Electrohydrodynamic coalescence of droplets using an embedded potential flow model

The coalescence, and subsequent satellite formation, of two inviscid droplets is studied numerically. The initial drops are taken to be of equal and different sizes, and simulations have been carried out with and without the presence of an electrical field. The main computational challenge is the tracking of a free surface that changes topology. Coupling level set and boundary integral methods with an embedded potential flow model, we seamlessly compute through these singular events. As a consequence, the various coalescence modes that appear depending upon the relative ratio of the parent droplets can be studied. Computations of first stage pinch-off, second stage pinch-off, and complete engulfment are analyzed and compared to recent numerical studies and laboratory experiments. Specifically, we study the evolution of bridge radii and the related scaling laws, the minimum drop radii evolution from coalescence to satellite pinch-off, satellite sizes, and the upward stretching of the near cylindrical protrusion at the droplet top. Clear evidence of partial coalescence self-similarity is presented for parent droplet ratios between 1.66 and 4. This has been possible due to the fact that computational initial conditions only depend upon the mother droplet size, in contrast with laboratory experiments where the difficulty in establishing the same initial physical configuration is well known. The presence of electric forces changes the coalescence patterns, and it is possible to control the satellite droplet size by tuning the electrical field intensity. All of the numerical results are in very good agreement with recent laboratory experiments for water droplet coalescence.

Figure 1

Three-dimensional sketch of the physical domain. The interior domain consisting of the two touching droplets is ${\mathrm{\Omega }}_1\left(t\right)$, ${\mathrm{\Omega }}_2\left(t\right)$ is the infinite exterior domain, and $\mathrm{\Gamma }\left(t\right)$ is the free surface that separates them. The outward normal vector to the free surface is $\mathbf{n}$, $\mathbf{R}(\mathbf{s},t)$ is the position vector of a fluid particle on the front, and $E_{\infty }$ is the far field electric field intensity.

Figure 2

Time stepping provides a regularization of blow-up variables at singular time $t=t_s$.

Figure 3

Initial stage evolution: reconnection event at $t=0.00155$.

Figure 4

Minimum neck radius evolution for the various grids.

Figure 5

Log-log plot of minimum neck radius evolution for two equal droplets coalescing. Base-e logarithms.

Figure 6

Comparison of the minimum neck radius evolution for two equal droplets coalescing: this work, Thoroddsen et al., and Paulsen et al. (The numbers from both authors have been read off from their figures, and therefore they cannot be considered as exact data.)

Figure 7

Left end drop axial coordinate evolution for two equal droplets coalescing.

Figure 8

Capillary wave amplitude. Focused front at $t=0.1$ and superimposed initial condition at $t=0$ (dashed). The dashed red line is the line $y=x$.

Figure 9

Fronts of two equal drops coalescing at indicated times. Taking $t_0=1720\mu \text{s}$ these dimensionless times correspond to $3.44,23.05,141.04,240.8,344,516,688,860,1032,1204,1376,1548\mu \text{s}$, respectively.

Figure 10

(a) Laboratory photographs taken from [13]. (b) Computational results for drop shape evolution during coalescence, at real times $0,100,200,400\mu \text{s}$. The top and bottom droplet radii are $R_b=0.573$ and $R_t=0.0.597$, and the scale bar is 1 mm long. The nondimensional computation times have been converted using the characteristic time scale $t_0=1720\mu \text{s}$.

Figure 11

Total volume and energy evolution; kinetic and surface energies evolution for two equal droplets coalescing. $t=2$ nondimensional elapsed time.

Figure 12

Log-log plot of bridge height $z_{\text{bridge}}\left(t\right)$ versus minimum neck radius for two equal droplets coalescing. The more persistent exponent is $\alpha =2$. Base-e logarithms.

Figure 13

Several snapshots and partial coalescence pinch-off pattern for $R_m=0.70$. The tail of the daughter droplet is long enough to generate a visible tertiary droplet.

Figure 14

Several snapshots and partial coalescence pinch-off pattern for $R_m=0.60$. Here the tertiary droplet is almost nonvisible.

Figure 15

Several snapshots and partial coalescence pinch-off pattern for $R_m=0.38$. The pinch-off of the daughter droplet occurs above the top of the father droplet.

Figure 16

Several snapshots and partial coalescence pinch-off pattern for $R_m=0.15$. The pinch-off of the daughter droplet occurs below the top of the father droplet.

Figure 17

Fronts of two drops, $R_1=1,R_2=0.1$, coalescing at indicated times.

