Effects of drive amplitude on continuous jet break-up

We develop a one-dimensional model of jet breakup in continuous inkjet printing to explore the nonlinear behavior caused by finite-amplitude modulations in the driving velocity, where jet stability deviates from classic (linear) “Rayleigh” behavior. At low driving amplitudes and high Weber numbers, the spatial instability produces drops that pinch-off downstream of the connecting filament, leading to the production of small satellite droplets between the main drops. On the other hand, we identify a range of driving amplitudes where pinching becomes “inverted,” occurring upstream of the filament connecting the main drops, rather than downstream. This inverted breakup is preferable in printing, as it increases the likelihood of satellite drops merging with the main drops. We find that this behavior can be controlled by the addition of a second harmonic to the driving signal. This model is in quantitative agreement with a full axisymmetric simulation, which incorporates nozzle geometry.We develop a one-dimensional model of jet breakup in continuous inkjet printing to explore the nonlinear behavior caused by finite-amplitude modulations in the driving velocity, where jet stability deviates from classic (linear) “Rayleigh” behavior. At low driving amplitudes and high Weber numbers, the spatial instability produces drops that pinch-off downstream of the connecting filament, leading to the production of small satellite droplets between the main drops. On the other hand, we identify a range of driving amplitudes where pinching becomes “inverted,” occurring upstream of the filament connecting the main drops, rather than downstream. This inverted breakup is preferable in printing, as it increases the likelihood of satellite drops merging with the main drops. We find that this behavior can be controlled by the addition of a second harmonic to the driving signal. This model is in quantitative agreement with a full axisymmetric simulation, which incorporates nozzle geometry.


I. INTRODUCTION
Inkjet printing is becoming a powerful manufacturing tool; it is ideal for a wide range of applications due to the advantages of being flexible, non-contact and scalable.
In particular, drop-on-demand (DoD) inkjets may be used as robotic pipettes to create micro arrays, fabricate three-dimensional objects or print electrical and optical devices 1 . Rayleigh 11 was first to establish that a liquid jet will be rendered unstable by surface tension provided that its wavelength exceeds its circumference. Linear stability analysis of the Navier-Stokes equations leads to a dispersion relation to describe how the growth rate of a disturbance depends on its wave-length. Maximising the dispersion relation reveals that the fastest growing wavelength is approximately 9 times the jet radius for low-viscosity liquids. However, linear theories based on Rayleigh's stability analysis do not predict the formation of satellite drops.
Consequently, weakly non-linear theories have been developed to investigate satellite formation. In particular, both Yeun et al. 12 and Lee et al. 13    show experimentally that this break-up behaviour can be controlled by forcing the jet with a suitable harmonic component added to the initial velocity profile. However, these studies do not explore the effect of changing Weber number, which has been shown by Vassallo et al. 20 to have a significant effect.
In order to control break-up and increase printing speed, CIJ printing exploits the effects of finite-amplitude modulations in the driving velocity profile. In particular, the driving amplitude for which inverted breakup is achieved is considered optimal for CIJ printing, since satellite drops can be elimi-  Denoting jet radius h(z, t) and velocity v(z, t), conservation of mass and momentum are given respectively by The (full nonlinear) curvature term is given by where the subscript denotes differentiation with respect to z and the dimensionless Ohnesorge number is for viscosity µ, density ρ and surface tension γ.
The jet velocity is non-dimensionalised with respect to the nozzle radius R and Rayleigh capillary time ρR 3 /γ so that the dimensionless velocity is defined in terms of the Weber number for mean jet velocity U . That is, the mean initial dimensionless velocity at the nozzle exit is given by The Reynolds' number is then given by Performing a linear stability analysis on Eqs. 1 yields the dispersion relation for dimensionless growth rate α and wavenumber k. The maximum growth rate is given by Thus, in the limit of small Ohnesorge number α * ≈ 1/3, and corresponds to wavenumber k * ≈ 0.7 or equivalently wavelength λ * ≈ 9, as shown in Ref. 23 .
The governing equations (1) are solved using a semi-implicit numerical scheme on a Eulerian grid for a range of boundary conditions chosen to replicate different driving methods, as discussed in the next section.
Further details of the numerical method are given in Ref. 23 . We then compare our results to Rayleigh's dispersion relation (Eq. 7).

B. Driving Profiles
In the frame-work of our 1D model, the details of the nozzle geometry are neglected and we consider dynamics outside the nozzle. In order to drive an instability, we can impose two different driving profiles: either a perturbation of the cross-sectional area at the nozzle exit, or a perturbation of the velocity profile at the nozzle exit.
Perturbations to the cross-sectional area mimic thermal fluctuations in the nozzle 1 , and a similar approach has been taken by van Hoeve et al. 7 . In our model, the crosssectional area is perturbed at the nozzle exit to induce a free-surface perturbation. Here ǫ is the driving amplitude and f is the driving frequency. In this case, the velocity profile is constant (unperturbed) at the nozzle exit, where v 0 is given by Eq. 5.
For small amplitudes (ǫ ≤ 0.01), a (sinusoidal) Rayleigh instability wave is propagated downstream from the nozzle exit, provided that the Weber number is sufficiently The jetting frequency is defined as for dimensionless wavelength λ. By choosing λ = λ * ≈ 9, the fastest growing disturbance dominates the flow, in the limit of small amplitude disturbances and low viscosity.
On the other hand, perturbations in the velocity profile mimic jets driven by a pressure modulation. In our model the velocity profile is perturbed at the nozzle exit via where v 0 is given by Eq. 5 and the jet radius fixed at Perturbations of the velocity profile do not necessarily translate to a sinusoidal variation in the free-surface height and so the instability is not necessarily related to a typical Rayleigh wave. Furthermore, industrial CIJ printers also typically operate at large modulation amplitudes (ǫ > 0.01) meaning that non-linear interactions are important.

