Local velocity variations for a drop moving through an orifice: Effects of edge geometry and surface wettability

We investigate velocity variations inside and surrounding a gravity-driven drop impacting on and moving through a confining orifice, wherein the effects of edge geometry (round vs sharp edged) and surface wettability (hydrophobic vs hydrophilic) of the orifice are considered. Using refractive-index matching and time-resolved particle image velocimetry, we quantify the redistribution of energy in the drop and the surrounding fluid during the drop's impact and motion through a round-edged orifice. The measurements show the importance of (a) the drop kinetic energy transferred to and dissipated within the surrounding liquid and (b) the drop kinetic energy due to internal deformation and rotation during impact and passage through the orifice. While a rounded orifice edge prevents contact between the drop and orifice surface, a sharp edge promotes contact immediately upon impact, changing the near-surface flow field as well as the drop passage dynamics. For a sharp-edged hydrophobic orifice, the contact lines remain localized near the orifice edge, but slipping and pinning strongly affect the drop propagation and outcome. For a sharp-edged hydrophilic orifice, on the other hand, the contact lines propagate away from the orifice edge and their motion is coupled with the global velocity fields in the drop and the surrounding fluid. By examining the contact line propagation over a hydrophilic orifice surface with minimal drop penetration, we characterize two stages of drop spreading that exhibit a power-law dependence with variable exponent. In the first stage, the contact line propagates under the influence of impact inertia and gravity. In the second stage, inertial influence subsides and the contact line propagates mainly due to wettability.

Figure 1

Schematic of the experimental setup and the drop generating mechanism: side view (left) and top view (right).

Figure 2

Image processing steps: (a) raw PIV image, (a) to (b) median filter, (b) to (c) standard deviation filter, (c) to (d) Canny edge detection, (d) to (e) morphological closing and filling, and (f) raw PIV image with detected interface.

Figure 3

Iterative morphological closing and filling operations performed on a preprocessed image. In each step, the size of the structuring element (SE) was incremented until the entire drop was filled.

Figure 4

Sample visualization image of a drop contact line at the hydrophilic orifice. The arrow indicates the contact angle obtained via image processing.

Figure 5

Isolation of velocity fields inside a drop (red) and in the surrounding fluid (blue) based on interface location.

Figure 6

(a) Axial velocity $u_z/U_t$ (left panel) and out-of-plane vorticity ${\omega }_{\theta }D/U_t$ (right panel) contours superposed with velocity vectors. Contour levels and vectors in the left panel are shown with respect to the laboratory reference frame. Velocity vectors in the right panel are shown with respect to the drop reference frame. (b) Kinetic energy of the surrounding fluid estimated over test windows of increasing size in $z\left({\text{KE}}_{s|\mathrm{\Delta }z}\right)$ and $r\left({\text{KE}}_{s|\mathrm{\Delta }r}\right)$ directions.

Figure 7

Variation of drop centroid velocity $u_{z,c}/U_t$ for a drop with $\text{Bo}=3.7$ moving through a round-edged orifice of $d/D=0.62$ (RHPB1 in Table 2) and outlines of the drop interface at six specified times.

Figure 8

Contours of axial velocity $u_z/U_t$ (left panel) and out-of-plane vorticity ${\omega }_{\theta }D/U_t$ (right panel) for a drop with $\text{Bo}=3.7$ impacting on a round-edged orifice with $d/D=0.62$ during the initial deceleration phase at times (a) $t=-0.02t_i^{*}$, (b) $t=0.22t_i^{*}$, and (c) $t=0.33t_i^{*}$.

Figure 9

Contours of axial velocity $u_z/U_t$ (left panel) and normalized out-of-plane vorticity ${\omega }_{\theta }D/U_t$ (right panel) for a drop with $\text{Bo}=3.7$ impacting on a round-edged orifice with $d/D=0.62$ during the reacceleration phase at times (a) $t=0.74t_i^{*}$, (b) $t=1.0t_i^{*}$, and (c) $t=1.3t_i^{*}$.

Figure 10

(a) Variation of total kinetic energy inside the drop $\left({\text{KE}}_d\right)$, total kinetic energy in surrounding oil $\left({\text{KE}}_s\right)$, drop deformation energy $\left(E_{\sigma }\right)$, the gravitational potential energy $\left({\text{GE}}_d\right)$, and the sum of these four terms $\left(E_{\text{total}}\right)$ above the orifice plate for a drop with $\text{Bo}=3.7$ moving through the round-edged orifice with $d/D=0.62$ and (b) decomposition into its translational component $\left({\text{KE}}_{d,T}\right)$ and the component due to local deformation and internal circulation $\left({\text{KE}}_{d,D}\right)$. Each quantity is normalized by the drop kinetic energy in its free-falling state.

Figure 11

Variation of drop centroid velocity $u_{z,c}/U_t$ for a drop with $\text{Bo}=3.7$ and $d/D=0.62$ moving through a sharp-edged hydrophobic (SHPB1 in Table 2) orifice superposed with the outlines of the drop interface at seven specified times. The dashed line indicates the plot from Fig. 7, included with time axis shifted to match the onset of penetration in the SHPB1 case.

Figure 12

Contours of axial velocity $u_z/U_t$ (left panel) and out-of-plane vorticity ${\omega }_{\theta }D/U_t$ (right panel) for a drop with $\text{Bo}=3.7$ impacting on a sharp-edged hydrophobic orifice with $d/D=0.62$ during the contact-line-initiated motion at (a) $t=0.5t_i^{*}$, (b) $t=1.24t_i^{*}$, and (c) $t=1.55t_i^{*}$.