Figure 18

(a) Three-dimensional jetting event and jet rupture into four tiny droplets. (b) Same event focused on the tiny droplets. $R_m=0.10$.

Figure 19

Maximum axial coordinate $z_{\text{max}}$ evolution for various mother droplet radius until pinch-off.

Figure 20

Minimum neck radius evolution for various mother droplet radius until pinch-off.

Figure 21

Three-dimensional snapshots at $t=0.1052,0.118,0.1224,0.1285$ of pinch-off profile and daughter droplet oscillations, $R_m=0.15$.

Figure 22

Three-dimensional focused snapshots at $t=0.1052,0.118,0.1224,0.1285$ of pinch-off profile and daughter droplet oscillations, $R_m=0.15$.

Figure 23

Complete sequence of two drops, $R_1=1,R_2=0.25$ coalescing at indicated times.

Figure 24

(a) Laboratory experiments for distilled water droplets, $R_f/R_m=2.72$, taken from [18]; coalescence starts on the second panel where subsequent images are shown at $0.27,0.67,0.93,1.2,1.8$, and $4.6\text{ms}$ later. The scale bar is $1\text{mm}$. (b) Computational result corresponds to $R_f/R_m=4$.

Figure 25

Minimum neck radius evolution when the bridge is expanding and scaling exponents for $R_m=0.25$. Base-e logarithms.

Figure 26

Minimum neck radius evolution when the bridge is expanding and scaling exponents for $R_m=0.38$. Base-e logaritms.

Figure 27

Minimum neck radius evolution when the bridge is expanding and scaling exponents for $R_m=0.50$. Base-e logarithms.

Figure 28

Minimum neck radius evolution before daughter droplet pinch-off and scaling exponents for $R_2=0.50$. Base-e logarithms.

Figure 29

Minimum neck radius evolution before daughter droplet pinch-off and scaling exponents for $R_2=0.70$. Base-e logarithms.

Figure 30

Normalized minimum neck radius, $r_{\text{min}}/R_m$, versus normalized time, $t/R_m^{3/2}$, for $R_m=0.10,0.25,0.38,0.50,0.60,0.70$. The collapse of all curves indicates the self-similar behavior of the minimum neck radius.

Figure 31

Normalized elevation with respect to $z=1$ $(z_{\text{max}}-1)/R_m$, versus normalized time, $t/R_m^{3/2}$, for $R_m=0.10,0.25,0.38,0.50,0.60,0.70$. The collapse of the curves for $0.25<R_m<0.60$ indicates the self-similar behavior of this variable in the partial coalescence pattern without tertiary droplet. The curve for $R_m=0.70$ departs above pointing out the tertiary droplet formation and $R_m=0.10$ departs below indicating total engulfment.

Figure 32

$M$ inviscid droplets interacting under electric forces.

Figure 33

Aspect ratio evolution for one drop of radius $r_o=0.25$ for various field intensities.

Figure 34

Aspect ratio evolution for two drops of radius $r_o=0.25$. The amplitude and oscillating period of the interacting drops are equal but distinct of these quantities if only the left drop were present.

Figure 35

Aspect ratio evolution for three drops of radius $r_o=0.25$. The amplitude and oscillating period of the central drop is distinct from left and right droplet oscillating parameters.

Figure 36

Electrical interaction of three droplets under $E_{\infty }=0.5$. Drop profiles at $t=0$, $t=0.275$, time at which the central droplet is at its maximum distortion, and $t=0.5$.

Figure 37

Minimum neck radius evolution for various electric field intensities until pinch-off, except for $E_{\infty }=0.3$ for which the drop will not pinch-off but burst.

Figure 38

Maximum axial coordinate evolution for various electric field intensities until pinch-off, except for $E_{\infty }=0.3$ for which the drop will not pinch-off but burst.

Figure 39

Superimposed droplet profiles at pinch-off for $R_m=0.38$ and various electric field intensities $E_{\infty }=0.,0.1,0.2$.

Figure 40

Snapshots of electrohydrodynamic partial coalescence for $R_m=0.38$ and $E_{\infty }=0.2$. At pinch-off time the electric field is turned off ($E_{\infty }=0$).

Figure 41

Snapshots of electrohydrodynamic partial coalescence for $R_m=0.38$ and $E_{\infty }=0.3$. The drop destabilizes and jet emission starts at $t=0.2367$.

Figure 42

Calculated ratios $r_{dm}=R_d/R_m$ versus electrical field intensity, $E_{\infty }$, and quadratic fit.