# One Dimensional Model for Droplet Ejection Process in Inkjet Devices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Droplet Formation Model

**T**, and ρ respectively, and different from previous 1D methods mentioned in the introduction; a shape function h(z, t) is introduced to describe the geometry of droplet.

**v**is the velocity vector of fluid, and $\nabla $ and Δ are gradient and Laplace operator, respectively. The stress balance and radial velocity field at interface give three boundary conditions are as follows:

**n**and

**t**are normal and tangential unit vectors, H is the mean curvature and the suffix of v represents the corresponding velocity component of fluid. Obviously, Equation (3) reflects the assumption of non-shear stress in free surface, Equation (4) is based on the Young–Laplace equation and the left-hand side (LHS) of Equation (5) is exactly the change rate of droplet shape. Due to the large aspect ratio of slender liquid column, lubrication approximation can be applied to simplify the governing equations. Therefore, axial velocity and pressure are expanded into Taylor series in the radial coordinate.

_{0}and capillary velocity scale ${v}_{c}$ = $\sqrt{\sigma /{\rho r}_{0}}$, the NS equations are transformed into two coupled 1D PDEs of motion:

#### 2.2. Nozzle Dynamics Model

_{z}is the axial pressure gradient. Note that, because the surface tension is not involved in the pipe flow, this equation has been made dimensionless with a series of scalers different than those in the droplet formation model. As a result, we use a different superscript, asterisk, to represent the dimensionless variable.

_{n}is the length of the nozzle. By performing a Laplace transform for Equation (11) with respect to t* and applying initial condition v*(0, r*) = 0, we get an ODE.

_{0}and the second kind Y

_{0}.

_{z}*(t*).

_{n}is the eigen value of J

_{0}(b

_{n}) = 0. The solution Equation (17) has been validated by Jiang [48] in comparison with the solutions based on the superposition of steady Poiseuille flow.

## 3. Numerical Method

#### 3.1. Solution Method for Droplet Formation

_{d}is the length of droplet in physical space. Note that, because of the linear relationship, nodes in physical space always have the same distribution as the computational space. Then the total time derivative of a scalar variable φ(z,t) can be expressed by the chain rule

_{begin}, and $\dot{{L}_{\mathrm{d}}}$ is the contraction rate of droplet v

_{end}-v

_{begin}. By applying Equation (19) for velocity v and shape function h, the equations of motion (9) and (10) are modified into the Lagrangian formulation.

_{0}< dz < 1.5dz

_{0}, in which dz is the current mesh size and dz

_{0}is the initial mesh size. If dz is out of this range, the droplet domain will be remeshed with a mesh size closest to the dz

_{0}. Due to the high sensitivity of surface tension to droplet shape, the shape-preserving piecewise cubic interpolation is used to evaluate these new nodes.

#### 3.2. Treatment of Droplet Ends and Droplet Breakup

^{+}= 0, Equation (22) is transformed by f

^{+}= h

^{2+}as expressed in Equation (23). A non-zero denominator is insured in the new expression of mean curvature (23) even if f

^{+}becomes zero at the ends of the droplet.

^{2+}in equations of motion (20) and (21) are replaced by f

^{+}in our numerical calculation. This practice also increases the accuracy of the shape function h by two orders of magnitude compared with directly using h in the finite-difference approximation.

_{0}, a breakup occurs there. To improve the numerical stability, instead of separating the droplet exactly at the point reaching the threshold, we create two new ends next to the breakup point and leave a space between them to guarantee a clear partition, as shown in Figure 3.

#### 3.3. Treatment of Droplet Coalescence

_{1}, v

_{2}, dt, and ΔL in Equation (24) are velocity of the tail node in the first droplet, velocity of the head node in the second droplet, time step, and distance between two droplets, respectively. The coefficient of 0.8 in Equation (24) is chosen from our numerical tests. Note that although mesh is always evenly distributed in a droplet, different droplets usually have different mesh size due to the extension and the contraction during fluid motion. As a result, the whole space occupied by the merged droplet needs to be first remeshed with a uniform mesh size (i.e., the original mesh size dz

_{0}), as shown in Figure 4b. Similar to the case of the remeshing, due to the severe contraction and the extension, the shape-preserving piecewise cubic interpolation is also used to evaluate height and velocity at new nodes. After that, two nodes nearest to the approaching ends can be identified, as shown in Figure 4c. We call these two nodes clip points, because the area between them, namely the blending area, will be clipped out for shape reconstruction. The smoothness of the blending area is paramount for numerical stability [41], therefore, a third-order polynomial curve is used to construct a smooth contour in the blending area. It means that at least two more equations are needed to determine this curve except the clip points. These two equations are given by a throat which is located in the center of blending area, with a height of 1% of orifice radius r

_{0}and a slope of zero. Figure 4d shows the result of drop coalescence with a highlight on the blending area and the throat. While this blending method is reliable and robust, it does add a small amount of mass into the system. We examined several instances, and found that the additional mass is less than 0.1% of the two merging droplets, and hence, is negligible.

