3.1 Coalescence and spreading dynamics

Figure 3.

(a–h) Side views of the sequence of coalescence and film spreading when an ethanol drop of radius $r_{d}=1.1~\text{mm}$ and 100 % concentration mixes with a water layer of $h_{l}=5~\text{cm}$ height. The point A shows the fraction of the drop being drawn apart by coalescence that forms the spreading film. The size of each image is $16.88~\text{mm}\times 8.4~\text{mm}$ . (i) Side-view visualisation of film spreading from a drop of $r_{d}=1.1~\text{mm}$ and $C_{e}=100\,\%$ with $55~\unicode[STIX]{x03BC}\text{m}$ polyamide particles in the substrate and $6~\unicode[STIX]{x03BC}\text{m}$ fluorescent particles in the drop. The dashed circles show the tip vortices at the periphery of the spreading film. The image shown is the negative of the obtained image for clarity and is of size $22.5~\text{mm}\times 9.08~\text{mm}$ . See supplementary movie 2.

Figure 4.

Variation of the film radius with time for different drop radii, ethanol concentrations and layer height. The hollow symbols represent $r_{d}=0.97~\text{mm}$ with varying ethanol concentrations in the drop: ○, $C_{e}=100\,\%$ ; ▫, $C_{e}=80\,\%$ ; ♢, $C_{e}=60\,\%$ ; ✩, $C_{e}=40\,\%$ ; $+$ , $C_{e}=20\,\%$ . The solid symbols represent experiments with 100 % concentration with varying $r_{d}$ : ▼, $r_{d}=0.9~\text{mm}$ ; ▲, $r_{d}=1.1~\text{mm}$ . The experiments with hollow and solid symbols had the water layer height $h_{l}=5~\text{mm}$ . An experiment with a larger layer height $h_{l}=75~\text{mm}$ , $C_{e}=100\,\%$ and $r_{d}=0.97~\text{mm}$ is shown by $\odot$ . The dashed line shows the $1/4$ slope of the data. The inset shows the $r_{f}$ versus $t$ data for $C_{e}=60\,\%$ and $r_{d}=0.97~\text{mm}$ along with the curve fit $r_{f}=11.89t^{0.232}$ , from which the velocity is calculated.

Figure 2 shows the top views of the sequence of spreading of an ethanol drop of $C_{e}=100\,\%$ and $r_{d}=0.97~\text{mm}$ . The local depression in surface tension caused at the point of contact of the drop with water results in radially outward Marangoni forces, which cause a thin film of ethanol–water mixture to spread. The expanding white circular region that is free of particles, seen in figure 2(a–c), shows the spreading film. We also notice the initiation and propagation of capillary waves ahead of the expanding film in figure 2(a–c). After some time (90 ms), the outer front of the film develops instabilities resulting in outward-moving fingers or plumes as seen in figure 2(d–f). Even though there is an inward flow in between these outward-growing fingers, the radius of the continuous film region, as well as the circular region covering the outer tips of the growing fingers, go on increasing in time, but with decreasing velocities (see supplementary movie 1). The length of the fingers increase along with the radius of the continuous film region, until the velocity of expansion of the continuous film reduces to zero, after which the continuous film region begins to retract inwards (figure 2 g–i) while the outer tips of the fingers remain approximately stationary in time. The total time of film expansion is approximately 0.4 s. Most of the drop remains at the centre of the expanding film during the whole time of film expansion, seen as the dark circular region at the centre of the expanding film in figure 2(a–f). This remnant part of the drop also expands, but at a much slower rate than the film, thins and then develops holes in it. The remnant drop eventually gets mixed with water by a combination of convection and diffusion, after the continuous film has started retracting, as seen in figure 2(g–i); the total time from contact to dissolution being approximately 0.6 s.

