Vorticity-induced flow-focusing leads to bubble entrainment in an inkjet printhead: Synchrotron x-ray and volume-of-fluid visualizations

The oscillatory flows present in an inkjet printhead can lead to strong deformations of the air-liquid interface at the nozzle exit. Such deformations may lead to an inward directed air jet with bubble pinch-off and the subsequent entrainment of an air bubble, which is highly detrimental to the stability of inkjet printing. Understanding the mechanisms of bubble entrainment is therefore crucial to improving print stability. In the present work, we use ultrafast x-ray phase-contrast imaging and direct numerical simulations based on the volume-of-fluid method to study the mechanisms underlying the bubble entrainment in a piezoacoustic printhead. We first demonstrate good agreement between experiments and numerics. We then show the different classes of bubble pinch-off obtained in experiments, and that those were also captured numerically. The numerical results are then used to show that the baroclinic torque, which is generated at the gas-liquid interface due to the misalignment of density and pressure gradients, results in a flow-focusing effect that drives the formation of the air jet from which a bubble can pinch off.

Figure 1

(a) Schematic of the experimental setup. (b) $t=0$ is when the first 472 ns x-ray pulse illuminates the printhead and the camera is triggered. The piezo in the printhead is triggered after a set delay $\mathrm{\Delta }t$, starting at $0\mu \mathrm{s}$. After $3.68\mu \mathrm{s}$ the next x-ray pulse illuminates the setup for the second frame in the recording. This process continues for 200 frames. The next recording has a delay $\mathrm{\Delta }t$ of $0.5\mu \mathrm{s}$ for the piezo actuation. This continues in steps of $0.5\mu \mathrm{s}$ until a maximum $\mathrm{\Delta }t$ of $3.5\mu \mathrm{s}$. (c) The trapezoidal piezo driving pulse. (d) Typical bubble entrainment phenomenon for a driving pulse with $V_{\text{max}}=140\mathrm{V}$ and $\tau =32\mu \mathrm{s}$. The edge of the nozzle is marked by the gray lines.

Figure 2

(a) Experimental snapshot where the black dash-dotted line is the axis of symmetry of the nozzle. (b) Edge detection result of the snapshot shown in (a). The total volume minus the volume in the orange area results in the volume of the blue region. (c) Extracted volume over time for $V_{\text{max}}=70\mathrm{V}$ and $\tau =34\mu \mathrm{s}$. (c) Flow rate ($Q$) as determined from the volume curve in (b). The experimentally obtained $Q\left(t\right)$ was used as an input to the numerical simulations shown in Fig. 4.

Figure 3

Schematic of the axisymmetric numerical domain (to scale), with the boundary conditions specified. The bottom boundary is the axis of symmetry, depicted by the dot-dashed line.

Figure 4

(a) Meniscus position during different time instants of the bubble entrainment process as observed in the experiment (grayscale images) and the numerical simulations (red curves) for $V_{\text{max}}=70\mathrm{V}$ and $\tau =34\mu \mathrm{s}$. (b) A zoomed-in image of the same bubble entrainment phenomenon. The outline of the nozzle is highlighted with gray lines.

Figure 5

(a) No bubble entrainment in the experiment with $V_{\max }=70\mathrm{V}$ and $\tau =32\mu \mathrm{s}$. (c) One bubble is entrained in the experiment at $V_{\max }=70\mathrm{V}$ and $\tau =34\mu \mathrm{s}$, which is the same experiment as in Fig. 4. (c) Two bubbles are entrained in the experiment at $V_{\max }=80\mathrm{V}$ and $\tau =34\mu \mathrm{s}$. (d) No bubble entrainment in the numerics with a viscosity of 2 mPas. (e) One bubble is entrained in the numerics with a viscosity of 1mPas, which is the same simulation as in Fig. 4. (f) Two bubbles are entrained in the numerics with a viscosity of 0.5 mPas. In all the numerical results, the nozzle dimensions are nondimensionalized by the nozzle radius $R_n=35\mu \mathrm{m}$.

Figure 6

Snapshots from numerical simulations of the meniscus oscillations leading to bubble entrainment. The velocity is depicted as a vector field, and the dimensionless vorticity ${\omega }^{*}=\omega \tau$ is illustrated in the background as a color code. In each snapshot, the interface is drawn in green. This is the $V_{\text{max}}=70\mathrm{V}$ and $\tau =34\mu \mathrm{s}$ case, where the fluid properties are as those in the experiment.

Figure 7

Illustration of the mechanism of meniscus oscillation and bubble entrainment. Four time instants are shown: $t/\tau \in \{0.55,1.08,1.22,1.32\}$. Each time point contains three panels, showing, from left to right, the dimensionless pressure $p^{*}=p\tau /\left({\rho }_l{R}_n^2\right)$ with isocontours, an explanatory schematic of the baroclinic torque generated at the interface, and the dimensionless vorticity ${\omega }^{*}=\omega \tau$, respectively.

Figure 8

Average acceleration of the liquid inside the nozzle with respect to time. The dashed line indicates the approximate average of the deceleration phase responsible of the topological change in Fig. 7.

Figure 9

Snapshots from numerical simulations of the meniscus oscillations leading to bubble entrainment. The velocity is depicted as a vector field, and the dimensionless vorticity ${\omega }^{*}=\omega \tau$ is illustrated in the background as a color code. In each snapshot, the interface is drawn in black. In this simulation, surface tension is artificially set to zero. The green dashed circles indicate the location of flow focusing at the interface due to the baroclinic torque.