Experimental characterization of the growth dynamics during capillarity-driven droplet generation

The transient dynamics of a growing droplet in a yarn is explored following the spatiotemporal evolution of the three-phase contact line as well as the liquid-air interface with the help of videographic techniques and subsequent image analyses. The spontaneous capillary flow of liquids in a porous network is used to generate a droplet on the freely hanging end of a yarn whose other end is dipped continuously in a liquid reservoir. The growing droplet initially moves upward due to surface tension until the attainment of a critical volume, beyond which gravity is able to pull it downward until detachment. Based upon the spatiotemporal trajectory of the three-phase contact line of the droplet, the entirety of the associated growth dynamics can be divided in three distinct regimes, namely, “radial growth,” “axial growth,” and “motion” stages. The transition from one to the other is governed by the subtle interplay between the capillary and the gravity forces. Several experimental fluids are considered to elucidate the effect of the fluid properties on the transient contact line and interfacial dynamics of drops. The kinetics of the three-phase contact line and the radius of the droplet is found to follow two distinct exponential scaling laws, developed through the combination of the relevant forces. A mathematical model has also been proposed to predict the critical volume of the growing droplet in relation to its final volume, beyond which gravity controls the transient dynamics.

Figure 1

Schematic diagram of the experimental setup with all the components.

Figure 2

Schematic diagram of the different stages involved in drop formation from a yarn, namely, (a) growth (“radial” and “axial”), (b) “motion,” (c) “necking,” and (d) “detachment.” The yellow arrows show the sequence of the occurrence of the events. The two red arrows in the image (a) show the position of two points of interest, namely, “tip” and “bottom,” referred for describing the growth dynamics of the droplets in yarn. The variables $l_d$ and $r_d$, shown in image (b) with respective positions of measurements, represent the length and radius of the droplet, respectively. The red dashed line drawn in the free end of the yarn in image (b) represents the position of the datum $z=0$ considered for various measurements during the study.

Figure 3

Spatiotemporal evolution of the droplet interface during the generation of water drop from a yarn. The corresponding experimental conditions are $\mathrm{We}=1.078\times {10}^{-6}$, $\mathrm{Oh}=0.0043$, $\mathrm{Fr}=3.76\times {10}^{-3}$. The red arrows mark the position of the three-phase contact line of the droplet (“tip”) at different instances, whereas the yellow dashed lines in the snapshots refer to the location of datum. The scale bar shows 1 mm in length. The events have been recorded at 25 frames per second using a SONY camcorder.

Figure 4

(a) The temporal variation of the position of tip $z_{tp}$ and bottom $z_{bt}$ points during the growth of a water drop. (b) The variation of length $l_d$ and radius $r_d$ of a water drop during its growth stage. Position of the tip and bottom below the datum, i.e., $z=0$ (yellow dashed line in Fig. 3) is considered as negative, while the position above it has been considered as positive. The dashed vertical lines delineate the three stages of drop growth in both (a) and (b). The corresponding experimental conditions are $\mathrm{We}=1.078\times {10}^{-6}$, $\mathrm{Oh}=0.0043$, $\mathrm{Fr}=3.76\times {10}^{-3}$. The uncertainties in the measurement of the position of the respective identities at each instant are marked by the error bars associated with each data point.

Figure 5

(a) Spatiotemporal evolution of the interface during the generation of (i) tween 20 and (ii) hexadecane droplet from a yarn. The red arrows mark the position of the three-phase contact line (tip) at different instants. The scale bar shows 1 mm in length. The corresponding experimental conditions are $\mathrm{We}=5.03\times {10}^{-8}$, $\mathrm{Oh}=0.0063$, $\mathrm{Fr}=5.73\times {10}^{-4}$, and $\mathrm{We}=2.90\times {10}^{-9}$, $\mathrm{Oh}=0.024$, $\mathrm{Fr}=1.36\times {10}^{-4}$ for rows (i) and (ii), respectively. The yellow dashed lines in the snapshots of rows (i) and (ii) show the datum. (b) The temporal variation in the position of the tip and bottom during the growth of droplets of tween 20 solution and hexadecane, in a yarn. The data of this plot correspond to the images of Fig. 5. The two different colors of the same symbols represent the data corresponding to tip and bottom for the two different experimental liquids. The differently colored dotted vertical lines delineate all the three stages seen during the drop growth period.

Figure 6

(a) The variation of the nondimensional tip position $z_{tp}^{*}$ with nondimensional time $t^{*}$ is plotted in a log-log scale for all the experimental conditions. The vertical shaded regimes correspond to the transition regimes separating the three stages of drop growth. (b), (c) Compare the experimental data of $z_{tp}^{*}$ with the derived asymptotic relation (dashed-dotted line) [Eq. (5)] for water (DIW) and 60% glycerol (Gly60) solution, respectively.

Figure 7

(a) Linear plot of variation of nondimensional radius $r_d^{*}$ with nondimensional time $t^{*}$ of a continuously growing droplet at different experimental conditions corresponding to Table 2. Logarithmic plots of $r_d^{*}$ with $t^{*}$ exhibit the good agreement between the derived scaling law (dashed-dotted line) [Eq. (6)] and the experimental data for (b) water and (c) 60% glycerol solution. respectively.

Figure 8

Force balance in case of a drop continuously growing on a yarn just prior to the initiation of sliding motion. The different geometrical parameters of the drop considered in the force balance are shown with the help of dimensional arrows. $F_g$ is the gravitational pull experienced by the droplet due to its own weight, $F_c$ is the capillary force along the contact line, and $F_{fc}$ is the capillary force due to the liquid film acting along the wetting length $l_{wt}$.

Figure 9

Comparison between the experimentally measured primary drop volume and theoretical law from Harkins and Browns [59].

Figure 10

(a) Schematic of an axisymmetric drop attached to a capillary with all the spatial and angular variables defining the shape at each instant. (b) Comparison between the numerically obtained droplet shape (yellow curve) with the shape obtained experimentally by fitting the former upon the latter. The axisymmetric equations are able to describe the shape well as the yellow curve fits completely the experimental shapes. The experimental conditions encountered are $Q^{\text{Twn20}}=8.68\times {10}^{-11}{\text{m}}^3/\mathrm{s}$, $\mathrm{We}=5.03\times {10}^{-8}$, $\mathrm{Oh}=0.0063$, and $\mathrm{Fr}=5.73\times {10}^{-4}$.