The side views of the film spreading process for a similar concentration drop with $r_{d}=1.1~\text{mm}$ , shown in figure 3, clarifies the critical role of the initial coalescence between the drop and the water layer in film formation. Regions marked as A in figure 3(a–f) show that a small fraction of the drop, from its bottom, gets pulled apart by the neck expansion during coalescence and becomes the source of the drop fluid in the film (see supplementary movie 2). As figure 3(a–f) shows, the major part of the drops goes down and then bounces back due to its buoyancy, while the initial bottom part of the drop gets pulled apart to form the spreading film. It could also be noticed that there is no substantial motion in the regions below the spreading film, except for a vertical redistribution of the particles at the interface seen as the dark region below the film in figure 3(g,h). Such a distribution of particles at the interface occurs because of the vertical momentum imparted to the interface as a result of the vertical oscillations of the drop due to the ligament retraction dynamics during pinch-off from the capillary. We now look at the quantitative variation of the radius of the expanding continuous circular film ( $r_{f}$ ) as a function of time ( $t$ ), measured as discussed in § 2, from images similar to that in figure 2.

3.2 Spreading radius and velocity

Figure 5.

Variation of the film expansion velocity with time for different drop radii, ethanol concentrations and layer height. The symbols are the same as in figure 4.

Figure 4 shows the measured values of $r_{f}(t)$ as a function of time $t$ for different $C_{e}$ (hollow symbols), with different $r_{d}$ (solid symbols) and with different $h_{l}$ (hollow circle with dot), the zero time being at the instant of contact of the drop with water. The horizontal error bars show the error in time measurement due to the uncertainty of the time of contact, discussed in appendix B. As discussed in appendix B, $r_{f}$ is the mean of three azimuthal measurements; the range of these three measured radii is shown in figure 4 for $C_{e}=100\,\%$ and $r_{d}=0.97~\text{mm}$ as the vertical bars. This range increases in the later measurements since the film develops azimuthal instabilities after expanding for some time to form fingers in its outer periphery, as we saw in figure 2(d). After this instability occurs, $r_{f}(t)$ is measured as the radius of the continuous film, not including the region with fingers. The plot includes measurements after the instability at the edge of the film occurs; no change in the rate of expansion of $r_{f}$ is seen after the instability occurs.

The radius $r_{f}$ increases as the continuous film region expands with time (figure 2 a–g) and then starts to decrease with time once the film front starts to retract (figure 2 g–i). The inset in figure 4 shows the curve fit that is used to calculate the film expansion velocity ( $u_{f}$ ) for one experiment, as discussed in § 2. The film expansion velocities ( $u_{f}$ ) decrease with time, as shown in figure 5. The initial ( $t\approx 0.03$  s) expansion velocities of the film are high ( ${>}10~\text{cm}~\text{s}^{-1}$ ), which reduces to zero, beyond which the velocities reverse their direction due to film retraction. Retraction occurs over a shorter time than expansion; for example, for $r_{d}=0.97~\text{mm}$ and $C_{e}=100\,\%$ , shown with ○ in figure 4, film expansion occurs over 0.4 s while retraction occurs over 0.2 s.

Figure 4 shows that, at any time, a larger $C_{e}$ in the drop results in a larger radius of the film. The slopes of $r_{f}$ versus $t$ for different ethanol concentrations are approximately the same, indicating the same power-law dependence of $r_{f}$ on $t$ for different ethanol concentrations; this power law is approximately $t^{1/4}$ as shown by the dashed line in figure 4. For the same $r_{d}$ , the higher the concentration of ethanol in the drop, the larger is the maximum $r_{f}$ and the larger is the time of spreading. It could also be noticed that, at any specific time, $r_{f}$ has a non-monotonic dependence on the initial radius of the drop $r_{d}$ . The hollow circles and the hollow circles with a centre dot in figure 4 are identical experiments except that the water layer height ( $h_{l}$ ) was 15 times larger in the latter. These two datasets fall on each other, implying that the radius $r_{f}$ at any time, the exponent of the power law of $r_{f}$ versus $t$ , as well as the velocity of spreading are all independent of the height of the underlying liquid layer; this independence of $h_{l}$ paves the way to our scaling analysis in § 4. We now proceed to find scaling relations for these dependences of $r_{f}$ on time, $r_{d}$ and the fluid properties. The scaling analysis presented in this paper is limited to the expanding radius of the spreading film, and does not include the retraction of the film.

