Instabilities, Traveling Waves, and Connections Between Viscoelastic Channel Flow and Kolmogorov Flow

Mixing at low Reynolds number is difficult, but viscoelasticity is known to enhance mixing via instabilities and turbulence, and these transitions are not well understood. 2D inertialess viscoelastic channel flow is known to be linearly stable and some evidence suggests that it cannot exhibit elastic turbulence. However, large perturbations can lead to stable traveling wave solutions known as “narwhals.” These narwhals resemble traveling waves that occur in viscoelastic Kolmogorov flow (doubly periodic flow driven by a unidirectional sinusoidal force). Kolmogorov flow does exhibit elastic turbulence, and these waves appear to be involved in the transition to turbulence. To explore the connection between channel flow and Kolmogorov flow, we study the dynamics of Kolmogorov flow with a low wave number driving force, which has not been examined thoroughly in the past, as well as wall-bounded flows driven by sinusoidal forcing. We find that the stability properties and ability to trigger elastic turbulence depend on both the wave number of the forcing and the presence of walls.