Flow transitions in two-dimensional foams

For sufficiently slow rates of strain, flowing foam can exhibit inhomogeneous flows. The nature of these flows is an area of active study in both two-dimensional model foams and three dimensional foam. Recent work in three-dimensional foam has identified three distinct regimes of flow [S. Rodts, J. C. Baudez, and P. Coussot, Europhys. Lett. 69, 636 (2005)]. Two of these regimes are identified with continuum behavior (full flow and shear banding), and the third regime is identified as a discrete regime exhibiting extreme localization. In this paper, the discrete regime is studied in more detail using a model two-dimensional foam: a bubble raft. We characterize the behavior of the bubble raft subjected to a constant rate of strain as a function of time, system size, and applied rate of strain. We observe localized flow that is consistent with the coexistence of a power-law fluid with rigid-body rotation. As a function of applied rate of strain, there is a transition from a continuum description of the flow to discrete flow when the thickness of the flow region is approximately ten bubbles. This occurs at an applied rotation rate of approximately $0.07{\mathrm{s}}^{-1}$.

Figure 1

(Color online) (a) A velocity profile (with $r_c=5.9\mathrm{cm}$) as a function of radial position for the system with outer radius of $9\mathrm{cm}$ and rotation rate of $0.09{\mathrm{s}}^{-1}$ (black symbols), with power-law fit where $n=0.515$ (blue line). (b) Same velocity profile (black symbols), with numeric fit to the Herschel-Bulkley model where $n=1.056$ (blue line).

Figure 2

(Color online) Comparison of the measured critical radius as a function of rotation rate for outer radii of $7\mathrm{cm}$ (black squares) and $9\mathrm{cm}$ (red circles). The open symbols correspond to calculation of $r_c$ using a Herschel-Bulkley model, and the solid symbols correspond to calculating $r_c$ assuming a power-law fluid coexisting with a solidlike state.

Figure 3

(Color online) (a) The exponent $n$ obtained from fitting the velocity curves to a power-law fluid coexisting with a solidlike state versus the external rotation rate for $R=7\mathrm{cm}$ (black squares) and $R=9\mathrm{cm}$ (blue circles). The dashed line is the median value. (b) The exponent $n$ obtained from fitting the velocity curves to a Herschel-Bulkley model versus the external rotation rate for $R=7\mathrm{cm}$ (black squares) and $R=9\mathrm{cm}$ (blue circles). The dashed line is the median value of the exponents. The solid line is $n=1$.

Figure 4

(Color online) (a) Plot of the scaled velocity profiles $\tilde{v}=(\Omega r∕r_c)-v_{\theta }∕r_c$ versus $r∕r_c$. Solid line is a guide to the eye. (b) Plot of $v_{\theta }∕\Omega r_c$ versus $r∕r_c$. Rates of strain are indicated in the legend. Both plots are for the system with an outer radius of $7\mathrm{cm}$.

Figure 5

(Color online) Critical rate of strain as a function of external rotation rate for $R=7\mathrm{cm}$ (blue triangles) and $R=9\mathrm{cm}$ (black squares).

Figure 6

(Color online) Probability distribution of measuring a particular value of $r_c$ based on averaging the velocity over finite time intervals. Four different time intervals are illustrated for a rotation rate of $0.07{\mathrm{s}}^{-1}$ with $R=7\mathrm{cm}$. The time intervals and mean for each distribution are indicated in the figure.

Figure 7

Illustration of the convergence of the measured value of $r_c$ as a function of the time interval over which the average velocity is computed is increased.