C. Break-up Criterion
We define a break-up criterion to be when h becomes less than a cut-off radius h c , which we typically set at 1% of the nozzle radius.    The main difference between the two driving methods is that driving the velocity profile generates a significantly shorter jet compared to driving the free-surface at the nozzle exit. It has been shown that the break-up length L of the fastest growing mode is given where α * is the growth rate of the fastest mode (Eq. 8  of the surface-driven jet (Fig.4). Thus, the prefactor A in Eq. 14 differs by a factor ∼ 2 depending on the driving mechanisms.
In fact, the predicted break-up length of the velocity-driven jet is equal to that obtained with a cross-sectional area perturbation of amplitude ǫ ∼ 0.15.
From linear stability analysis of the fastest growing mode, the prefactor A is determined by whereh = 1 is the mean jet radius, and h ′ is the amplitude of the disturbance wave. Furthermore, for a velocity perturbation of amplitude v ′ , it is found that the amplitude of the resulting free-surface disturbance is given by B. Non-linear (ǫ > 0.01) behaviour As shown earlier (Fig.3) In order to highlight this non-linear behaviour, Fig.8 shows how the predicted break-up length of a velocity-driven jet with ǫ = 0.05 diverges from the inverse of Rayleigh's dispersion relation (Eq.7); breakup length increases with wavelength, rather than obeying the linear theory as seen for smaller driving amplitudes (Fig.6). More- over, the fastest growing disturbance wave is found to have wavelengths smaller than the classic Rayleigh wave (λ * ≈ 7, rather than λ * ≈ 9). This non-linear effect has also been observed in experiments 27 .
It is also worth noting that if we apply enough forcing this non-linear jet does not appear to stabilise at λ ≈ 2π. Furthermore, the model predicts a similar deviation from the linear theory for larger amplitudes ǫ = 0.1 and 0.15 (not shown). Finally, the stability of a surface-driven jet also deviates from Rayleigh's theory. However in contrast to the the velocity-driven instability the fastest growing disturbance wave shifts to slightly larger wavelength λ * ≈ 10 ( Fig.8).
At this large amplitude, the surface variation produced by modulating the driving velocity is not similar to a sinusoidal wave; the jet velocity is distorted due to the non-linear advection term appearing in conservation on Due to this non-linearity, the peak of the velocity profile travels faster than the trough, so that the pulse becomes accumulatively more like a sawtooth wave and generates a 'shock' in the velocity profile, as sketched in Fig.9.
Consequently, fluid upstream of the shock moves faster than the fluid downstream causing steep bulges to form on the uniform thread. This behaviour is evident is the freesurface predicted by the 1D model shown in jets is restricted to a narrow operating window, as discussed earlier (Fig.2).

C. Full axisymmetric simulations
Full axisymmetric simulations were performed using the method of Harlen et al 28 to validate the 1D model. The code uses a Eulerian-Lagrangian finite-element method 29 to capture the evolving free-surface shape and has previously been used to study jet break-up in drop-on-demand printing for both Newtonian 30 and viscoelastic fluids 28 and CIJ printing of Newtonian fluids 6 .
The software uses a moving-mesh, finiteelement method to solve the Navier-Stokes for axisymmetric jet velocity u = (u r , 0, u z ), pressure p and stress tensor σ along with the incompressibility condition By allowing the finite elements to deform with the fluid velocity, the Newtonian con- stitutive equation is solved in the co-deforming frame for viscosity µ and velocity gradient tensor K ij = ∂u i ∂x j . At the fluid-air interface the boundary condition is defined to be wheren is the unit vector normal to the interface, γ is surface tension and R 1 , R 2 are the principle radii of curvature. For free-surface problems such as inkjet printing, this method naturally captures the freesurface shape. Further details of the numerical scheme can be found in Refs. 28,29,31 .
The shape of the simulated print head is chosen to replicate the shape of nozzle used in experiments 6 , which is similar to that of a CIJ nozzle, while simplifying the interior of the actual print head behind the nozzle by assuming axisymmetry (real print heads are typically non-axisymmetry). The initial finite-element grid, with a nozzle aspect ratio 1, is shown in Fig. 10 and has previously been described by Casterjon-Pita et al. 6 . For an unmodulated jet with mean velocity U at the nozzle outlet, the magnitude of the velocity applied at the inlet is where R is the nozzle radius and A in is the surface area of the print head inlet surface.
To simulate a jet with modulation of frequency f and amplitude ǫ, the inlet velocity is prescribed in terms of time t as u noz (t) = u 0 (1 + ǫ sin(2πf t)).  To understand this secondary instability wave, we decompose the free-surface profile where f is the Rayleigh frequency. The magnitude of the Fourier coefficients enables the magnitude of the secondary harmonic to be compared over a range of amplitudes.
Moreover, for the case ǫ = 0.15 the filament radius is h f ≈ 0.5 (Fig.3). Thus, the stability criterion k 4 h f > 1 is satisfied, causing break-up to occur downstream of the filament region.
The addition of harmonics to the initial velocity profile can have a significant effect on the break-up behaviour of a continuous inkjet. In particular, Chaudhary et al 19 have shown theoretically that the formation of satellites can be controlled by forcing the jet with a suitable harmonic added to the fundamental.
As an example, we examine the effect of adding a secondary harmonic to the driving velocity profile such that v(0, t) = v 0 (1 + ǫ sin(2πf t) + ǫ sin(4πf t + θ)).
The phase of this second harmonic is given by θ and its amplitude is equal to that of the fundamental ǫ, as in the work of Chaudhary et al. 19 . We observe a distinct change that is found during inversion relative to the fundamental solution (see Fig.14).