Drop impact on thin film: Mixing, thickness variations, and ejections

The impact of drops on a thin liquid film is a phenomenon encountered in industrial applications, but also of particular interest in nature. Examples include the growth of stalagmites in karstic caves, a case where the drop feeds the film with ions that will subsequently precipitate. The mixing upon impact, which is witnessed both in the film and in the ejections, remains poorly understood. In the case of stalagmites, this short-term mixing directly affects the ion distribution in the film between impacts. In this work we investigate the mixing and ejection processes occurring during the impact of a free-falling drop on a thin, horizontal film of miscible liquid. We perform laboratory experiments and record side and top views of high-speed movies of impacts in a range of parameters close to impacts observed on stalagmites in caves. We observe that several outcomes arise from these impacts depending on the initial film thickness and Weber number. We relate the geometry of the splashing crown growth to the four scenarios observed. Additionally, the postimpact mixing patterns and film thickness variations are analyzed through an original colorimetry-based technique. From there we infer the size of the stain and quantity of water left by the drop in the film, as well as the total volume ejected away during the impact.

Figure 1

Experimental methodology. (a) Experimental setup used to release droplets from a given height onto a thin underlying film and record the impacts from both top and side views. (i) Plastic tube in which the drops fall. (ii) High-speed color camera used to record impacts from the top. (iii) Falling droplet of radius $r_{\mathrm{d}}$ and impact velocity $u_{\mathrm{d}}$, both measured from the side view [see panel (b)]. (iv) Four lamps placed in the corners of the balance. (v) Stages used to displace the needle above the film, horizontally and vertically. The vertical stage is motorized and automated. (vi–viii) Needle and balance used to take pointwise manual thickness measurements [see panel (c)] of the liquid film spread on a horizontal hydrophilic tape. (ix) High-speed monochrome camera used to record impacts from the side. (b i) Side-view geometrical measurements of the crown: crown height $h$, top radius $r_{\mathrm{t}}$, cavity radius in the film $r_{\mathrm{c}}$ and inclination $\alpha$. The drop schematic of (a iii) has a size corresponding to the actual drop that produced the crown shown in this case. (b ii) Four examples of raw data graphs of $\alpha \left(t\right)$, $h\left(t\right)$, $r_{\mathrm{t}}\left(t\right)$, and $r_{\mathrm{c}}\left(t\right)$, respectively, during a period of twice the capillary time $t_{\mathrm{c}}$ defined in Sec. 2c ($15\mathrm{m}\mathrm{s}$), for $\delta =150\mu \mathrm{m}$ and $z=1\mathrm{m}$. In each graph, the green area shows the portion on which measurements are averaged as described in the text, and the average is represented by the red line. (c) Needle with glued aluminum sphere used for measuring the local film thickness, entering (i–ii) and leaving (iii–iv) a green film of thickness $\delta =170\mu \mathrm{m}$. (i) The needle and sphere right before the sphere touches water. (ii) The meniscus formed when the sphere touches the film. This event produces a decrease in the mass read by the balance since a part of the film weight is supported by the needle. The needle is moved further downward and the mass increases once the sphere touches the bottom surface of the film. The needle is then moved upward. (iii) The shape of the meniscus right before it separates from the sphere. (iv) The droplet left on the sphere 4 s afterwards. (v) A zoom on this droplet hanging on the bottom half of the aluminum sphere. Both appear green because of light reflections from the green film.

Figure 2

Example of an impact experiment (a) and measurements obtained with the colorimetry-based algorithm (b, c): (a) Top view picture of the spot left by the drop in the film about $1\mathrm{s}$ after the impact. The initial film thickness was $\delta =103\mu \mathrm{m}$ and the drop had a fall of $2\mathrm{m}$. The white dot in the center of the photograph corresponds to the impact point. (b) Red dye proportion left in the film after the impact, normalized by the proportion in the drop, $p_r/p_{r,\mathrm{d}}$. (c) Film thickness difference between after and before the drop impact on the film, ${\delta }^{\prime }-\delta$ (in $\mu \mathrm{m}$).

Figure 3

Colorimetry calculation (${·}_c$) of the thickness and red dye proportion of an arbitrary calibration film: (a) Initial photograph taken with the Phantom Miro M110 high-speed camera. (b) Red proportion $p_{r,\mathrm{c}}$ computed at each point in the frame of (a), divided by the actual proportion $p_{r,\mathrm{m}}=2.5\%$. (c) Thickness ${\delta }_{\mathrm{c}}$ computed at each point of panel (a). Green dye proportion was set in every point to $p_g=5\%$.

