Comparison of low-amplitude oscillatory shear in experimental and computational studies of model foams

A fundamental difference between fluids and solids is their response to applied shear. Solids possess static shear moduli, while fluids do not. Complex fluids such as foams display an intermediate response to shear with nontrivial frequency-dependent shear moduli. In this paper, we conduct coordinated experiments and numerical simulations of model foams subjected to boundary-driven oscillatory planar shear. Our studies are performed on bubble rafts (experiments) and the bubble model (simulations) in two dimensions. We focus on the low-amplitude flow regime in which T1 events, i.e., bubble rearrangement events where originally touching bubbles switch nearest neighbors, do not occur, yet the system transitions from solid- to liquidlike behavior as the driving frequency is increased. In both simulations and experiments, we observe two distinct flow regimes. At low frequencies $\omega$, the velocity profile of the bubbles increases linearly with distance from the stationary wall, and there is a nonzero total phase shift between the moving boundary and interior bubbles. In this frequency regime, the total phase shift scales as a power law $\Delta \sim {\omega }^n$ with $n\approx 3$. In contrast, for frequencies above a crossover frequency $\omega >{\omega }_p$, the total phase shift $\Delta$ scales linearly with the driving frequency. At even higher frequencies above a characteristic frequency ${\omega }_{nl}>{\omega }_p$, the velocity profile changes from linear to nonlinear. We fully characterize this transition from solid- to liquidlike flow behavior in both the simulations and experiments and find qualitative and quantitative agreements for the characteristic frequencies.

Figure 1

A typical snapshot of the 2D bubble raft used in experiments with packing fraction $\varphi \approx 0.86$ and $N\approx 3700$ bubbles. The bubble raft is composed of a bidisperse distribution of bubble sizes: a 4 to 1 ratio of $2.5\pm 0.3\text{-mm}$- to $5.3\pm 0.5\text{-mm}$-diameter bubbles.

Figure 2

(Color online) Normalized horizontal velocity profiles $v_x\left(y\right)/v_x\left(L_y\right)$ at $t=0$ for the (a) pure solid and (b) pure liquid obtained from solutions to Eq. (1) as a function of the driving frequency $\omega$. We show $\omega /{\omega }_{nl}=0.1$ (squares), 1.1. (circles), 2.1 (upward triangles), 4.1 (downward triangles), 6.1 [diamonds, only in (b)], and 20.1 [left triangles, only in (b)]. When referring to the solid (liquid), ${\omega }_{nl}$ corresponds to ${\omega }_{nl}^s$ $\left({\omega }_{nl}^l\right)$.

Figure 3

(Color online) Normalized horizontal velocity profiles $v_x\left(y\right)/v_x\left(L_y\right)$ at $t=0$ for a viscoelastic material with $\beta =G\rho L_y^2/{\eta }^2=1$ obtained from solutions to Eq. (1) as a function of the driving frequency $\omega$. We show $\omega /{\omega }_{nl}=0.1$ (squares), 1.1 (circles), 2.1 (upward triangles), 4.1 (downward triangles), 6.1 (diamonds), 8 (left triangles), and 20 (right triangles). ${\omega }_{nl}$ corresponds to ${\omega }_{nl}^l={\omega }_{nl}^s$.

Figure 4

(Color online) The total phase shift $\Delta =\delta \left(0\right)-\delta \left(L_y\right)$ for viscoelastic materials with different values of $\beta =G\rho L_y^2/{\eta }^2$ versus the normalized driving frequency $\omega /{\omega }_{nl}^l$. The curves for all $\beta$ scale linearly with $\omega$ at sufficiently high frequencies. We include $\beta =0$ (solid line), 0.0005 (squares), 0.005 (upward triangles), 0.05 (downward triangles), and 0.5 (filled circles). The crossover frequency ${\omega }_p$ can be obtained by locating the intersection of the low-frequency and high-frequency power-law scaling forms for $\Delta$. The slope of the black solid (dashed) line is 1 (3).