Figure 13

Contours of axial velocity $u_z/U_t$ (left panel) and out-of-plane vorticity ${\omega }_{\theta }D/U_t$ (right panel) at an equivalent time $(t=1.75t_i^{*}$ in Fig. 11) for a drop with $\text{Bo}=3.7$ impacting on (a) a round-edged orifice and (b) a sharp-edged orifice with $d/D=0.62$.

Figure 14

Contours of axial velocity $u_z/U_t$ (left panel) and normalized axial strain $\alpha D/U_t$ (right panel) for a drop with $\text{Bo}=3.7$ impacting and partially releasing through a sharp-edged hydrophobic orifice with $d/D=0.62$ at times (a) $t=2.27t_i^{*}$, (b) $t=2.57t_i^{*}$, (c) $t=3.02t_i^{*}$, and (d) $t=3.16t_i^{*}$.

Figure 15

Schematic a of drop penetrating a sharp-edged hydrophobic orifice demonstrating pinning force $F_p$ and inertial force $F_i$ acting on the trailing drop fluid, and angles of contact: Young's angle of the leading contact line ${\text{CL}}_1$, the experimentally observed apparent contact angle ${\mathrm{\Theta }}_a$ of the trailing contact line ${\text{CL}}_2$, the wedge angle $\varphi$, and the canthotaxis regime $({\mathrm{\Theta }}_Y+\varphi -\pi ,{\mathrm{\Theta }}_Y)$ marked by dashed lines.

Figure 16

Contours of axial velocity $u_z/U_t$ (left panel) and radial velocity $u_r/U_t$ (right panel) for drops with $\text{Bo}=5.3$ and $d/D=0.62$ impacting (a)–(c) sharp-edged hydrophobic and (d)–(f) sharp-edged hydrophilic orifices at times (a) and (d) $t=0.17t_i^{*}$, (b) and (e) $t=0.35t_i^{*}$, and (c) and (f) $t=0.42t_i^{*}$.

Figure 17

High-resolution contours of radial velocity $u_r/U_t$ superposed with velocity vectors in the surrounding fluid near the contact line for the same velocity field shown in Fig. 16 for the sharp-edged hydrophilic orifice case.

Figure 18

Contours of axial velocity $u_z/U_t$ (left panel) and the axial strain $\alpha D/U_t$ (right panel) for a drop with $\text{Bo}=5.3$ and $d/D=0.62$ impacting and completely releasing through a sharp-edged hydrophobic orifice at times (a) $t=0.75t_i^{*}$ and (b) $t=0.92t_i^{*}$ and partially releasing through a hydrophilic orifice at times (c) $t=0.75t_i^{*}$ and (d) $t=1.17t_i^{*}$.

Figure 19

Schematic of a drop penetrating a sharp-edged hydrophilic orifice showing various angles of contact: Young's angle ${\mathrm{\Theta }}_Y$ of the leading contact line ${\text{CL}}_1$, the experimentally observed apparent contact angle ${\mathrm{\Theta }}_a$ of the trailing contact line ${\text{CL}}_2$, the wedge angle $\varphi$, and the canthotaxis regime $({\mathrm{\Theta }}_Y+\varphi -\pi ,{\mathrm{\Theta }}_Y)$ marked by dashed lines. Thick dashed lines with arrows indicate the concept of interface deformation before breakup.

Figure 20

Contours of axial velocity $u_z/U_t$ (left panel) superposed with velocity vectors and out-of-plane vorticity ${\omega }_{\theta }{U}_t/D$ (right panel) at (a) $t=0.07t_i^{*}$, (b) $t=0.21t_i^{*}$, and (c) $t=0.35t_i^{*}$ for a drop with $\text{Bo}=4.6$ impacting on a sharp-edged hydrophilic orifice with $d/D=0.44$. (d)–(f) Corresponding local radial velocity $u_r/U_t$ contours superposed with velocity vectors near the contact line.

Figure 21

Contours of axial velocity $u_z/U_t$ (left panel) superposed with velocity vectors and out-of-plane vorticity ${\omega }_{\theta }{U}_t/D$ (right panel) at (a) $t=0.5t_i^{*}$, (b) $t=0.67t_i^{*}$, and (c) $t=0.95t_i^{*}$ for a drop with $\text{Bo}=4.6$ impacting on a sharp-edged hydrophilic orifice with $d/D=0.44$. (d)–(f) Corresponding local radial velocity $u_r/U_t$ contours superposed with velocity vectors near the contact line.

Figure 22

(a) Variation of normalized spreading distance $2\mathrm{\Delta }r/D$ from the orifice edge, (b) contact line velocity $u_{\text{CL}}/U_i$ based on the cubic-spline fit with respect to time $(t-t_c)/t_i^{*}$ measured from the time of initial contact $t_c$, (c) dynamic angle of contact ${\mathrm{\Theta }}_d$ with respect to $2\mathrm{\Delta }r/D$, and (d) example snapshots of contact line location and orientation above the plate surface for a drop with $\text{Bo}\approx 5$ impacting a hydrophilic orifice with $d/D=0.44$ (frames A–C) and 0.62 (frames D–F). The triangles in (a) demonstrate power-law variations during early inertia-dominated $\left(\kappa =4/5\right)$ and later viscous-dominated $\left(\kappa =1/10\right)$ spreading.