4.1 Mass and momentum balance

Figure 6.

Schematic of film spreading from a lighter miscible drop on a deep liquid layer showing the symbols used.

We assume the spreading to be as shown in figure 6. As we saw in § 3.2, the film spreading is rapid, with the spreading being over in 0.4 s for the 100 % ethanol drop. Further, as could be seen from figure 2, only a small fraction of the drop spreads as the film during the time of film spreading. Owing to the small characteristic time of spreading ( $t_{\unicode[STIX]{x1D707}f}\sim r_{f}/u_{f}$ ), we assume that the film is well mixed during its spreading so that the density of the film remains a constant over the short time of spreading. This assumption implies that the loss of ethanol from the film during spreading due to evaporation and mixing with the underlying water layer is small. As shown in appendix C, the evaporation velocities are of the order of $10^{-5}~\text{m}~\text{s}^{-1}$ while $u_{f}\sim 5\times 10^{-2}~\text{m}~\text{s}^{-1}$ ; evaporative flux in $t_{\unicode[STIX]{x1D707}f}$ will be negligible compared to the longitudinal flux in the film. Since convective mixing is absent (see figure 3), the downward mixing velocities, of the order of diffusive velocities $D/h_{f}\sim 10^{-4}~\text{m}~\text{s}^{-1}$ , are also negligible in time $t_{\unicode[STIX]{x1D707}f}$ . When fraction $G$ of the volume of the drop spreads as a film of constant density $\unicode[STIX]{x1D70C}_{f}$ , with a time-dependent film height $h_{f}(t)$ , mass balance of the drop and the film implies that

(4.2) $$\begin{eqnarray}G{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}r_{d}^{3}\unicode[STIX]{x1D70C}_{d}=\unicode[STIX]{x03C0}r_{f}^{2}h_{f}\unicode[STIX]{x1D70C}_{f}.\end{eqnarray}$$

The change in the radius of the drop fluid reservoir, seen as the dark centre region in figure 2(a–f), is small while the film spreads. Further, the value of $t_{\unicode[STIX]{x1D707}f}$ is small. For these reasons, we assume the change in volume of the drop during $t_{\unicode[STIX]{x1D707}f}$ to be negligible; the change in density of the drop is also then negligible in $t_{\unicode[STIX]{x1D707}f}$  ( $\text{d}\unicode[STIX]{x1D70C}_{d}/\text{d}t\approx 0$ when $t<t_{\unicode[STIX]{x1D707}f}$ ). Since the mass and momentum changes of the drop reservoir over $t_{\unicode[STIX]{x1D707}f}$ are then negligible, as shown in appendix D, the mass balance of the drop and the film reduces to

(4.3) $$\begin{eqnarray}2\unicode[STIX]{x03C0}r_{f}u_{f}h_{f}\unicode[STIX]{x1D70C}_{f}+\unicode[STIX]{x03C0}r_{f}^{2}\unicode[STIX]{x1D70C}_{f}\frac{\text{d}h_{f}}{\text{d}t}=0.\end{eqnarray}$$

Similarly, as shown in appendix D, using (4.3) in the momentum balance for the drop and the film and neglecting terms involving change in mass and momentum of the drop reservoir implies that

(4.4) $$\begin{eqnarray}2\unicode[STIX]{x03C0}r_{f}(\unicode[STIX]{x1D70E}_{l}-\unicode[STIX]{x1D70E}_{f})-\unicode[STIX]{x1D707}_{f}\frac{u_{f}}{h_{f}}\unicode[STIX]{x03C0}r_{f}^{2}=\unicode[STIX]{x03C0}r_{f}^{2}h_{f}\unicode[STIX]{x1D70C}_{f}\frac{\text{d}u_{f}}{\text{d}t}.\end{eqnarray}$$

4.2 Scaling of spreading radius and velocity

Figure 7.

