Velocity Profile inside Piezoacoustic Inkjet Droplets in Flight: Comparison between Experiment and Numerical Simulation

Inkjet printing deposits droplets with a well-controlled narrow size distribution. This paper aims at improving experimental and numerical methods for the optimization of drop formation. We introduce a method to extract the one-dimensional velocity profile inside a single droplet during drop formation. We use a novel experimental approach to capture two detailed images of the very same droplet with a small time delay. The one-dimensional velocity within the droplet is resolved by accurately determining the volume distribution of the droplet. We compare the obtained velocity profiles to a numerical simulation based on the slender jet approximation of the Navier-Stokes equation and we find very good agreement.

Figure 1

In the experimental setup, the laser pulse from the Nd:YAG laser illuminates the diffusor, which emits a fluorescent light pulse of 8 ns at high intensity. The fluorescence is focused by an aspherical condenser lens onto the droplets. A notch filter, placed between the condenser lens and the droplets, prevents any remaining laser light from reaching the CCD camera. The images are recorded using an inline assembly microscope with a dual frame camera.

Figure 2

Time series of droplets recorded with single-flash photography. From left to right, multiple images of single droplets with a delay of $3\mu \mathrm{s}$ between these individual droplets. The width of the droplet is $23\mu \mathrm{m}$, the tail is about $4\mu \mathrm{m}$, and the secondary tail has a width below $1\mu \mathrm{m}$. The figure illustrates the imaging quality of the setup, and the absence of motion blur due to the use of the 8 ns iLIF.

Figure 3

Accurate subpixel edge detection, independent of the contrast in the image, is required. On the left a region of interest of the drop image is shown. First, the edge is determined using a standard threshold method (data not shown). From this trace the normal lines along the boundary of the drop are calculated (dotted green line). By calculating the inflection point along the normal lines the intensity independent contour is determined (blue). The red line shows the corrected position of the surface, which is determined by volume conservation arguments, see Sec. 5. The right part shows the intensity curve along the normal line (green dotted line).

Figure 4

A spherical droplet is imaged while gradually increasing the intensity of the illumination over the dynamic range of the camera. The radius is determined using a threshold method (red open dots) and by calculating the inflection point along the normal lines (blue filled dots). The radii from the threshold images change by 20% for increasing intensity, while the proposed method shows less than 1% variation. The remaining variation is primarily caused by the residual variation in the reproducibility of the drop formation (as shown in the inset of Fig. 6)

Figure 5

For each droplet the contour was determined with subpixel accuracy. To calculate the volume distribution the drop is assumed to be axisymmetric. The volume ($v_i$) of element i of the drop is estimated by truncated cones with radii $r_i$ and $r_{i+1}$ and a height $\mathrm{\Delta }{x}_i$

Figure 6

The total volume $V_{\text{drop}}$ of each ejected droplet, as extracted from the recordings, given as a function of the delay time. For each delay the information of two image pairs, i.e., four images, are used. The second pair is recorded one full period after the first to confirm stability at longer time scale. After a delay of $10\mu \mathrm{s}$ the meniscus starts moving outward, and at approximately $60\mu \mathrm{s}$ the droplets are detached from the meniscus. The blue data give the uncorrected volume following the inflection point method, while the red data give volume after correcting using volume conservation. The inset shows the calculated corrected volume in frame A against the corrected volume in frame B, which illustrates the volume reproducibility ($V_{\text{drop}}=11.02\pm 0.13\mathrm{pl}$) and the accuracy of the method ($3\sigma =17\mathrm{fl}$).

Figure 7

The volume distribution is calculated for frames A and B. The volume per element of the first time step is mapped to the volume elements in the second time step. The displacement of the corresponding center of masses divided by the time interval between the recordings gives the mean velocity of the volume elements.

Figure 8

The different analysis steps in the experimental procedure. Left: First two recordings of the same droplet are made, with a delay of 600 ns between the two frames. Middle: For both images the volume distribution over the droplet is calculated with subpixel accuracy. Right: The velocity inside the droplet is determined by calculating the displacement of the volume elements. The velocity is represented by the color scale.

Figure 9

Comparison between the experimental and numerical result. At different times, we show both the experimental result (left) and the numerical result (right). The experimental result is obtained from a different droplet each time, whereas the numerical droplet is calculated from the initial volume and velocity distributions from the experiment shown at $t=0\mu \mathrm{s}$ (see also Fig. 10). The velocity is represented by the color scale.

Figure 10

Detailed comparison between the experimental result (blue) and the numerical result (dashed red) at five times instances. The top row shows the velocity distribution, whereas the bottom row shows the profiles. The experimental and the numerical profiles are nearly identical. The Rayleigh-Plateau instability causes the velocity inside the tail to vary from droplet to droplet (also visible in Fig. 11). During the contraction of the droplet there is very good agreement between the velocities of the head and tail droplet.

Figure 11

The figure shows the spatial and temporal dependance of the momentum, calculated by multiplying the velocity distribution with the mass distribution. The drop flies from top to bottom. The upper figure shows the experiments, and the lower one the numerical simulation. It can be observed how the tail of the droplet is pulled towards the main drop by capillary forces. In the experimental plot it can be seen that the position of the tail and the momentum in the tail vary slightly between different droplets due to the Rayleigh breakup. This leads to the visible jitter in the curve of the tail.

Figure 12

Comparison between the experimental and numerical result for the case of colliding droplets. We show the experimental evolution (left) and the numerical results (right) at different times. The numerical droplet evolution is calculated from the experimental volume and velocity distributions at $t=0\mu \mathrm{s}$. The color scale represents the velocity.

Figure 13

The momentum distribution for the colliding droplets in experiment (top) and numerical simulation (bottom) showing overall good agreement but with an apparent lower experimental reproducibility in droplet behavior.