Figure 1

(a) Coalescence cell and measurement circuit. We apply an ac voltage $V_{\mathrm{in}}$ (Hewlett-Packard, HP3325A) across known circuit elements ($R_k$ and $C_k$) and coalescing liquid drops. We read voltages $V_1$ and $V_2$ into Labview (National Instruments) to calculate the cell impedance (dotted box). $Z_{CR}$ is from the coalescence region (dashed box); $Z_t$ and $Z_b$ are due to the top and bottom nozzles; $C_p$ is a small cell capacitance in parallel with $Z_{CR}$; and $R_{\mathrm{in}}=50\Omega$ is the input impedance of a high-speed digitizer (National Instruments, NI PCI-5105). The nozzles are brought together to measure $Z_t+Z_b$. (b) Mean value of $R_{CR}-R_0$ for six coalescences of aqueous NaCl ($\gamma =88\mathrm{mN}/\mathrm{m}$, $\rho =1180\mathrm{kg}/{\mathrm{m}}^3$, and $\mu =1.9\mathrm{mPa}\mathrm{s}$). Vertical error bars are the standard deviation of the points in each logarithmically spaced bin. Horizontal error bars are $\pm 1/(2f)$. Asymptotic behavior is consistent with $1.3\times {10}^{-3}{\tau }^{-1}$ at early times (dotted line) and $0.90{\tau }^{-1/2}$ at late times (dashed line). Inset: Measurements of the bridge radius from the electric method (○) probe much earlier than high-speed imaging (×) but extend just as far to long times.Reuse & Permissions

Figure 2

Bridge radius $r$ versus $\tau$ for glycerol-water-NaCl mixtures of various viscosities. At each viscosity, measurements from 5 or more runs are logarithmically binned and averaged. (a) At two viscosities, data for $r(\tau )$ are compared with Eq. (2), with logarithmic corrections given by Eq. (3) (dashed lines) and with $C_0=1$ (solid lines). The data fit better to $C_0=1$. (b) Inset: $r(\tau )$ for viscosities ranging from 1.9 to $230\mathrm{mPa}\mathrm{s}$. Main: Data are collapsed by rescaling the horizontal and vertical axes by ${\tau }_c$ and $r_c$ for each viscosity. Limiting behavior is proportional to $\tau /{\tau }_c$ (dotted line) at early times and $\sqrt{\tau /{\tau }_c}$ (dashed line) at late times. The collapsed data are well described by Eq. (5) (solid line).Reuse & Permissions

Figure 3

(a),(b) Measured dimensionless scaling-law prefactors $C_0$ and $D_0$ versus viscosity. In (a), the dashed line is $C_0=1$. In (b), the dashed line is $D_0=1.62$, the value obtained from the simulation [2]. Error bars show the range in which ${\tau }_c$ and $r_c$ can be varied without affecting the quality of the collapse. (c) Viscous to inertial crossover times versus viscosity, obtained via the collapse in Fig. 2. The solid line, ${\tau }_c=0.3{\mu }^2$, is based on our proposed Reynolds number [i.e., Eq. (6) with $\rho =1200\mathrm{kg}/{\mathrm{m}}^3$, $\gamma =65\mathrm{mN}/\mathrm{m}$, and $D_0=1.62$] and fits the data well. The dashed line shows ${\tau }_c={\mu }^3/\rho {\gamma }^2=0.2{\mu }^3$ obtained from the Reynolds number proposed in the literature [1, 2]. Clearly, this is a poor fit to the data.Reuse & Permissions

Figure 4

Length and velocity scales for coalescence. (a) In the inertial regime, a fluid bridge of radius $r$ has height $r^2/A$. (b) As the neck expands radially, fluid fills in the neighboring region with a flow, which occurs on a scale set by half the gap size, $L\approx r^2/2A$, that is essentially axial in direction (vertical arrows). The speed of this flow, $U$, can be related to $dr/d\tau$. The volume sandwiched between the drops out to a radius $r$ (shaded region) is given by $V=\frac{\pi }{2A}{r}^4$. As the bridge expands, $dV/d\tau =\frac{2\pi }{A}{r}^{3}dr/d\tau$. This fluid is supplied through two annular regions of radius $r$ and width $r^2/A$, above and below the gap (dotted lines), of total area $\frac{4\pi }{A}{r}^3$. Setting the volume flow rate, $\frac{4\pi }{A}{r}^{3}U$, equal to $dV/d\tau$, we get $U\approx \frac{1}{2}dr/d\tau$.Reuse & Permissions