Auto-ejection of liquid from a nozzle

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle. We test the LB model by conducting some simulations, and find that the numerical results agree well with the theoretical and experimental data. We further consider the effects of contraction ratio, length ratio, contact angle, and nozzle structure on the auto-ejection, and observe some distinct phenomena during the ejection process, including the deformation of meniscus, capillary necking, and droplet pinch off. Finally, the results reported in the present work may play an instructive role on the design of droplet ejectors and the understanding of jetting dynamics in microgravity environment.

Figure 1

Schematic of auto-ejection and boundary conditions.

Figure 2

Schematic of interpolation schemes [fluid nodes (circle dots), solid nodes (square dots), and boundary nodes (diamond dots)].

Figure 3

Schematic of fluid nodes (circle dots), solid nodes (square dots), and boundary nodes (diamond dots) around a curved boundary.

Figure 4

Schematic of droplet spreading on a solid sphere.

Figure 5

The equilibrium states of the droplet obtained by the axisymmetric LB method [(a) $\theta ={30}^{{}^{\circ }}$, (b) $\theta ={60}^{{}^{\circ }}$, (c) $\theta ={90}^{{}^{\circ }}$, and (d) $\theta ={120}^{{}^{\circ }}]$.

Figure 6

The imbibition height and meniscus velocity along the centerline of the tube #35 [(a) the position of meniscus, (b) the velocity of the meniscus].

Figure 7

The evolutions of meniscus at specific time stage in tube #35 [(a) point $a$, (b) point $b$, (c) point $c$, (d) point $d$, (e) point $e$, (f) after point $e$].

Figure 8

The ejected droplets of different PDMS fluids in tube #35 [(a) 2cS PDMS fluid, (b) 5cS PDMS fluid].

Figure 9

The auto-ejection process of 2cS PDMS fluid in tube #36 (left: numerical results, right: experimental results [19]).

Figure 10

The meniscus velocities along the centerline and imbibition heights at different contraction ratios [(a) meniscus velocity, (b) meniscus position].

Figure 11

The maximum velocities and heights under different contraction ratios.

Figure 12

The evolutions of flow field and interface of the case with $L_t=4.5\mathrm{mm}$ and $C=0.25$.

Figure 13

The evolutions of flow field and interface of the case with $L_t=4.5\mathrm{mm}$ and $C=0.275$.

Figure 14

The evolutions of flow field and interface of the case with $L_t=4.5\mathrm{mm}$ and $C=0.37$.

Figure 15

The velocity profiles and imbibition heights of the cases with $K=0.09, K=0.15$, and $K=0.21$.

Figure 16

The maximum velocities and heights under different length ratios.

Figure 17

The evolutions of flow field and interface of the case with $L_t=7.5\mathrm{mm}$ and $C=0.25$.

Figure 18

The evolutions of flow field and interface of the case with $L_t=10.5\mathrm{mm}$ and $C=0.25$.

Figure 19

The meniscus velocities along the centerline and the imbibition heights under different values of contact angle [(a) meniscus velocity, (b) imbibition height].

Figure 20

The maximum velocities and heights under the different contact angles.

Figure 21

The evolutions of flow field and interface of the case with $L_t=10.5\mathrm{mm}$ and $\theta ={10}^{{}^{\circ }}$.

Figure 22

The evolutions of flow field and interface of the case with $L_t=10.5\mathrm{mm}$ and $\theta ={15}^{{}^{\circ }}$.

Figure 23

The effects of length ratio ($K$), contraction ratio ($C$), and contact angle ($\theta$) on the number of ejected droplets.

Figure 24

The schematic of different nozzle structures.

Figure 25

The meniscus velocities along the centerline and the imbibition heights under different nozzle structures [(a) meniscus velocity, (b) meniscus velocity].

Figure 26

The maximum velocities and heights under the different contraction radius.

Figure 27

The flow fields and interfaces at $t=1.3\mathrm{s}$ [(a) $R_{t_1}=6.0\mathrm{mm}$, (b) $R_{t_1}=4.5\mathrm{mm}$ and $R_{t_2}=8.5\mathrm{mm}]$.