the induced elastic stresses, rather than the detail behavior of molecular/microscopic variables. These stresses, resulting from the average of molecular behaviors, can be described in many situations by the moments of distribution function of molecular configurations. Notice that the PDF of the molecular configurations carries all the microscopic information of the system. Among different molecular models, the two most used well-known models are the Hookean dumbbell model, which is related to the Oldroyd-B viscoelasticity [1, 12], and the finite-extensible-nonlinear-elastic (FENE) dumbbell model [1, 2]. While the Hookean models are the best understood one and form the basis for most analytic studies, in this paper we focus on the FENE spring dumbbell model