Figure 4

(a) Parity plot between spatially averaged film thickness $\overline{{\delta }_{\mathrm{c}}}$ computed from colorimetry measurements and corresponding film thickness manually measured and averaged, $\overline{{\delta }_{\mathrm{m}}}$, prior to the impact. Symbol color is greener (redder) for markers located closer to (further away from) the axes bisector. The left inset of the graph in panel (a) shows a histogram representing the number of experiments per range of relative error between manual and numerical measurements, $|\overline{{\delta }_{\mathrm{m}}}-\overline{{\delta }_{\mathrm{c}}}|/\overline{{\delta }_{\mathrm{m}}}$. The bins are $2\%$ wide. The second inset in panel (a) shows the coefficient of variation (c.v.) of ${\delta }_{\mathrm{c}}$, i.e., over different cells of the same film, as a function of $\overline{{\delta }_{\mathrm{m}}}$. The c.v. can be seen as a measure of the spatial heterogeneity of the film, and therefore as a characteristic error bar on $\overline{{\delta }_{\mathrm{m}}}$. The shaded area in this inset shows the interquartile range and the green horizontal line the median of the c.v. of ${\delta }_{\mathrm{c}}$. (b) Examples of colorimetry and manual measurements, sometimes leading to larger errors in the average film thickness estimation because of, e.g., the visibility of the puddle edge (see i), with correspondence in the graph from panel (a).

Figure 5

Phase diagram (${\delta }^{★}$, $\text{We}$) of the reported experiments and correspondence with the various scenarios discussed in Sec. 3. For each, the monochrome side view shows the typical shape of the crown during impact, while the color top view shows the postimpact mixing pattern. (a) Very thin film (${\delta }^{★}<1$), crown fragmentation and circular red spot: (i) Low $\text{We}$ (${\delta }^{★}=0.57$, $\text{We}=525$), (ii) High $\text{We}$ (${\delta }^{★}=0.61$, $\text{We}=2740$). (b) Intermediate regime (${\delta }^{★}\gtrsim 1$), crown retraction and randomlike mixing pattern: (i) High $\text{We}$ (${\delta }^{★}=1.31$, $\text{We}=2740$), (ii) Intermediate $\text{We}$ (${\delta }^{★}=1.31$, $\text{We}=1675$). (c) Thick film (${\delta }^{★}\gtrsim 1.75$), low $\text{We}$: postimpact central jet protrusion (${\delta }^{★}=1.81$, $\text{We}=525$). (d) Thick film (${\delta }^{★}\gtrsim 1.75$), high $\text{We}$: crown folding (${\delta }^{★}=4.37$, $\text{We}=2250$). Background color grading of the central graph indicates that scenario transitions are continuous. The shaded area in the bottom right of the central plot represents the range of values covered in Ersoy and Eslamian [39]. The dashed black line show the splashing criterion developed by Cossali et al. [33]: $\text{We}/{\text{Oh}}^{2/5}=2100+29{\left({\delta }^{★}\right)}^{1.44}$. The scale bars are $1\mathrm{cm}$.

Figure 6

Side (monochrome, upper pictures, $s$) and top (color, bottom pictures, $t$) views of the impact sequences described in Sec. 3. For panels (a–d), (i) shows the crown growth ($st$) and (iv) the convection-induced mixing pattern ($t$). (a) Very thin film (${\delta }^{★}<1$), intermediate $\text{We}$: (ii) crown break-up into ligaments ($st$), (iii) ligament fragmentation ($st$), (v) fingeringlike pattern in the film ($t$). (b) Transitional regime (${\delta }^{★}\approx 1$), high $\text{We}$: (ii) crown maximum extension ($st$), (iii) retraction ($st$). (c) Thick film (${\delta }^{★}>1$), low $\text{We}$: (ii) crown decline ($st)$, (iii) postimpact central jet formation ($st$). (d) Thick film (${\delta }^{★}>1$), high $\text{We}$: (ii–iii) crown folding ($st$), (v) crown capillary waves ($s$), (vi) fingering in the film ($t$). The scale bars are $1\mathrm{cm}$. Videos corresponding to the pictures shown are available in the Supplemental Material [58].

