Figure 1

(a) Schematic representation of the three main steps in a T1 event. The bubbles labeled 1 and 2 are initially neighbors. As the bubbles are sheared, all four bubbles meet at a vertex. After the event, the bubbles labeled 3 and 4 are neighbors. (b)–(d) are three images illustrating an actual T1 event in the bubble raft. The location of one T1 event is highlighted by artificially coloring the bubbles involved white. The images are taken $3.2\mathrm{s}$ apart and the white scale bar in (b) is $2\mathrm{mm}$ long.Reuse & Permissions

Figure 2

(a) A schematic drawing of the apparatus showing a side view. The main elements of the apparatus are the knife edge disk that is supported by a torsion wire and that serves as the inner cylinder for the bubbles. There is a separate fixed inner cylinder in the fluid (in gray). There is a segmented outer cylinder for generating flow, and there is a fixed dish that holds the fluid. The bubbles sit on top of the fluid, as indicated by the circles. (b) An image from the top of the bubbles in the apparatus that shows a portion of both the outer and inner cylinder. The black scale bar in the lower left corner is $5\mathrm{mm}$.Reuse & Permissions

Figure 3

Stress on the inner cylinder versus strain for the initial period of shear. The curve is divided into three regions labeled $A$, $B$, and $C$. Region $A$ is the only period for which no T1 events are observed.Reuse & Permissions

Figure 4

A plot of $v\left(r\right)∕r$ vs radial position for the bubble motion during interval $A$ in Fig. 3. The solid squares are data averaged over all bubbles at a given radial position. The solid line is a fit to Eq. (4).Reuse & Permissions

Figure 5

(Color online) Five images representing typical bubble deviations from elastic flow during the initial stress rise. The arrow in image (a) gives the direction of rotation of the outer cylinder. Images (a)–(c) are taken from the corresponding three strain intervals in Fig. 3 and show typical strain intervals in which no T1 events occur. Image (d) shows a typical localized T1 event from region $B$ in Fig. 3. Image (e) shows an event composed of multiple T1 events in which two rows of bubbles slip pass each other. This event is taken from region $C$ in Fig. 3. The circles indicate the location of a subset of bubbles that have been tracked. The sizes of all the circles are the same, independent of actual bubble size, for clarity. They are grayscaled (color-coded) based on the deviation from elastic displacements, as defined in the text. White represents deviations less than $0.05\mathrm{cm}$. The grayscale (color) bar gives the scale for deviations greater than $0.05\mathrm{cm}$. The solid line associated with bubbles gives the direction of the deviation from elastic displacements. The squares represent the location of T1 events, where the grayscale (color) represents the time relative to the start of the strain interval used to calculate $\Delta$. The scale bar in image (a) is $0.5\mathrm{mm}$.Reuse & Permissions

Figure 6

Scatter plot of the number of T1 events during a period of stress increase vs the effective elastic modulus $G$ for the same period of strain. The solid line is a guide to the eye.Reuse & Permissions

Figure 7

The individual points are the radial position of T1 events as a function of strain. The solid curve is the stress as a function of strain (measured using images of the inner cylinder).Reuse & Permissions

Figure 8

The solid line is the same stress vs strain curve as shown in Fig. 7. The bars summarize the data in Fig. 7 by plotting only the number of T1 events∕bubble in a strain interval of 0.013.Reuse & Permissions

Figure 9

The squares are the average velocity of the bubbles normalized by $\Omega r$ as a function of radial position. The bars give the total number of T1 events∕bubble as a function of radial position. The solid lines are guides to the eye. The horizontal line is $v\left(r\right)∕\Omega r=1$, which corresponds to the motion of a rigid body. The vertical line is the location of the shear discontinuity, as reported in Ref. 39.Reuse & Permissions

Figure 10

This is two strain intervals from Fig. 7, showing both the location of T1 events (solid squares) and the stress (solid line, as measured by the magnetic method) as a function of strain. Each interval is further divided into smaller strain intervals by the vertical lines. The labels in (b) correspond to the images in Fig. 11.Reuse & Permissions

Figure 11

(Color online) Five images representing typical bubble deviations from elastic flow during the corresponding strain intervals as indicated in Fig. 10b. The arrow in image (a) gives the direction of rotation of the outer cylinder. The circles indicate the location of a subset of bubbles that have been tracked. The size of the circle is the same for all bubbles for clarity. The circles are grayscaled (colored) based on the deviation from elastic displacements, as defined in the text. White represents deviations less than $0.05\mathrm{cm}$. The bar gives the scale for deviations greater than $0.05\mathrm{cm}$. The solid line associated with bubbles gives the direction of the deviation from elastic displacements. The squares represent the location of T1 events, where the grayscale (color) equals the time relative to the start of the strain interval used to calculate $\Delta$. The scale bar in image (a) is $0.5\mathrm{mm}$.Reuse & Permissions

Figure 12

The average angular displacement of the bubbles normalized by $\Omega \Delta t$ as a function of radial position. The different symbols are for the different strain intervals in Fig. 10(b): (a) open circles; (b) solid squares; (c) solid circles; (d) open squares; and (e) solid triangles. Here $\Delta t$ is the time interval for each strain interval. The solid line is the fit to elastic behavior from Fig. 4.Reuse & Permissions