#### 3.4. Coupling of Nozzle Dynamics and Droplet Formation

_{orifice}produced by the outer flow, i.e., meniscus and droplets, are taken into consideration for the calculation of the pressure gradient in the nozzle region [32]. p

_{orifice}is the capillary pressure and can be determined by the shape of the outer flow. The non-dimensional pressure gradient in the nozzle region can be obtained by

_{center}and the capillary pressure p

_{orifice}at the orifice plane.

_{N}and R

_{T}and a in Equation (27) are the normal and tangential principle radius of meniscus, as well as the extension of vertex. Equation (28) is the Young–Laplace equation.

_{N}and R

_{T}are given in Equation (29).

_{mean}.

## 4. Numerical Tests

#### 4.1. Infinite Microthread

^{+}= 2πr

_{0}/λ and amplitude ratio ε, the shape of the microthread is described by

_{c}is applied into the Re number.

^{+}= t/t

_{c}(capillary time t

_{c}= (ρr

_{0}

^{3}/σ)

^{1/2}) of breakup are obtained with different k

^{+}and Re under ε = 0.05. Table I lists the breakup times predicted from our simulations, as well as Ashgriz’s [55] and Furlani’s [42] simulations. A very good agreement between our study and these published studies can be found from Table 1. Figure 7 shows the evolution of the droplet profile at the breakup moment and post-breakup for k

^{+}= 0.7 and Re = 10.

#### 4.2. Free Flying Droplet

_{0}= 10 m/s, r

_{0}= 10 μm, and Oh = 0.01. To check the mesh convergence, three mesh sizes (i.e., the number of nodes are 51, 101, and 201, respectively) are applied. The time histories of the momentum and the mass of the flying droplet are plotted in Figure 8c,d, respectively. Note that the y axis is the deviation compared with analytical solution with unit of ‰. We can find that even for the coarsest mesh, the error is less than 0.3‰. The momentum increases very slightly at the beginning and then becomes stable. When the mesh size is refined further, the momentum is almost strictly conserved in the whole motion. The sphere shape of droplet is also very well maintained in the entire process for all mesh sizes. The shape at t

^{+}= 1 and t

^{+}= 2 for mesh size of 101 nodes are shown in Figure 8a,b. It is clear from Figure 8 that the droplet shape is identical to the initial shape.

#### 4.3. Liquid Filament Contraction

_{0}/2r

_{0}(where L

_{0}and r

_{0}is the length and radius of the filament, respectively), and the Oh number. In this test, a short filament and a long filament are examined respectively, and three Oh numbers are applied in the long filament. The results are compared with the 2D axisymmetric simulations which are carried out using the modified open-source code Gerris [27].

^{+}= L/r

_{0}. It is clear that the 1D simulation result agrees very well with the 2D simulation during the contracting phase. The result from the 1D model begins to diverge from that predicted by the 2D model during the oscillation stage. This is because of the large vorticity generated during oscillation, which makes the 1D assumption invalid [44,58].

^{+}= 16 and 17. This is caused by the intrinsic limitation of the 1D method, i.e., the inability to describe a multivalued curve. The comparisons of the filament length predicted by the 1D and 2D simulations for different Oh numbers are shown in Figure 10b,f,h. It is observed that the red line with circle mark (1D) nearly overlaps the blue line with triangle mark (2D) in most of the solution time.

#### 4.4. Step Driving Pressure

_{0}= 10 μm, L

_{n}= 50 μm. Ink properties are ρ = 1.135 g/cm

^{3}, σ = 67.26 dyn/cm, μ = 0.0615 P. A simplified step driving pressure (shown in Table 2 and Figure 11) is employed in the test, which is used by Fromm [34] and Adams [32], as well. The results are compared with the 2D simulation.

