Stochastic Processes, Polymer Dynamics, and Fluid Mechanics

One can attempt to achieve a theoretical understanding of polymer fluid dynamics on two different levels: continuum mechanics and kinetic theory. Continuum mechanics deals with the formulation and solution of a system of macroscopic equations for the density, velocity, temperature, and possibly other fields describing the fluid structure, which are related to conservation laws for mass, momentum, energy, and maybe other quantities associated with the additional fields. In order to obtain a closed system of macroscopic equations one needs to supplement the fundamental conservation laws by certain empirically or microscopically founded equations of state for the fluxes of the conserved quantities. Such equations of state, which are characteristic for a given material, are often referred to as constitutive equations. The formulation of suitable constitutive equations and of general admissibility criteria for such constitutive equations is a central part of continuum mechanics. In polymer kinetic theory, one attempts to understand the polymer dynamics, the constitutive equation for the momentum flux or stress tensor, and eventually polymer fluid dynamics by starting from coarse-grained molecular models. Excellent reviews of the state of the art in both continuum mechanics and kinetic theory are given in the two volumes of the comprehensive introductory textbook Dynamics of Polymeric Liquids by R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager [1, 2]. The more recent literature in continuum mechanics and in kinetic theory has been reviewed in [3] and [4], respectively.