Figure 1

Sketch of a 1-D array of water droplets dragged slowly by a faster flow of oil in a microfluidic channel, as in the experiment of Ref. [6], which had a height of 10 $\mu$m. The droplets are approximated as having a cylindrical disk shape, with a radius of $R=10$ $\mu$m. In the experiment, there was probably a thin layer of oil and surfactant that provides a lubricating effect between water droplets and the surfaces of the floor and ceiling of the channel [2].Reuse & Permissions

Figure 2

Droplet positions, at three times in the simulation, shown in the inertial frame of the array of droplets. Only a portion of a total of 256 droplets is shown here. (a) At the time $t=0$, the initial positions of the droplets are specified. We then integrate the fully nonlinear equation of motion, Eqs. (1, 2, 3), to advance the droplet position and velocity each time step. (b) At $t=5$ s, a droplet fluctuates about its equilibrium position. (c) At $t=9.6$ s, the displacements of the droplets have grown as large as $a$, due to the instability.Reuse & Permissions

Figure 3

Power spectra of longitudinal (a) and transverse (b) waves, with the logarithm of spectral power shown as a function of frequency and wave number. Spectral power is mostly concentrated in narrow ranges, which are centered on the dispersion relations for a linear theory [Eq. (7)], shown as dashed lines. The feature near $ka\approx \pm 0.8$ and $\omega \approx \pm 10$ s${}^{-1}$, marked $\gamma$ in (b), is not predicted by the linear theory and is interpreted here as an indication of nonlinear effects, possibly resonant wave interaction of the waves marked $\alpha$ and $\beta$.Reuse & Permissions

Figure 4

Correlation strength $\left|C\right|$, as defined in Eq. (11), for longitudinal $k_L$ and transverse $k_T$ waves. We find a strong correlation, for certain combinations of $k_L$ and $k_T$. The contour scale has a threshold of 0.2 to suppress noise.Reuse & Permissions

Figure 5

Interpretation of correlation data in Fig. 4. Wave numbers that exhibit high correlations in Fig. 4 obey matching conditions for wave numbers shown with a dot-dash curve with $k_{L}a=-k_{T}a$ and dashed curves with $k_{L}a=k_{T}a\pm (\pi -\epsilon )$ where $0\le \epsilon <0.5$. We also find a matching condition for frequency by comparing to the dispersion relations, Eq. (7), which are plotted along the edges of the correlation graph. As an example of these matching conditions, we consider a longitudinal wave $k^{\prime }$ as marked with an open circle. Following the horizontal line $H1$ across the diagram, strong correlation occurs for the transverse waves with wave numbers $-k^{\prime }$ and $k^{\prime }+\pi -\delta$, which have the same frequency as the longitudinal wave.Reuse & Permissions

Figure 6

Space-time diagrams for velocities (a) $v_x$ and (b) $v_y$. To make the spacetime diagram easier to examine, we present the data for only a portion of the array of droplets, $146\le m\le 196$. The sloped stripes in (a) and (b) indicate that longitudinal and transverse wave packets propagate in opposite directions. Note that the wave growth is spatially localized and that the high-amplitude wave packets at $t>8$ s include only three or four droplets at the location marked $*$.Reuse & Permissions

Figure 7

Sketch of dispersion relation. The slope, which corresponds to the phase speed $\omega /k$, varies with $k$ because of the curvature of the dispersion relation. Two cases of interest are shown: for long wavelength (small $k$), $\omega /k$ is the sound speed $C_S=250$ $\mu$m/s, while for a shorter wavelength (larger $k$), $\omega /k$ is slower. In Fig. 6 we observe a phase speed of 100 $\mu$m/s, corresponding to the lesser slope indicated here.Reuse & Permissions

Figure 8

Instability growth, as measured by the time series of the mean squared velocity, averaged over the entire length of the 1-D array. The simulation was run three ways for this test. The solid curves are four simulation runs for the fully nonlinear case using Eq. (3) when integrating the equation of motion, as in Figs. (1)–(6). The dashed curve is for the linear case using Eq. (4). The open circles are results for the fully nonlinear simulation with droplets constrained to move only longitudinally (i.e., in the $x$ direction). We find that for the nonlinear case there is rapid nonexponential growth, but for the other two cases there is no growth, indicating that the instability requires nonlinearity and transverse motion. All the data in other figures of this paper are from run 1.Reuse & Permissions

Figure 9

Contours of $\Delta$ as defined in Eq. (A1), assuming the dispersion relations in Eq. (7). Darker colors indicate smaller values of $\Delta$. We find that the minima of these contours (the darkest colors) coincide with the dark features in Fig. 4, demonstrating that the matching criterion is modeled accurately by Eq. (14). A portion of a circle is shown, with a center at $(2.058\pi ,-2.058\pi )$ and a radius of $2.309\pi$, as determined by a manual fit. The agreement of this circle with the locus of the minima motivates the forms of Eqs. (15) and (16).Reuse & Permissions