Investigation of the vortex instability in a two-dimensional inkjet print-zone using numerical analysis

A numerical model was employed to investigate the vortex instability in a two-dimensional inkjet print-zone. The simulation models the entrainment effect of the droplets on the airflow via a dispersed-phase continuum method that, due to the separation of length scales, treats the force exerted by the main droplets as a continuum smooth field. The trajectory and speed of the main droplets are also assumed to be unaffected by the airflow. The results indicate the existence of a dimensionless droplet density threshold ($N_c$) at which the vortex shifts from steady to oscillatory. This demonstrates that the two-dimensional instability has a supercritical Hopf type of bifurcation, i.e., the shift from stable to unstable is continuous but not smooth and the amplitude of oscillation follows the asymptotic square-root behavior characteristic of this type of bifurcation. Further, as shown by tests with stationary paper and no induced cross-flow, the mechanism of instability cannot be attributed to the interaction between the incoming cross-flow and the entrained airflow. Characterizing the two-dimensional airflow instabilities and their mechanism provides a better understanding of the airflow dynamics in the print gap of inkjet printers.

Figure 1

Schematic of the physics in the print gap, where $H$ is the print gap height, $W$ is the droplet velocity, $V_p$ is the paper speed, $b$ is the breadth of the print-zone, and the origin of the coordinate system is marked by the white square.

Figure 2

Nondimensionalized velocity magnitude ($\tilde{u}=u/W$) and planar vorticity (${\tilde{\omega }}_x={\omega }_{x}H/W$) fields: $b/H=1/3, N=1.57\times {10}^{-3}$, and $V_p/W=0.027$ where the white square represents the origin of the coordinate system and the white cross represents the probe location.

Figure 3

Nondimensionalized velocity ($\tilde{u}=u/W$) oscillation signal: $b/H=1/3, N=1.57\times {10}^{-3}$, and $V_p/W=0.027$.

Figure 4

Discrete Fourier transform of the case with $b/H=1/3, N=1.57\times {10}^{-3}$, and $V_p/W=0.027$. $\text{St}=fH/W$, where $f$ is the frequency of oscillation.

Figure 5

Nondimensionalized velocity magnitude ($\tilde{u}=u/W$) and planar vorticity (${\tilde{\omega }}_x={\omega }_{x}H/W$) fields: $b/H=1/3, N=1.33\times {10}^{-3}$, and $V_p/W=0.027$ where the square represents the origin of the coordinate system and the white cross represents the probe location.

Figure 6

Nondimensionalized velocity ($\tilde{u}=u/W$) oscillation signal: $b/H=1/3, N=1.33\times {10}^{-3}$, and $V_p/W=0.027$.

Figure 7

Nondimensionalized velocity magnitude ($\tilde{u}=u/W$) and planar vorticity (${\tilde{\omega }}_x={\omega }_{x}H/W$) fields: $b/H=1/3, N=0.31\times {10}^{-3}$, and $V_p/W=0.027$ where the square represents the origin of the coordinate system and the white cross represents the probe location.

Figure 8

Bifurcation diagram for $V_p/W=0.027$ and $b/H=1/3$. Top figure: Nondimensionalized amplitude of oscillation (${\tilde{r}}^2=r^2/W^2$) vs dimensionless number density ($N$). Bottom figure: Initial dimensionless velocity (${\tilde{u}}_0=u_0/W$) vs number density ($N$).

Figure 9

Discrete Fourier transform for cases $N=1.3\times {10}^{-3}, N=1.33\times {10}^{-3}, N=1.41\times {10}^{-3}$, and $N=1.49\times {10}^{-3}$.

Figure 10

Nondimensionalized velocity magnitude ($\tilde{u}=u/W$) and planar vorticity (${\tilde{\omega }}_x={\omega }_{x}H/W$) fields: $b/H=1/3, N=1.57\times {10}^{-3}$, and $V_p/W=0$ where the square represents the origin of the coordinate system and the white cross represents the probe location.

Figure 11

System of vortices created by the entrained flow: main vortices (A) and secondary recirculation regions (B).

Figure 12

Nondimensionalized velocity ($\tilde{u}=u/W$) oscillation signal: $b/H=1/3, N=1.57\times {10}^{-3}$, and $V_p/W=0$.

Figure 13

Bifurcation diagram: Nondimensionalized amplitude of oscillation (${\tilde{r}}^2=r^2/W^2$) vs dimensionless number density ($N$) for $V_p/W=0$ and $b/H=1/3$.

Figure 14

Nondimensionalized velocity magnitude ($\tilde{u}=u/W$) and planar vorticity (${\tilde{\omega }}_x={\omega }_{x}H/W$) fields: $b/H=1/3, N=1.57\times {10}^{-3}$, and $V_p/W=0.041$ where the white square represents the origin of the coordinate system and the white cross represents the probe location.

Figure 15

Bifurcation diagram: Nondimensionalized amplitude of oscillation (${\tilde{r}}^2=r^2/W^2$) vs dimensionless number density ($N$) for $V_p/W=0.041$ and $b/H=1/3$.

Figure 16

Bifurcation diagram: Nondimensionalized amplitude of oscillation (${\tilde{r}}^2=r^2/W^2$) vs dimensionless number density ($N$) for $V_p/W=0.027, b/H=1/3.3$, and $b/H=1/3.6$.

Figure 17

Strouhal number ($\text{St}=fH/W$) vs number density ($N$) for different printing conditions.