A blessing of viscoelastic narwhals*

Recent years have seen significant progress in our understanding of chaotic motion of dilute polymer solutions. The key component of these advances is the discovery of a linear instability in a viscoelastic pressure-driven channel flows [1]. While the linear instability itself only exists at ultralow polymer concentrations and rather large values of the Weissenberg number, the ensuing 2D non-linear states have been shown to extend sub-critically to a wide range of experimentally relevant material parameters, both in the elasto-inertial [2] and purely-elastic regimes [3]. In 2D, these 'narwhal' states -  named to reflect the spatial organisation of the associated polymer stress - take the form of stable travelling-wave solutions, while in 3D they are linearly unstable [4] and are implicated in organising the dynamics of purely elastic turbulence [5].

Here, we investigate the spatial extent of narwhal states in the absence of inertia. We demonstrate that they are localised along the flow direction and we map their typical size for various material parameters. We further probe their localised nature by studying a group of travelling-wave states simultaneously occupying the same channel - a blessing of narwhals. We show that beyond a particular distance, narwhal states are insensitive to each other's presence and can co-exist. We speculate how these findings are connected to the dynamics of localised disturbances in 3D purely-elastic turbulence.

[1] M. Khalid et al., Phys. Rev. Lett. 127, 134502 (2021)

[2] J. Page et al., Phys. Rev. Lett. 125, 154501 (2020)

[3] A. Morozov, Phys. Rev. Lett. 129, 017801 (2022)

[4] M. Lellep et al., J. Fluid Mech. 959, R1 (2023).

[5] M. Lellep et al. (under review)