Initial regime of drop coalescence

When two drops are slowly brought together and first touch, a microscopic liquid neck or a bridge forms between them. The expansion of the neck is controlled by the capillary (Laplace) pressure which diverges when the curvature of the interface is infinite at the point where the drops first touch. The change in topology and the flows that ensue as time advances and the bridge grows from microscopic to macroscopic scales, and the two drops merge into one, are intimately coupled to this singularity in the dynamics. Despite the large volume of work dedicated to this problem, currently experiment, theory, and computation are not in complete agreement with respect to the earliest times following the initial contact of the two drops. Experiments, supported by simulations, report an initial regime where the radius of the connecting bridge grows linearly in time before a transition to either a Stokes regime or an inertial regime where either viscous or inertial force balances capillary (surface tension) force. In the initial linear regime, referred to as the inertially limited viscous (ILV) regime, all three forces are thought to be important. This is in contrast to theory which predicts that all coalescence events begin in the Stokes regime where inertia is negligible. Here we use high-accuracy numerical simulations to show that the ILV regime is only realized when the two coalescing drops are initially separated by a finite distance. Moreover, for two drops that initially just touch at a point, coalescence always begins in the Stokes regime. It is demonstrated that the linear ILV regime is more akin to a Taylor-Culick-type regime whose existence and duration are purely consequences of the use of an initial bridge of finite size that poorly approximates the point contact condition that is a cardinal feature of the coalescence singularity.

Figure 1

The onset of the coalescence of two equal-sized drops in a dynamically passive exterior fluid, e.g., a gas such as air: definition sketch and initial conditions. At the initial instant $\tilde{t}=0$, two spherical drops, each of radius $\tilde{R}$, are connected by a microscopic neck or bridge of radius ${\tilde{R}}_0$ and half height ${\tilde{Z}}_0$, a zoomed-in view of which is also shown (zoomed-in view not to scale). The fluid within the drops is quiescent at $\tilde{t}=0$.

Figure 2

Portion of an example mesh during the coalescence of two drops of $\text{Oh}=0.001$ at the instant in time when $R_{\min }\approx 0.18$. The mesh is heavily concentrated both radially and axially near the neck compared to the bulk of the drop(s). The dividing surface or boundary, marked in red, moves according to an algebraic condition so that its position changes dynamically in time in response to the evolution of the bridge radius $R_{\min }\left(t\right)$, as discussed in the text. Actual simulations typically use a mesh that is two to four times more dense than the example shown here.

Figure 3

Scaling behavior of three quantities obtained from simulations. (a) The variation of the minimum neck radius $R_{\min }$ with the viscous coalescence time ${\tau }_v$ (main figure) and the variation of the magnitude of twice the mean curvature $2\mathcal{H}$ with $R_{\min }$ (inset), both of which are shown as log-log plots. (b) The variation of the velocity of the growing neck or the bridge, which has been rescaled by the visco-capillary velocity, with $R_{\min }$ and which is shown as a semilog plot. In (a)–(b), it is demonstrated that the scaling laws inferred from the simulations closely follow the predictions obtained from Stokes theory for $R_{\min }, 2\mathcal{H}$, and $u_v$ rather than those that would be expected in a linear ILV regime. Here $\text{Oh}=0.6, R_0={10}^{-6}$, and $Z_0=(1/2)R_0^2=5\times {10}^{-13}$.

Figure 4

Variation of the appropriately scaled profiles of the bridge or neck connecting two coalescing drops as a function of the minimum radius $R_{\min }$ and hence time. The profiles are seen to collapse nicely onto a single similarity profile as $R_{\min }\to 0$. The inset shows a blow-up of the interface profiles shown in the main figure by focusing in on the region in the vicinity of the location where the bridge radius is a minimum. Here $\text{Oh}=0.6, R_0={10}^{-6}$, and $Z_0=(1/2)R_0^2=5\times {10}^{-13}$.

Figure 5

The effect of the initial half bridge height $Z_0$ on the variation with time of the instantaneous value of the minimum radius $R_{\min }$ (main figure) and the variation of the instantaneous value of the half bridge height $Z_b$ with $R_{\min }$ (inset). Here all simulations begin with an initial bridge of radius $R_0={10}^{-3}$ but the initial bridge height in the different simulations is systematically increased from the case where two drops virtually touch at a point, viz. $Z_0=(1/2)R_0^2$, to values where $Z_0$ approaches $0.1R_0$. Lines/curves represent simulation results when $\text{Oh}=0.6$, whereas symbols denote the results of a Stokes flow simulation in which $Z_0=0.1R_0={10}^{-4}$. A line of slope one in the main plot shows that a linear regime in radial scaling is followed until the axial scale $Z_b$ begins to vary as $R_{\min }^2$ as shown in the inset.

