Comparison between step strains and slow steady shear in a bubble raft

We report on a comparison between stress relaxations after an applied step strain and stress relaxations during slow, continuous strain in a bubble raft. A bubble raft serves as a model two-dimensional foam and consists of a single layer of bubbles on a water surface. For both step strains and continuous strain, one observes periods of stress increase and decrease. Our focus is on the distribution of stress decreases, or stress drops. The work is motivated by apparent disagreements between quasistatic simulations of flowing foam and simulations of continuous strain for foam. Quasistatic simulations have reported larger average stress drops than the continuous strain case. Also, there is evidence in quasistatic simulations for a general divergence of the average size of the stress drops that only appears to occur in steady strain near special values of the foam density. In this work, applied step strains are used as an approximation to quasistatic simulations. We find general agreement in the dependence of the average stress drop on rate of strain, but we do not observe evidence for a divergence of the average stress drop.

Figure 1

A plot of the stress versus time for step strain measurements with a step time of $1\text{second}$ at a rotation rate of $0.1\mathrm{rad}∕\mathrm{s}$ and a relaxation time of $20\text{seconds}$.

Figure 2

A closer view of the stress vs time graph for step strains shown in Fig. 1. The dotted lines mark the end of the step and the onset of a relaxation. The multiple relaxations and plateaus that can be seen following a step strain indicate a complex relaxation time scale.

Figure 3

A distribution of the change in stress defined as the difference between one step strain value and the next. Negative changes in stress correspond to stress increases while positive ones correspond to stress drops. The triangles are a waiting time of $1\text{second}$, the squares are $10\text{seconds}$ and the circles are $20\text{seconds}$. As the waiting time is increased across this regime the stress distribution broadens and becomes more asymmetric.

Figure 4

Probability distribution for only the stress drops for both a series of step strains (closed symbols) and for continuous strain (open symbols). For long enough waiting times $(>10\mathrm{s})$ the overall shape of the distribution between the two types of flows are similar. The three waiting times for the step strain experiments are $1\mathrm{s}$ (▴), $20\mathrm{s}$ (∎), and $60\mathrm{s}$ (▾). The two continuous rotation rates are $\Omega =0.005\mathrm{rad}∕\mathrm{s}$ (◻) and $\Omega =0.002\mathrm{rad}∕\mathrm{s}$ (엯). The results highlight the similarities of the step strain and continuous strain distributions for large stress drops.

Figure 5

Plot of the average stress drop as a function of ${\Omega }_{\mathrm{eff}}$ for the step strain experiments (∎) and as a function of $\Omega$ for the continuous strain experiments (▴). Also shown are the results for the step strain experiments with an alternative definition of a stress drop that counts each individual decrease during an entire relaxation event (엯), and the alternative definition of stress drops for continuous strain, as discussed in the text (▿). The results highlight the differences between the two types of strain for the average stress drop measurements, as well as the impact of different definitions of a stress drop.

Figure 6

Plot of $⟨\sigma \left(t\right)-\sigma (t+\tau )⟩$ versus $\tau$ for two different rotation rates. The circles are $\Omega =0.005\mathrm{rad}∕\mathrm{s}$, and the squares are $\Omega =0.001\mathrm{rad}∕\mathrm{s}$. In both cases, the curves plateau for $\tau$ on the order of $10\mathrm{s}$, with the exact value of the plateau proportional to the rotation rate.

Figure 7

Plotted here is the system size dependence of the average stress drop for two different waiting times. The squares are for a waiting time of $60\mathrm{s}$, and the triangles are for a waiting time of $2\mathrm{s}$. The open symbols correspond to the top axis, which gives the average number of bubbles in the radial direction. The closed symbols are relative to the bottom axis. In all cases, no observable system size dependence was measured.