Stress-gradient induced migration of polymers in corrugated channels

Abstract

We study the flow of a dilute polymer solution in a wavy channel under steady-state flow conditions by employing the nonequilibrium thermodynamics two-fluid model [Mavrantzas and Beris, Phys. Rev. Lett. 69, 273–276 (1992)], allowing for the coupling between polymer concentration and polymer stresses. The resulting highly complex system of partial differential equations describing inhomogeneous transport phenomena in the fluid are solved with an efficient implementation of the mixed finite-element method. We present numerical results for polymer concentration, stress, velocity, and fluxes of polymer as a function of the nondimensional parameters of the problem (the Deborah number De, the Peclet number Pe, the Reynolds number Re, the ratio of the solvent viscosity to the total fluid viscosity β, and the constriction ratio of the channel width cr). We find that the constricted part of the wall is depleted of polymer, when the polymer diffusion length scale, expressed by the ratio of De/Pe, increases. The migration is more pronounced for macromolecules characterized by longer relaxation times and takes place toward the expanding part of the channel or toward the centerplane. Migration is also enhanced by the width variability of the channel: The more corrugated the channel, the stronger the transfer of polymer to the centerplane. This increases the spatial extent of polymer depletion near the wall or induces a zone of sharp variation in polymer stress and concentration, which moves away from the channel wall, especially in lower polymer concentration. The development of a polymer-depleted layer smooths out the boundary layer which is known to arise with Boger fluids at the walls of such corrugated channels or tubes and gives rise to an “apparent” slip in the constricted section of the wall and to a very low value of the drag force on the wall. When and where boundary layers arise, they scale as (1/De) for the stresses and as (De/Pe)1/3 for the concentration.