Upstream Wall Vortices - Soft Matter

The movies below accompany the Supplementary Information section of our paper entitled "Upstream Wall Vortices in Viscoelastic Flow Past a Cylinder", Soft Matter, 2022 — Currently under revisions. Link when available.

The movies illustrate the time-dependent velocity fields for the flow of the wormlike micellar solution past the microcylinder at the Weissernberg numbers Wi = Uλ/R, where U is the average flow speed, λ is the terminal relaxation time of the fluid, and R = 102 μm is the radius of the cylinder. For reference, Wi_c1 is the critical Wi for the onset of the upstream bending streamline instability, Wi_c2 is the critical Wi for the formation of upstream wall-attached vortices, Wi_c2 is the critical Wi for the onset of time dependence, and Wi_c4 is the critical Wi for the significant growth of the upstream cylinder vortex.

Movie S1 — Particle images and calculated velocity field at Wi_c1 < Wi = 38 < Wi_c2, at the z = 0 plane, the same case as shown in Fig. 3 (b) in the main text. This is a time-steady flow within the regime where there are bending streamlines upstream of the cylinder but no wall vortices.

Movies S2-S5 are at the same Wi as shown in Fig. 7 in the main text. These videos show the time-dependent velocity fields along with the time series of the probe velocity \(|v_{p}/U|\), the middle position of the wall vortices χ, their widths \(\mathcal{W}\), and the length of the cylinder vortex \(\mathcal{L}\) (if present).

 Movie S2 — Wi_c2 < Wi = 90 < Wi_c3 at the z = 0 plane. Time-steady with small wall vortices.

Movie S3 — Wi_c3 < Wi = 148 < Wi_c4 at the z = 0 plane. Time-dependent with small wall vortices at a Wi near the initial transition to time dependence.

Movie S4 — Wi_c3 < Wi = 190 < Wi_c4 at the z = 0 plane. Time-dependent with asymmetric wall vortices at a higher Wi than in Movie S3.

Movie S5 — Wi = 507 > Wi_c4 at the z = 0 plane. Time-dependent at a Wi within the cylinder vortex regime.

Movies S6-S8 show the time-dependent velocity fields recorded in the \(x-z\) plane at the positions \(y \approx \pm 1.8R\) to illustrate how the wall vortices are shaped and behave in the \(z\) direction. The movies correspond to the time-averaged velocity fields shown in Fig. S1 in the Supplementary Information. Note that the videos were not recorded coincidentally at a given Wi, so it is not possible to directly compare the shape of the vortices and time-dependent behaviour between the two planes in each video.

Movie S6 — Wi_c2 < Wi = 97 < Wi_c3 at the \(y \approx \pm 1.8R\) planes. Approximately time-steady small wall vortices.

Movie S7 — Wi_c3 < Wi = 145 < Wi_c4 at the \(y \approx \pm1.8R\) planes. Time-dependent asymmetric wall vortices.

Movie S8 — Wi_c3 < Wi = 175 < Wi_c4 at the \(y \approx \pm 1.8R\) planes. Time-dependent asymmetric wall vortices at slightly higher Wi than in Movie S7.