Stress-gradient-induced polymer migration: Perturbation theory and comparisons to stochastic simulations

Abstract

While phenomenological expressions have been given in the past for polymer flux due to gradients in polymer stress or gradients in velocity gradient, we here provide a systematic perturbation theory for the coupled equations for polymer rheology, fluid dynamics, and polymer mass transport. Analytical results are obtained for the steady-state concentration of dilute dumbbells in the absence of hydrodynamic interactions, as a function of the Weissenberg number Wi, Peclet number Pe, and the “gradient number” Gd, where the latter is the ratio of polymer equilibrium size to the characteristic distance over which velocity gradients change. At lowest order in all three perturbation variables, the theory yields a Poisson's equation with the driving term given by 2c¯Dτ2 vj,k,j vj,j,k, where c¯ is the average concentration, D is the dumbbell diffusivity, τ is the relaxation time, and v is velocity with Einstein notation for the subscripts. We find using Brownian dynamics (BD) simulations in a simple Taylor vortex f...