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, Wic1 is the critical Wi for the onset of the upstream bending streamline instability, Wic2 is the critical Wi for the formation of upstream wall-attached vortices, Wic2 is the critical Wi for the onset of time dependence, and Wic4 is the critical Wi for the significant growth of the upstream cylinder vortex.

Movie S1 — Particle images and calculated velocity field at Wic1 < Wi = 38 < Wic2, 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 — Wic2 < Wi = 90 < Wic3 at the z = 0 plane. Time-steady with small wall vortices.
Movie S3 — Wic3 < Wi = 148 < Wic4 at the z = 0 plane. Time-dependent with small wall vortices at a Wi near the initial transition to time dependence.
Movie S4 — Wic3 < Wi = 190 < Wic4 at the z = 0 plane. Time-dependent with asymmetric wall vortices at a higher Wi than in Movie S3.
Movie S5 — Wi = 507 > Wic4 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 — Wic2 < Wi = 97 < Wic3 at the \(y \approx \pm 1.8R\) planes. Approximately time-steady small wall vortices.
Movie S7 — Wic3 < Wi = 145 < Wic4 at the \(y \approx \pm1.8R\) planes. Time-dependent asymmetric wall vortices.
Movie S8 — Wic3 < Wi = 175 < Wic4 at the \(y \approx \pm 1.8R\) planes. Time-dependent asymmetric wall vortices at slightly higher Wi than in Movie S7.