## 5. Example of Droplet Ejection Simulation

^{3}, σ = 40 dyn/cm, μ = 0.2 cP, 1 cP, 2 cP corresponding to Oh = 0.01, 0.05, 0.1. The nozzle dimensions are r

_{0}= 10 μm, L

_{n}= 50 μm. The waveform starts from a negative pressure with the minimum of −1.23 Mpa and the duration of 1.1 μs. At 2.0 μs, the first waveform is amplified by a higher pressure, reaching the extremum of 2.19 Mpa and −1.76 Mpa at 2.3 μs and 4.0 μs, respectively. Finally, the pressure waves are damped to zero.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematics of inkjet systems: (

**a**) continuous inkjet (CIJ) inkjet; (

**b**) drop-on-demand (DOD) inkjet.

**Figure 4.**Numerical procedure of droplet coalescence: (

**a**) Two approaching droplets with different mesh sizes; (

**b**) Remesh the whole space occupied by the merged droplet with a uniform mesh size; (

**c**) Find clip points and blending area; (

**d**) A merged droplet after the reconstruction of blending area.

**Figure 5.**Different stages of meniscus: (

**a**) Hemi-ellipsoid: the extension of vertex is less than the negative orifice radius, due to negative driving pressure at the start of a pulse; (

**b**) Segment of sphere: the extension of vertex is shorter than the positive orifice radius; (

**c**) First coupling moment: the extension of vertex reaches positive orifice radius, and the hemisphere is discretized by a uniform staggered mesh. The boundary condition of droplet formation model is the mean velocity at nozzle orifice. The blue dashed line is the position of orifice plane, and the red circle is the position of vertex.

**Figure 7.**Evolution of an infinite microthread under harmonic perturbation (k

^{+}= 0.7, Re = 10, ε = 0.05): (

**a**) Breakup moment; (

**b**–

**d**) Post-breakup moments. The dash line represents initial shape of the microthread.

**Figure 8.**Single sphere droplet flying without disturbance: (

**a**) Droplet shape at t

^{+}= 1; (

**b**) Droplet shape at t

^{+}= 2; (

**c**) Momentum history; (

**d**) Mass history.

**Figure 9.**Short filament contraction (α = 4.5, Oh = 0.1): (

**a**) Comparison of initial and current shape; (

**b**) Comparison of time histories of the filament length between 1D and 2D simulations.

**Figure 10.**Contraction of long liquid filament: (

**a**,

**c**,

**d**,

**e**,

**g**) Shape evolution with different Oh; (

**b**,

**f**,

**h**) Filament length vs time with different Oh.

**Figure 13.**Comparison between 1D and 2D methods: (

**a**) Accumulative liquid volume from the orifice plane; (

**b**) Mean velocity.

**Table 1.**Comparison of scaled breakup times from our 1D simulation (third row) with Ashgriz’s (first row) 2D simulation and Furlani’s 1D simulation (second row).

ε = 0.05 | ||||
---|---|---|---|---|

k^{+} = 0.2 | k^{+} = 0.45 | k^{+} = 0.7 | k^{+} = 0.9 | |

Re = 200 | 25.213 | 12.911 | 10.035 | 14.495 |

25.0 | 12.6 | 9.7 | 11.0 | |

25.036 | 12.722 | 9.767 | 11.098 | |

Re = 10 | 26.683 | 14.299 | 11.623 | 14.83 |

27.9 | 14.3 | 11.4 | 14.4 | |

27.005 | 14.306 | 11.480 | 14.523 | |

Re = 0.1 | 230.6 | 243.2 | 311.9 | 628.2 |

227 | 238 | 305 | 634 | |

234.025 | 245.748 | 313.740 | 642.686 |

$\mathbf{t}/\left({\mathit{r}}_{0}\sqrt{\frac{\mathit{\rho}{\mathit{r}}_{0}}{\mathit{\sigma}}}\right)$ | $\mathbf{P}/\left(\frac{{\mathit{r}}_{0}}{\mathit{\sigma}}\right)$ |
---|---|

0.00–0.21 | −60 |

0.21–0.82 | +80 |

0.82–1.43 | +60 |

1.43– | +1 |

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Jiang, H.; Tan, H.
One Dimensional Model for Droplet Ejection Process in Inkjet Devices. *Fluids* **2018**, *3*, 28.
https://doi.org/10.3390/fluids3020028

**AMA Style**

Jiang H, Tan H.
One Dimensional Model for Droplet Ejection Process in Inkjet Devices. *Fluids*. 2018; 3(2):28.
https://doi.org/10.3390/fluids3020028

**Chicago/Turabian Style**

Jiang, Huicong, and Hua Tan.
2018. "One Dimensional Model for Droplet Ejection Process in Inkjet Devices" *Fluids* 3, no. 2: 28.
https://doi.org/10.3390/fluids3020028