Figure 5

Plot of ${\omega }_p$ (solid line) and ${\omega }_d$ (stars) versus $\beta =G\rho L_y^2/{\eta }^2$ for the viscoelastic Kelvin-Voigt model. Note that ${\omega }_p<{\omega }_d$ over a wide range of $\beta$.

Figure 6

Average shear stress ${\Sigma }_{xy}$ plotted versus the applied shear rate $\dot{\gamma }$ for a system with $\varphi =0.86$ and $b^{\ast }=2$ undergoing steady planar shear flow in 2D simulations of the bubble model. The crossover shear rate at which the system transitions from quasistatic to power-law behavior is ${\dot{\gamma }}_c\approx {10}^{-4}$.

Figure 7

A schematic of the experimental setup that applied oscillatory planar shear to the bubble rafts. The $U$ joint highlighted in the figure is used to convert the rotary motion of the driving motor into oscillatory planar shear.

Figure 8

(Color online) Shear stress ${\Sigma }_{xy}$ versus the applied shear rate $\dot{\gamma }$ for the same bubble raft system described in Fig. 1 undergoing steady planar shear. We estimate the crossover frequency ${\dot{\gamma }}_c\approx 0.065{\text{s}}^{-1}$ above which the system transitions from the quasistatic to the power-law flow regime.

Figure 9

(Color online) The normalized average horizontal velocity $v_x\left(y\right)/v_x\left(L_y\right)$ (averaged over times when the boundary velocity is maximum) as a function of distance $y$ from the stationary bottom wall for several driving frequencies $\omega$ in the bubble raft experiments. At frequencies $\omega <{\omega }_d\approx 4.0{\text{s}}^{-1}$, the profiles are roughly linear, while above this characteristic frequency, liquidlike nonlinear profiles are observed.

Figure 10

(Color online) Total phase difference $\Delta$ plotted as a function of the driving frequency $\omega$ for the bubble raft experiments (black squares). The two triangles represent an upper limit for the measured phase-shift given the resolution of our experiment at the corresponding frequencies. The solid and dashed lines have slopes 1 and 2, respectively. The intersection of these two lines provides a rough estimate for the crossover frequency, ${\omega }_p\approx 1.9\pm 0.9{\text{s}}^{-1}$.

Figure 11

(Color online) The normalized average velocity $v_x\left(y\right)/v_x\left(L_y\right)$ (averaged over times when the boundary velocity is maximum) as a function of distance $y$ from the stationary wall plotted for several driving frequencies $\omega$ in the bubble model simulations. For $\omega \gtrsim {\omega }_d\approx 0.02$, the velocity profiles display liquidlike nonlinear decay.

Figure 12

(Color online) Total phase difference $\Delta$ plotted as a function of the driving frequency $\omega$ for the bubble model simulations at $\varphi =0.86$ and $b^{\ast }=2$. From the intersection of the solid red and dashed blue lines with slopes 1 and 3, respectively, we estimate ${\omega }_p\approx 0.015$.

Figure 13

(Color online) Summary of the measurements of the normalized frequencies ${\omega }_d/{\omega }_c$ (top panels) and ${\omega }_p/{\omega }_c$ (bottom panels) for the bubble model simulations (solid squares) and bubble raft experiments (solid lines). ${\omega }_d$ and ${\omega }_p$ are normalized by ${\omega }_c$, which is the frequency at which steady planar shear flows crossover from highly fluidized to quasistatic behavior. For the simulations, we show ${\omega }_d/{\omega }_c$ and ${\omega }_p/{\omega }_c$ as functions of $b^{\ast }$ at fixed $\varphi =0.86$ (left panels) and $\varphi$ at fixed $b^{\ast }=2$ (right panels). Because it is difficult to define $\varphi$ and $b^{\ast }$ precisely in experiments, the results for the bubble rafts are presented as solid lines (in between dashed lines, which give error bars for the experimental measurements) to identify the range of $\varphi$ and $b^{\ast }$ that best fit the experiments.