Impact of boundaries on velocity profiles in bubble rafts

Under conditions of sufficiently slow flow, foams, colloids, granular matter, and various pastes have been observed to exhibit shear localization, i.e., regions of flow coexisting with regions of solidlike behavior. The details of such shear localization can vary depending on the system being studied. A number of the systems of interest are confined so as to be quasi two-dimensional, and an important issue in these systems is the role of the confining boundaries. For foams, three basic systems have been studied with very different boundary conditions: Hele-Shaw cells (bubbles confined between two solid plates); bubble rafts (a single layer of bubbles freely floating on a surface of water); and confined bubble rafts (bubbles confined between the surface of water below and a glass plate on top). Often, it is assumed that the impact of the boundaries is not significant in the “quasistatic limit,” i.e., when externally imposed rates of strain are sufficiently smaller than internal kinematic relaxation times. In this paper, we directly test this assumption for rates of strain ranging from ${10}^{-3}\text{to}{10}^{-2}{\mathrm{s}}^{-1}$. This corresponds to the quoted rate of strain that had been used in a number of previous experiments. It is found that the top plate dramatically alters both the velocity profile and the distribution of nonlinear rearrangements, even at these slow rates of strain. When a top is present, the flow is localized to a narrow band near the wall, and without a top, there is flow throughout the system.

Figure 1

(a) Schematic representation of a $T1$ event illustrating two neighboring bubbles (A and B) switching to next nearest neighbor, and two next nearest neighbors (C and D) becoming neighbors. (b),(c),(d) A sequence of images that have been thresholded to bubbles are black taken from a bubble raft illustrating a $T1$ event.

Figure 2

(a) A schematic of the apparatus as viewed from the top. The details are described in the text. Highlighted in the figure are the driving bands (C) that are used to generate flow. A close up photograph of a driving band is given in (b). The spacing for the bands is $4\mathrm{mm}$.

Figure 3

(a) Image of a typical set of bubbles in the case with no top on the system. The scale bar represents $7\mathrm{mm}$. (b) Image of a typical set of bubbles for the case of a top on the system. The scale bar represents $7\mathrm{mm}$.

Figure 4

(Color online) Illustration of identifying $T1$ events using the Voronoi construction. Images (a) and (b) are the Voronoi reconstruction for two sequential images. Bubbles involved in $T1$ events are colored.

Figure 5

(Color online) The scaled velocity profile $[U\equiv \left(\text{bubble}\text{velocity}\right)∕\left(\text{belt}\text{velocity}\right)]$ as a function of scaled position $(y∕⟨d⟩)$ across the trough for three different rates of strain for a system without a top, where $⟨d⟩$ is the average bubble diameter. The scale of the position axis is set so that it extends from the edge of one belt to the other belt.

Figure 6

(Color online) The scaled velocity profile $[U\equiv \left(\text{bubble}\text{velocity}\right)∕\left(\text{belt}\text{velocity}\right)]$ as a function of scaled position $(y∕⟨d⟩)$ across the trough for three different rates of strain for a system with a top, where $⟨d⟩$ is the average bubble diameter. The scale of the position axis is set so that it extends from the edge of one belt to the other belt.

Figure 7

The natural log of the scaled velocity profile $\left[\ln \left(U\right)\right]$ as a function of scaled position $(y∕⟨d⟩)$ across the trough for two rates of strain in the system with a top (solid squares are a rate of strain of $0.014{\mathrm{s}}^{-1}$ and open triangles are $0.0014{\mathrm{s}}^{-1}$). The lines represent linear fits to the $0.014{\mathrm{s}}^{-1}$ data, indicating an exponential decay of the velocity in that regime.

Figure 8

Positions of $T1$ events in the central portion of the system (a fraction of the $x$ direction) without a confining top. Both the $x$ and $y$ positions are scaled by the average bubble diameter $⟨d⟩$.

Figure 9

Positions of $T1$ events in the central portion of the system (a fraction of the $x$ direction) without a confining top. Both the $x$ and $y$ positions are scaled by the average bubble diameter $⟨d⟩$.

Figure 10

(Color online) Histogram of the probability of a $T1$ event $\left[P\left(T1\text{event}\right)\right]$ as a function of $y∕⟨d⟩$ computed using in the range $-22⩽x∕⟨d⟩⩽22$ for both the case with a top (solid red bars) and without a top (hashed open bars). The histogram shows the absence of $T1$ events from the center of the system with a top present, and the relative flatness of the distribution in the center for flow without a top.