Figure 7

Crown geometry parameters, as described in Sec. 2 and in Fig. 1. (a) Average inclination $\alpha$ of the crown during the growth phase (solid symbols). The dotted line corresponds to $\alpha =90{}^{\circ }$. A comparison is made with data from Fedorchenko and Wang [35] for ${\delta }^{★}=0.4$ and ${\delta }^{★}=4$ (hollow symbols). (b) Ratio between the radii of the top rim and the cavity expanding in the film, $r_{\mathrm{t}}/r_{\mathrm{c}}$ (solid symbols). The hollow symbol shows the value obtained by Cossali et al. [28] for ${\delta }^{★}=25.4$ with $r_{\mathrm{t}}=5.3r_{\mathrm{d}}$ and $r_{\mathrm{c}}=6.2r_{\mathrm{d}}$. The dotted line shows the case where $r_{\mathrm{t}}=r_{\mathrm{c}}$. The insets respectively represent $r_{\mathrm{t}}$ and $r_{\mathrm{c}}$ normalized by the drop radius $r_{\mathrm{d}}$. (c) Maximum height $h$ reached by the top rim of the crown at various $\text{We}$, normalized by the drop radius $r_{\mathrm{d}}$ (solid symbols). The inset shows the measurement relative to ${\delta }^{★}=4.4$ from scenario C, compared to data from Cossali et al. [28] (hollow symbols), obtained for ${\delta }^{★}\in \{11.1,25.4,43\}$. (d) Maximum length $j$ of the jet emitted following the crown retraction at low $\text{We}$, normalized by the drop radius $r_{\mathrm{d}}$. A comparison is made with data from Fedorchenko and Wang [35] for ${\delta }^{★}=46.3$ at various $\text{We}$. All measurements are shown as a function of the nondimensional film thickness ${\delta }^{★}$. The legend from panel (d) is the same for all four graphs. Symbols correspond to various intervals of $\text{We}$: circles for $\text{We}<1000$, squares for $1000<\text{We}<2000$, and diamonds for $\text{We}>2000$. The color of the symbols refers to the scenarios described in Sec. 3.

Figure 8

Crown shape factor and retraction velocity components. (a) Examples of crown shapes shown in the graph of (b): (i) ${\delta }^{★}=0.56$, $\text{We}=1675$, shape factor of 1.39, (ii) ${\delta }^{★}=1.1$, $\text{We}=1675$, shape factor of 1.09, (iii) ${\delta }^{★}=4.3$, $\text{We}=525$, shape factor of $-2.24$ and average velocity ratio of 1.98, and (iv) ${\delta }^{★}=4.3$, $\text{We}=2740$, shape factor of 0.37 and average velocity ratio of 0.95. Two frames showing the main retraction direction of the crown are added in (iii) and (iv), with a correspondence in the graph from panel (c). The scale bars are 1 cm. (b) Crown shape factor $\psi$, and (c) ratio of crown retraction velocity components $\xi$, both computed as a function of the nondimensional film thickness ${\delta }^{★}$. The dashed line from panel (b) at $\psi =0$ corresponds to a change of sign when $\alpha =90{}^{\circ }$. The other dashed line at $\psi =1$ corresponds to the change from a crown convex profile to a straight profile. In panel (c) the two dashed lines are drawn at $\xi =1$ and $\xi =2$. The symbol colors correspond to the scenarios identified in Sec. 3. The legend is the same in both graphs.

Figure 9

Nondimensional equivalent radius $R_{\mathrm{eq}}$ of the red stain left by the drop in the film, defined in Eq. (2): (a) Measurements of $R_{\mathrm{eq}}$ as a function of the nondimensional initial film thickness ${\delta }^{★}$, divided in three bins of $\text{We}$. The gray and dotted bands represent the maximum spatial extension $R_{\text{dry}}/r_{\mathrm{d}}$ reached by a drop on a dry wall between $\text{We}=500$ and $\text{We}=3000$. These boundaries are estimated using the results from Laan et al. [43] (dots) and Gordillo et al. [45] (plain), respectively. The symbol colors correspond to the scenarios observed in Sec. 3. The roman lowercase letters show the location of the examples from panel (b) in the graph. (b) Examples of radii computed according to Eq. (2) and superimposed with the corresponding initial photographs, for the following ranges of parameters: (i) very thin film, high $\text{We}$, (ii) thin film, intermediate $\text{We}$ and (iii) thick film, low $\text{We}$. Picture (i) also shows the $(r,\theta )$ coordinates used to compute the integral from Eq. (2). The equivalent radius and drop impacting point are represented in each photograph by a white dashed circle and a white dot, respectively. The scale bars are $1\mathrm{cm}$.

Figure 10

Nondimensional second order moment ${\sigma }_{\mathrm{eq}}$ of the red proportion in the film postimpact, defined in Eq. (3): (a) measurements of ${\sigma }_{\mathrm{eq}}$ as a function of the nondimensional initial film thickness ${\delta }^{★}$, for three bins of $\text{We}$. The colors represent the various scenarios described in Sec. 3. The roman lowercase letters show the locations of the examples from panel (b) in the graph. The inset represents the second-order moment divided by the equivalent radius, ${\sigma }_{\mathrm{eq}}/R_{\mathrm{eq}}$. The dashed line in the inset corresponds to the theoretical uniform distribution value of $1/\left(2\sqrt{2}\right)$. (b) Examples of radii $R_{\mathrm{eq}}$ (dashed, black) and $R_{\mathrm{eq}}\pm {\sigma }_{\mathrm{eq}}$ (dashed, white) superimposed with the corresponding initial photographs, for the following ranges of parameters: (i) very thin film, high $\text{We}$, (ii) thin film, intermediate $\text{We}$, and (iii) thick film, low $\text{We}$. The white dots show the impact points. The scale bars are $1\mathrm{cm}$.