Figure 6

Scaling of the curvature in the Stokes and the ILV regimes. The variation of the magnitude of twice the mean curvature $2\mathcal{H}$ with $R_{\min }$ determined from simulations. In both cases, $\text{Oh}=0.6$ and $R_0={10}^{-3}$. In one of the cases (red curve), the simulations have been carried out with an initial condition of an ideal bridge or a virtually point contact initial condition where $Z_0=(1/2)R_0^2=5\times {10}^{-7}$. In the other case (blue curve), a finite-size bridge is used as an initial condition where $Z_0=0.1R_0={10}^{-4}$. In the first case, the dynamics follows that which is expected in the Stokes regime (black line of slope of $-3$.) In the second case, the dynamics follows ILV scaling at early times where the curvature is nearly constant (dashed black line) for some time. In the latter case, the dynamics ultimately transitions to the Stokes regime at late times (and/or for large values of $R_{\min }$) where $\left|2H\right|$ once again scales as $R_{\min }^{-3}$.

Figure 7

Scaling behavior or the variation with the viscous coalescence time ${\tau }_v$ of the minimum neck radius $R_{\min }$ on a log-log plot for a range of values of $\text{Oh}$ indicated by the various line types in the legend of the main figure. The simulation results show that that in all cases (except when $\text{Oh}={10}^{-3}$ where the dynamics is already in the inviscid regime), coalescence begins in the Stokes regime and the variation of $R_{\min }$ with ${\tau }_v$ closely follows that observed in the Stokes flow simulation (indicated by the square data points) until the crossover radius $R_c$ is reached. Thereafter, the dynamics transitions to the inviscid regime where $R_{\min }$ exhibits a power law scaling in ${\tau }_v$ with a slope of $1/2$. The data points that are circles correspond to a straight line of slope $1/2$ on the log-log plot as indicated in the legend. The results for $\text{Oh}=440$, 20, and 3 are virtually indistinguishable from the results of the Stokes flow simulation, i.e., the curves representing these large but finite $\text{Oh}$ cases lie on top of the data points corresponding to the Stokes flow case, over the entire time range shown. A relation for $R_c$ is shown in the inset with both the linear dependence of Ref. [28] $(U=\gamma /\mu )$ and the logarithmic corrections suggested here that are derived from theory $[U=-(\gamma /\mu )ln\left(R_{\min }\right)]$ and simulation data $[U=U(\mathrm{sim}\left)\right]$. Here all simulations start with an initial bridge of $R_0={10}^{-3}$ and $Z_0=(1/2)R_0^2=5\times {10}^{-7}$.

Figure 8

Instantaneous flow fields and pressure contours within the drops at the instant in time when $R_{\min }=0.03$ for the situations in which (a) $\text{Oh}={10}^{-3}$, (b) $\text{Oh}=3\times {10}^{-2}$, and (c) $1/\text{Oh}=0$ or $\text{Oh}=\infty$ (Stokes flow). Although there are only two distinct scaling regimes governing the time evolution of the neck in the vicinity of the space-time singularity, the global dynamics can exhibit three qualitatively different types of flows. (d) The existence of three different types of global flow fields is further demonstrated by monitoring the instantaneous value of the axial velocity of the back of the drop(s), $V_{b.o.d.}$. Here simulation results for a range of $\text{Oh}$ (shown by different symbols) are compared to predictions from Stokes flow theory (shown by the black line). All simulations start with an initial bridge of $R_0={10}^{-3}$ and $Z_0=(1/2)R_0^2=5\times {10}^{-7}$.

Figure 9

Variation with $\text{Oh}$ of the appropriately scaled profiles of a growing bridge connecting two coalescing drops as $\text{Oh}\to \infty$ at the instant in time when $R_{\min }=8\times {10}^{-3}$. As $\text{Oh}\to \infty$, the profiles are seen to collapse or tend nicely onto a single similarity profile that results in the limit of Stokes flow. When $\text{Oh}\ll 1$, the scaled interface shape takes on a starkly different profile with a characteristic bulge indicative of the inviscid regime. The inset shows a zoomed-in view of the neck or the tip of the receding air film. Here $R_0={10}^{-3}$ and $Z_0=(1/2)R_0^2=5\times {10}^{-7}$ in all simulations.

Figure 10

Profiles of a growing bridge between two coalescing drops for the same set of $\text{Oh}$ and at the same instant in time as in Fig. 9. (a) Interface profiles are shown after the radial coordinate has been shifted as explained in the text. Here both the shifted radial coordinate and the axial coordinate are then normalized by $R_{\min }^2$ to highlight the blob in the situation in which the Ohnesorge number is smallest ($\text{Oh}=0.001$) and that the blob is of nearly equal $O\left(1\right)$ extent in both the radial and the axial directions. The inset shows a zoomed-in view of the neck or the tip of the receding air film. (b) The interface profile when $\text{Oh}=0.001$ that further highlights the blob that forms in the inviscid limit.

Figure 11

Phase diagram in the space of Ohnesorge number and crossover radius, $(\text{Oh},R_c)$, of drop coalescence in a dynamically passive fluid, e.g., a gas such as air, showing the regimes of coalescence and the conditions for transitions between them. Following the occurrence of the space-time singularity when two drops have just touched at a point, coalescence will always begin in the Stokes regime and the dynamics will either transition to the inviscid regime when the minimum radius of the neck connecting the drops exceeds the crossover radius or terminate with the two drops having fully coalesced.