Variation of the dimensionless numbers of the film with time for an experiment with $C_{e}=100\,\%$ and $r_{d}=0.97~\text{mm}$ : ○, $Re_{f}$ ; ▫, $We_{f}$ ; ♢, $Oh_{f}$ ; $+$ , $Ca_{f}$ .

We now estimate the range of dimensionless numbers of the film during its spreading so as to simplify (4.4). We assume that the spreading occurs fast enough to neglect the mixing of the film with the underlying water layer, so that the film properties are the same as the drop properties, implying $\unicode[STIX]{x1D70C}_{f}=\unicode[STIX]{x1D70C}_{d}$ , $\unicode[STIX]{x1D70E}_{f}=\unicode[STIX]{x1D70E}_{d}$ and $\unicode[STIX]{x1D707}_{f}=\unicode[STIX]{x1D707}_{d}$ . Figure 7 shows the variation of the dimensionless numbers of the film as a function of time in a typical experiment of 100 % ethanol drop of $r_{d}=0.97~\text{mm}$ . The Weber numbers based on film thickness ( $We_{f}=\unicode[STIX]{x1D70C}_{f}u_{f}^{2}h_{f}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ ) are much less than one ( $0<We_{f}<0.005$ ), implying that the surface tension effects are much more than the inertia effects in film spreading. Similarly, since the Reynolds numbers based on film thickness ( $Re_{f}=\unicode[STIX]{x1D70C}_{f}u_{f}h_{f}/\unicode[STIX]{x1D707}_{f}$ ) are in the range $0.06<Re_{f}<1.4$ , viscous effects are greater than inertia effects, except at the beginning of the film spreading when $Re_{f}$ is of order one. The capillary numbers ( $Ca_{f}=\unicode[STIX]{x1D707}_{f}u_{f}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ ) are very small ( $0.0006<Ca_{f}<0.0035$ ); surface tension forces dominate over viscous resistance. The values of Ohnesorge numbers ( $Oh_{f}=\unicode[STIX]{x1D707}_{f}/\sqrt{\unicode[STIX]{x1D70C}_{f}h_{f}\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}}$ ) are in the range of $0.049<Oh_{f}<0.099$ ; surface tension is more predominant than viscous effects.

Based on the order of these dimensionless numbers, the momentum equation (4.4) can be simplified as follows. Since $\text{d}u_{f}/\text{d}t\sim u_{f}/t_{\unicode[STIX]{x1D707}f}$ with $t_{\unicode[STIX]{x1D707}f}\sim r_{f}/u_{f}$ , the ratio of the term on the right-hand side of (4.4) to the first term on the left-hand side is $We_{f}$ ; as seen in figure 7, $We_{f}\ll 1$ . Similarly, the ratio of the term on the right-hand side of (4.4) to the second term on the left-hand side of (4.4) is $Re_{f}(h_{f}/r_{f})$ . Since $Re_{f}<1$ , as shown in figure 7, and $h_{f}/r_{f}\ll 1$ , since the film thickness is much smaller than its radius, we have $Re_{f}(h_{f}/r_{f})\ll 1$ . At the same time, the ratio of the terms on the left-hand side of (4.4) is $h_{f}/Ca_{f}r_{f}$ , which is of order one, since $Ca_{f}\ll 1$ (figure 7) and $h_{f}/r_{f}\ll 1$ . Hence, to leading order, the term on the right-hand side of (4.4) can be neglected. Further, assuming the film properties are the same as the drop properties, the momentum balance reduces to

(4.5) $$\begin{eqnarray}2\unicode[STIX]{x03C0}r_{f}\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D707}_{d}\frac{u_{f}}{h_{f}}\unicode[STIX]{x03C0}r_{f}^{2}.\end{eqnarray}$$