Figure 11

Drop, film, dye proportions, and volume distribution nomenclature. (a) Impacting drop containing both red and green dyes in respective proportions $p_{r,\mathrm{d}}$ and $p_g$. The drop has a radius $r_{\mathrm{d}}$, a final velocity $u_{\mathrm{d}}$ and a volume $V_{\mathrm{d}}$ split into two contributions: $V_{\mathrm{d}\to \mathrm{e}}$ which goes into the ejections, and $V_{\mathrm{d}\to {\mathrm{f}}^{\prime }}$ which remains in the film after the impact. (b) Initial film containing only green dye in proportion $p_g$, of supposedly uniform thickness $\delta$ and of volume $V_{\mathrm{f}}$ divided into $V_{\mathrm{f}\to \mathrm{e}}$, the ejected part, and $V_{\mathrm{f}\to {\mathrm{f}}^{\prime }}$, the remaining part. (c) Postimpact ejected droplets of unknown red volume fraction and of total added volume $V_{\mathrm{e}}$ coming from the sum of $V_{\mathrm{d}\to \mathrm{e}}$, the ejected volume initially in the drop, and $V_{\mathrm{f}\to \mathrm{e}}$, the ejected volume initially contained in the film. (d) Postimpact film of both unknown thickness ${\delta }^{\prime }$ and red proportion $p_r$, and of known green proportion $p_g$. Its volume $V_{\mathrm{f}{}^{\prime }}$ corresponds to the addition of $V_{\mathrm{d}\to {\mathrm{f}}^{\prime }}$, the volume going from the impacting drop into the film, and $V_{\mathrm{f}\to {\mathrm{f}}^{\prime }}$, the part already in the film before the impact and which has not been ejected.

Figure 12

Ejected ratio ${\phi }_{\mathrm{e}}$ from Eq. (4), corresponding to the total ejected volume $V_{\mathrm{e}}$ over the impacting drop volume $V_{\mathrm{d}}$, as a function of ${\delta }^{★}$ and for various bins of $\text{We}$. The dashed lines represent limit values exhibited by this ratio, i.e., ${\phi }_{\mathrm{e}}=0.6$ and ${\phi }_{\mathrm{e}}=1$. The colors correspond to the different scenarios observed and described in Sec. 3. The black crosses represent data from Yarin and Weiss [65], adapted to our experimental range by using $\text{Oh}=1.7\times {10}^{-3}$ and $\text{We}=1500$.

Figure 13

(a) Proportion of the volume coming from the drop left in the film after the impact, ${\phi }_{\mathrm{d}\to \mathrm{f}{}^{\prime }}$, as defined in Eq. (5). The dashed line shows the limit case for which ${\phi }_{\mathrm{d}\to \mathrm{f}{}^{\prime }}=0.5$. (b) Ratio of the film volume lost in the ejections over the initial drop volume, ${\phi }_{\mathrm{f}\to \mathrm{e}}$, as defined in Eq. (6). The dashed line shows the case for which ${\phi }_{\mathrm{f}\to \mathrm{e}}=0.1$. Measurements are shown as a function of the film thickness ${\delta }^{★}$ and for three bins of $\text{We}$. The colors correspond to the different scenarios observed and described in Sec. 3. The legend is the same in both graphs.

Figure 14

Cutting of pictures in square cells and numbering.

Figure 15

Lighting coefficients ${\ell }_j\left(\mathit{x}\right)$ relative to Eq. (B6), for (a) $j=R$, (b) $j=G$, and (c) $j=B$. Coefficients were obtained by solving Eq. (B7) in the least-square sense with a Newton-Raphson technique.

Figure 16

Possible rejected experiments: (a) Impact occurring too close to an edge, in this case at a distance of $8\mathrm{m}\mathrm{m}$ of the upper border of the film in the field of view. The parameters are ${\delta }^{★}=1.3$ and $\text{We}=2750$. The maximum radius $r_{\mathrm{c}}$ of the cavity in the film is $13\mathrm{m}\mathrm{m}$. (b) Air entrapment in the case of an impact for which ${\delta }^{★}=4.3$ and $\text{We}=1675$. The scale bars are $1\mathrm{cm}$.