Scattering of liquid droplets from axisymmetric targets

Droplets that glide along a bath of the same fluid are seen to scatter from a cylindrical meniscus analogous to Rutherford scattering of like-charged atomic particles. We define the impact parameter, $b$, as the distance between the tangent line to the initial trajectory of the droplet and the parallel radial line from the center of the target. The scattering angle, ${\theta }_S$, and the distance of closest approach, $r_{\mathrm{C}}$, are measured as functions of $b$. Using numerical methods to model the experimental results, we confirm previous theoretical results that the approximate profile of the meniscus is given by a Bessel function for regions distant from the contact line of the meniscus.

Figure 1

Parameters defined. (a) Droplets initially move along a line parallel with and a distance $b$ away from a radial line extending out from the target axis. This radial line is chosen to be the $x$ axis with $y$ perpendicular to $x$ and also in the plane of motion. The distance $b$ is called the impact parameter and is varied to achieve different scattering angles, ${\theta }_S$, measured from the initial direction. The position, $r$, of the droplet is measured relative to the center of the target. (b) The droplet rides up the meniscus around the cylindrical target, reaching a height $z$ dependent on the initial velocity of the droplet, $v_0$.

Figure 2

Side-view image sequence of a droplet interacting with the target meniscus for $b=0$. Time is measured with $t=0$ being the moment of closest approach. The vertical dashed lines guide comparisons for the droplet's position at symmetric incoming and outgoing times. Droplet radius in the horizontal plane is 1.18 mm.

Figure 3

Experimental setup. An $\mathsf{L}$-shaped glass tube (A) is supported by a two-axis translation stage and positioned so that one leg of the tube extends vertically through the free surface of the bath. A syringe pump (B) pushes fluid through a pipette tip connected by flexible Tygon tubing. The pipette tip is positioned so as to gently deposit droplets onto the meniscus formed on an angled glass slide. The droplet accelerates away from the glass slide $\left(-\widehat{x}\right)$ (C) toward the glass tube, scattering from the meniscus formed on the tube. The fluid is held in a clear container which allows video imaging from above, below, or sides with backlit illumination provided by a bank of diffuse LED lights.

Figure 4

Droplet trajectories. (a) Droplet trajectories for varying $b$. Positions are normalized by the sum of the droplet and target radii, $(x^{*},y^{*})=(x,y)/(R_T+R_D)$. (b) The direction of droplet motion as a function of time, given by an the angle $\theta \left(t\right)$ measured clockwise from $-\widehat{x}$. $t=0$ is defined by the time of closest approach. Each of the four curves correspond to the trajectories in panel (a) matched by line shading. The horizontal dashed line is shown for reference to see the decreasing value of $\theta$ after reaching a maximum.

Figure 5

Droplet-meniscus interactions for $b=0$. (a) Droplet speed as a function of radial position, $r^{*}$, for the same four different values of $b$ as in Fig. 4. The arrows indicate which portions of the graph represent motion toward the target (decreasing $r$) and away (increasing $r$). (b) Trajectory of the droplet (black line) and the profile of the unperturbed meniscus (gray line). For better comparison with the meniscus profile, the droplet's center-of-mass trajectory has been translated vertically downward until the outgoing path of the droplet lies along the static meniscus. Vertical position is measured relative to the meniscus height, $z^{*}=z/h_0$. Inset: Same droplet trajectory without the translation showing the true position of the droplet's center of mass above the static meniscus.

Figure 6

Scattering angle, ${\theta }_S$ (a), and distance of closest approach, $r_C*$ (b), as functions of $b^{*}$. Gray lines in panels (a) and (b) are best fits using the Bessel function model, whereas the dotted black lines are best fits using the exponential model. The dashed black line in panel (b) has unit slope.

Figure 7

Droplet trajectories with model fits. Experimental trajectories (black lines) with solutions to both the Bessel function (solid gray line) and exponential (dotted black line) models to the meniscus profile.