Velocity Profiles in Slowly Sheared Bubble Rafts

Measurements of average velocity profiles in a bubble raft subjected to slow, steady shear demonstrate the coexistence between a flowing state and a jammed state similar to that observed for three-dimensional foams and emulsions [P. Coussot et al., Phys. Rev. Lett. 88, 218301 (2002)]. For sufficiently slow shear, the flow is generated by nonlinear topological rearrangements. We report on the connection between this short-time motion of the bubbles and the long-time averages. We find that velocity profiles for individual rearrangement events fluctuate, but a smooth, average velocity is reached after averaging over only a relatively few events.

Figure 1

Stress versus time for $\Omega =5\times {10}^{-3}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ [gray line labeled (1)] and $\Omega =8\times {10}^{-4}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ [black line labeled (2)]. The solid horizontal line is ${\tau }_o$ determined from a fit using data from Ref. 16 to a Herschel-Bulkley fluid model. The dashed horizontal line is $\sigma \left(r_i\right)$ when $\sigma \left(6.7\mathrm{c}\mathrm{m}\right)={\tau }_o$ (black line) or $\sigma \left(6.3\mathrm{c}\mathrm{m}\right)={\tau }_o$ (gray line). The dotted line is $\sigma \left(r_i\right)$ when $\sigma \left(R\right)={\tau }_o$. The vertical lines indicate the stress drops presented in Fig. 3 for $\Omega =8\times {10}^{-4}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$.

Figure 2

$v\left(r\right)=v_{\theta }\left(r\right)/\left(r\Omega \right)$ versus $r$ for $\Omega =5\times {10}^{-3}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ averaged over time from $t=250\mathrm{s}$ to the end of the run (●) and for $\Omega =8\times {10}^{-4}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ averaged over time from $t=650\mathrm{s}$ to the end of the run (■). The solid lines are fits to a power-law model for viscosity (see discussion in text). The dashed line is the $v=1$ line. Also shown are $v\left(r\right)$ for $\Omega =5\times {10}^{-3}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ averaged over a single event (□), four events (○), 10 events (△), and 20 events (▽). The inset shows the averages over the entire run in the region of the discontinuity for both shear rates (same symbols).

Figure 3

Average velocity during the stress drops indicated in Fig. 1 for $\Omega =8\times {10}^{-4}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$. The insets are images of a subset of the bubbles, with gray circles corresponding to $v\left(r\right)$ in the same direction as $\Omega$, black circles corresponding to $v\left(r\right)$ opposite $\Omega$, and white circles corresponding to $\left|v\right(r\left)\right|\le 6\times {10}^{-4}\mathrm{c}\mathrm{m}/\mathrm{s}$.