Coherent Structures in Plane Channel Flow of Dilute Polymer Solutions with Vanishing Inertia

When subjected to sufficiently strong velocity gradients, solutions of long, flexible polymers exhibit flow instabilities and chaotic motion, often referred to as elastic turbulence. Its mechanism differs from the familiar, inertia-driven turbulence in Newtonian fluids and is poorly understood. Here, we demonstrate that the dynamics of purely elastic pressure-driven channel flows of dilute polymer solutions are organized by exact coherent structures that take the form of two-dimensional traveling waves. Our results demonstrate that no linear instability is required to sustain such traveling wave solutions and that their origin is purely elastic in nature. We show that the associated stress profiles are characterized by thin, filamentlike arrangements of polymer stretch, which is sustained by a solitary pair of vortices. We discuss the implications of the traveling wave solutions for the transition to elastic turbulence in straight channels and propose ways for their detection in experiments.

Figure 1

Exact traveling-wave solutions. The trace of the conformation tensor (color) and the flow streamlines (solid lines) for $\beta =0.8$ and $\mathrm{Wi}=26$ (a), $\mathrm{Wi}=40$ (b), and $\mathrm{Wi}=80$ (c). The mean flow is from left to right. The streamlines represent velocity deviation from the mean streamwise profile ${\overline{v}}_x(y)$. (d) The speed of the traveling wave solutions determined by tracking the position of the maximum of $\mathrm{Tr}\mathit{c}$ as a function of time. (e) The relative kinetic energy of the traveling wave solutions tracing the upper branch of the corresponding bifurcation from infinity. (f) The saddle-node values ${\mathrm{Wi}}_{\mathrm{sn}}$ (black symbols) and their rheological counterparts ${\mathrm{Wi}}_{\mathrm{sn}}^{(\mathrm{rheo})}$ (red symbols) determined as the lowest value of the Weissenberg number at which the solution can be sustained.

Figure 2

Characterization of the TWS velocity field for $\beta =0.8$ and $\mathrm{Wi}=80$. (a) The velocity deviation from the mean profile $[v_x(x,y)-{\overline{v}}_x(y),v_y(x,y)]$ (vectors), and the out-of-plane component of the vorticity ${\partial }_x{v}_y(x,y)-{\partial }_y[v_x(x,y)-{\overline{v}}_x(y)]$ (color). (b) Comparison between the laminar and the mean streamwise velocity profiles.