Eliminating $h_{f}$ from (4.2) and (4.5) gives

(4.6) $$\begin{eqnarray}r_{f}^{3}\dot{r_{f}}=\frac{8}{3}Gr_{d}^{3}\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D707}_{d}}.\end{eqnarray}$$

On integrating (4.6), we get

(4.7) $$\begin{eqnarray}r_{f}^{\ast }=c_{1}(Gt_{\unicode[STIX]{x1D707}d}^{\ast })^{1/4}\end{eqnarray}$$

as the scaling of the dimensionless film radius $r_{f}^{\ast }=r_{f}/r_{d}$ on the dimensionless time $t_{\unicode[STIX]{x1D707}d}^{\ast }=t/t_{\unicode[STIX]{x1D707}d}$ , with the characteristic time of spreading,

(4.8) $$\begin{eqnarray}t_{\unicode[STIX]{x1D707}d}=\frac{\unicode[STIX]{x1D707}_{d}r_{d}}{\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}},\end{eqnarray}$$

and the constant of integration $c_{1}=(32/3)^{1/4}$ . Since, $\dot{r_{f}}=u_{f}$ , the radial velocity of film spreading, equation (4.6) can also be rewritten in dimensionless form as

(4.9) $$\begin{eqnarray}Ca_{d}=c_{2}G{r_{f}^{\ast }}^{-3},\end{eqnarray}$$

where the capillary number $Ca_{d}=u_{f}\unicode[STIX]{x1D707}_{d}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ is the dimensionless spreading velocity and $c_{2}=8/3$ . In the scaling laws (4.7) and (4.9), $G$ is the initial fraction of the drop that mixes to form the film, which is still unknown. We now obtain an expression for $G$ based on the coalescence dynamics of the drop and the liquid layer.

4.3 Estimation of the initial spreading fraction G

Figure 8.

Schematic of the initial stages of coalescence of a drop with a liquid layer.

Figure 8 shows the schematic of the initial stage of coalescence of the drop with the liquid layer, which is a schematic of the zoomed view close to the interface in figure 3(b). The fraction of the volume of the drop $G$ that is pulled apart by the neck expansion in the coalescence time $t_{co}$ is

(4.10) $$\begin{eqnarray}G=\frac{2\unicode[STIX]{x03C0}r_{n}\unicode[STIX]{x1D6FF}u_{r}t_{co}}{{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}{r_{d}}^{3}},\end{eqnarray}$$

where, as shown in figure 8, $\unicode[STIX]{x1D6FF}$ is the thickness of the neck region, $r_{n}$ is the neck radius and $u_{r}$ is the neck retraction velocity.

The coalescence time $t_{co}$ in (4.10) depends on the Bond number of the drop, $Bo_{d}=\unicode[STIX]{x1D70C}_{d}gr_{d}^{2}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ . For low $Bo_{d}$ , the coalescence is purely capillary-driven with $t_{co}$ equal to the capillary time scale $t_{c}=\sqrt{\unicode[STIX]{x1D70C}_{d}r_{d}^{3}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}}$ , while for large $Bo_{d}$ , $t_{co}$ is equal to the gravity time scale $t_{g}=\sqrt{r_{d}/g}$ (Chen, Mandre & Feng Reference Chen, Mandre and Feng2006). As shown in table 1, the range of values of $Bo_{d}$ for the present experiments is $0.12<Bo_{d}<0.26$ ; the present experiments fall in an intermediate $Bo_{d}$ range. Chen et al. (Reference Chen, Mandre and Feng2006) have shown that, for such a capillary–gravity regime of drop coalescence,

(4.11) $$\begin{eqnarray}t_{co}=0.77\sqrt{\frac{\unicode[STIX]{x1D70C}_{d}r_{d}^{3}}{\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}(1+Bo_{d})}},\end{eqnarray}$$

which tends to $t_{c}$ as $Bo_{d}\rightarrow 0$ and to $t_{g}$ as $Bo_{d}$ becomes large.

