Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

TL;DR

This paper develops a two-fluid kinetic model for dilute polymer solutions in non-homogeneous flows, incorporating distinct velocities for polymer and solvent phases, and derives inertialess limits for the governing equations.

Contribution

It introduces a rigorous derivation of a two-fluid kinetic model with phase-specific velocities for dilute polymer solutions, extending standard models to strongly non-homogeneous flows.

Findings

Derivation of a two-fluid model with separate velocities for polymer and solvent.

Inertialess limits for the Fokker-Planck and friction force equations.

Model applicability to strongly non-homogeneous flow conditions.

Abstract

We propose a two fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier-Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker-Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case.