Nonlinear dynamics of the viscoelastic Kolmogorov flow

Abstract

The weakly nonlinear dynamics of large-scale perturbations in a viscoelastic flow is investigated both analytically, via asymptotic methods, and numerically. For sufficiently small elasticities, dynamics is ruled by a Cahn–Hilliard equation with a quartic potential. Physically, this amounts to saying that, for small elasticities, polymers do not alter the purely hydrodynamical mechanisms responsible for the nonlinear dynamics in the Newtonian case (i.e. without polymers). The approach to the steady state is quantitatively similar to the Newtonian case as well, the dynamics being ruled by the same kink–antikink interactions as in the Newtonian limit. The above scenario does not extend to large elasticities. We found a critical value above which polymers drastically affect the dynamics of large-scale perturbations. In this latter case, a new dynamics not observed in the Newtonian case emerges. The most evident fingerprint of the new dynamics is the slowing down of the annihilation processes which lead to the steady states via weaker kink–antikink interactions. In conclusion, polymers strongly affect the large-scale dynamics. This takes place via a reduction of drag forces we were able to quantify from the asymptotic analysis. This suggests a possible relation of this phenomenon with the dramatic drag-reduction effect taking place in the far turbulent regime.