The thickness of the neck $\unicode[STIX]{x1D6FF}$ in (4.10) can be estimated in the following way. From the geometry of figure 8,

(4.12) $$\begin{eqnarray}w=r_{d}(1-\cos \unicode[STIX]{x1D703}),\end{eqnarray}$$

which, on substitution of $\cos \unicode[STIX]{x1D703}\approx 1-\unicode[STIX]{x1D703}^{2}/2$ for small $\unicode[STIX]{x1D703}$ , becomes equal to

(4.13) $$\begin{eqnarray}w=\frac{r_{d}\unicode[STIX]{x1D703}^{2}}{2}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D703}\approx r_{n}/r_{d}$ for small $\unicode[STIX]{x1D703}$ from figure 8, (4.13) becomes

(4.14) $$\begin{eqnarray}w=\frac{r_{n}^{2}}{2r_{d}}.\end{eqnarray}$$

Mass balance of the retracting rim resulting in a bulge of diameter $\unicode[STIX]{x1D6FF}$ at the tip of the rim would imply

(4.15) $$\begin{eqnarray}\int _{0}^{r_{n}}2\unicode[STIX]{x03C0}r_{n}w\,\text{d}r_{n}=\unicode[STIX]{x03C0}\left(\frac{\unicode[STIX]{x1D6FF}}{2}\right)^{2}2\unicode[STIX]{x03C0}r_{n}.\end{eqnarray}$$

Evaluating the integral in (4.15), after substituting for $w$ from (4.14), and simplifying, we get

(4.16) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}=\sqrt{\frac{r_{n}^{3}}{2\unicode[STIX]{x03C0}r_{d}}}.\end{eqnarray}$$

From scaling arguments, Eggers, Lister & Stone (Reference Eggers, Lister and Stone1999) obtained the same relation, without the prefactor, for drop coalescence.

The neck retraction velocity $u_{r}$ in (4.10) is a resultant of the balance of inertia $\unicode[STIX]{x1D70C}_{d}u_{r}^{2}/2$ and surface tension force $\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}/r_{n}$ to give

(4.17) $$\begin{eqnarray}u_{r}=c_{3}\sqrt{\frac{2\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}_{d}r_{n}}},\end{eqnarray}$$

where $c_{3}$ is a constant prefactor, whose value is chosen later to obtain the limiting value of $G$ as $Bo_{d}\rightarrow 0$ . Substituting (4.11), (4.16) and (4.17) into (4.10) and noting that $r_{n}/r_{d}\approx \unicode[STIX]{x1D703}$ for small $\unicode[STIX]{x1D703}$ in figure 8, we get

(4.18) $$\begin{eqnarray}G\approx c_{4}\sqrt{\frac{\unicode[STIX]{x1D703}^{4}}{1+Bo_{d}}},\end{eqnarray}$$

where $c_{4}=0.65c_{3}$ . To eliminate $\unicode[STIX]{x1D703}$ from (4.18), we need one more relation of $G$ in terms of $\unicode[STIX]{x1D703}$ , which can be obtained as follows. From the geometry of figure 8,

(4.19) $$\begin{eqnarray}G\approx {\textstyle \frac{1}{2}}\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}r_{n}^{2}/{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}r_{d}^{3}.\end{eqnarray}$$

Replacing $\unicode[STIX]{x1D6FF}$ in (4.19) from (4.16) and noting that $r_{n}/r_{d}\approx \unicode[STIX]{x1D703}$ for small $\unicode[STIX]{x1D703}$ , we get

(4.20) $$\begin{eqnarray}G\approx {\textstyle \frac{3}{20}}\unicode[STIX]{x1D703}^{7/2}.\end{eqnarray}$$

Replacing $\unicode[STIX]{x1D703}$ in (4.18) in terms of $G$ from (4.20) results in

(4.21) $$\begin{eqnarray}G\approx \frac{1}{2(1+Bo_{d})^{7/6}},\end{eqnarray}$$

where we have chosen the prefactor $c_{3}=0.39$ so as to get $G=0.5$ when $Bo_{d}\rightarrow 0$ , as found by Chen et al. (Reference Chen, Mandre and Feng2006).

