On the compatibility between various macroscopic formalisms for the concentration and flow of dilute polymer solutions

Abstract

A detailed comparison is made of three different formulations of the macroscopic equations governing the concentration changes and the rheology in flows of dilute polymer solutions. More specifically, the governing equations obtained using an inhomogeneous kinetic theory, a continuum two‐fluid Hamiltonian model, and a body‐tensor continuum formalism are compared on a term by term basis. To allow the comparison, an improved kinetic theory‐based model for inhomogeneous polymer solutions is developed in which all terms up to and including second‐order derivatives in space are consistent retained in the Taylor expansions. The analysis is carried out both for the elastic Hookean dumbbell and the bead‐spring Rouse chain models. With both mechanical models, it is seen that the new approach leads to the same equation for the polymer chain number density, which is also identical to that derived by the two continuum models. Regarding the stress evolution equation all formulations are found to lead to identical expressions, provided that second‐ and higher‐order terms (with respect to the deformation gradient) are neglected.