4.4 $Bo_{d}$ dependence and validation

The scaling of the dimensionless film radius (4.7) and the dimensionless film velocity (4.9) can now be expressed in terms of the drop Bond number ( $Bo_{d}$ ) by writing $G$ in these equations in terms of $Bo_{d}$ using (4.21). By doing such a substitution, we obtain

(4.22) $$\begin{eqnarray}fr_{f}^{\ast }={t_{\unicode[STIX]{x1D707}d}^{\ast }}^{1/4}\end{eqnarray}$$

and

(4.23) $$\begin{eqnarray}4f^{4}Ca_{d}={r_{f}^{\ast }}^{-3},\end{eqnarray}$$

where the function

(4.24) $$\begin{eqnarray}f(Bo_{d})=\frac{3^{1/4}}{2}(1+Bo_{d})^{7/24}.\end{eqnarray}$$

Since $r_{f}^{\ast }$ in (4.23) is given by (4.22), (4.23) can also be rewritten in terms of $t_{\unicode[STIX]{x1D707}d}^{\ast }$ , so as to obtain a decreasing dependence of dimensionless film velocity on time in the form $4fCa_{d}={t_{\unicode[STIX]{x1D707}d}^{\ast }}^{-3/4}$ . Equations (4.22) and (4.23) are the proposed scalings that are expected to capture the dependence of spreading radii and velocities on time, property ratios between the drop and the liquid layer, and the initial drop radius; we now verify these relations with our experimental data shown in figure 4.

Figure 9.

Variation of the dimensionless film radius with the dimensionless time. The symbols are as in figure 4.

Figure 10.

Variation of the dimensionless film velocity with dimensionless film radius. The symbols are as in figure 4.

Figure 9 shows the variation of the dimensionless spreading radius $r_{f}^{\ast }$ , scaled by the function $f$ , with the dimensionless time $t_{\unicode[STIX]{x1D707}d}^{\ast }$ , plotted using the data shown in figure 4 for different $C_{e}$ , $r_{d}$ and $h_{l}$ . The part of the data that shows an increasing film radius collapses fairly well onto

(4.25) $$\begin{eqnarray}fr_{f}^{\ast }=0.83{t_{\unicode[STIX]{x1D707}d}^{\ast }}^{1/4},\end{eqnarray}$$

shown by the dashed line in figure 9, the prefactor being close to the expected value of 1 from (4.22). The variation of the dimensionless film expansion velocity $Ca_{d}$ , scaled by $4f^{4}$ , with the dimensionless spreading radius $r_{f}^{\ast }$ is shown in figure 10. Similar to that in figure 9, the data collapse fairly well onto

(4.26) $$\begin{eqnarray}4f^{4}Ca_{d}=0.5{r_{f}^{\ast }}^{-3}.\end{eqnarray}$$

The data show a $-3$ power-law exponent except for a slight deviation at small $r_{f}^{\ast }$ . This deviation is expected to be due to the non-negligible inertial effects in film spreading for small $r_{f}$ when the film spreads very fast (see figure 5) so that $We_{f}$ becomes close to one, as seen in figure 7. The present analysis neglected the inertial effects in film spreading and is only valid when $We_{f}<1$ . Further, the present analysis assumed that $r_{d}\ll r_{f}$ ; this assumption also breaks down at small $r_{f}$ when the film radius is of the same size as the drop radius. The reason for the prefactors in (4.25) and (4.26) to be slightly less than 1 could be the slight inaccuracy of the prefactors assumed in the scaling relationships used in the derivation of (4.22) and (4.23).