Active control of jet breakup and droplet formation using temperature modulation

Using a slender-jet approach, we numerically investigate the control of jet breakup using temperature modulation at the nozzle with a specified frequency and amplitude. Our results show that temperature modulation does lead to instability through capillary and Marangoni stresses, providing control of the droplet formation in terms of intact length and resultant drop size distribution, which is otherwise irregular due to inevitable presence of background noise. For understanding the mechanisms underpinning the breakup of a thermally modulated jet in the presence of noise, it is useful to decompose the surface tension forces into a contribution from curvature-gradient forces and a contribution from surface tension-gradient forces, associated with axial variations in the jet curvature and the temperature-dependent surface tension coefficient, respectively. We show that in the limit of slow axial heat diffusion and slow cooling to the ambient, as considered here, the breakup of a thermally modulated jet is governed by the ratio of the surface tension-gradient force to the imposed random perturbation force at nozzle exit. This so-called “thermal modulation strength number” depends on the amplitude and frequency of the thermal modulation, the sensitivity of the surface tension coefficient to variations in temperature, the Weber number, and the strength of the Gaussian white noise added to the nozzle exit velocity. We show that the thermal modulation strength number governs the shift in breakup characteristics from forward to rear pinchoff for increasing modulation strength as well as the nature of the instability. When thermal modulation is weak, the surface tension-gradient forces act only as a trigger, and curvature-gradient forces soon take over and grow exponentially downstream from the jet due to inertio-capillary growth. When thermal modulation is strong, the surface tension-gradient forces not only act as a trigger, but remain significant until breakup. The thermal modulation strength number is thus useful to the design of thermal modulation in practical applications as a possible alternative to often-used mechanical excitation mechanisms to control jet breakup.

Figure 1

Illustration of the temperature-controlled jet breakup where the temperature at the nozzle, $T_{\mathrm{nozzle}}$, is modulated to perturb the capillary stress, $\gamma \left(T\right)\kappa$, and the Marangoni stress, ${\nabla }_s\gamma$.

Figure 2

A schematic of the staggered grid.

Figure 3

Random velocity fluctuations imposed at the nozzle, the standard deviation $\sigma$ scales with $A\mathrm{\Delta }t$, where $\mathrm{\Delta }t$ is the adaptive time step, and $A$ is a free constant that adjusts the strength of the fluctuation.

Figure 4

A comparison of the isothermal reference case (black) and a temperature-modulated case (red) for ${\varepsilon }_T=0.05, {\mathrm{\Pi }}_{\gamma }=0.05$, and $\omega =0.7$. Parts of the jet where $R<R_c$ are not shown to avoid confusion with the beads-on-a-string structure seen in viscoelastic liquids.

Figure 5

The intact length as a function of modulation frequency and ${\mathrm{\Pi }}_{\gamma }$; the temperature modulation amplitude is fixed at ${\varepsilon }_T=0.05$. The gray shaded region covers one standard deviation above and below the mean intact length for the reference case (i.e., ${\mathrm{\Pi }}_{\gamma }=0$).

Figure 6

Comparison of the drop size distributions of the isothermal reference case and temperature-modulated case shown in Fig. 4: (a) at the tip of the intact jet, $z=L_b$, and (b) downstream, $z=280R_0$. The parameters for the perturbed case are ${\varepsilon }_T=0.5, \omega =0.7$, and ${\mathrm{\Pi }}_{\gamma }=0.05$.

Figure 7

Average intact and merging lengths of the jet with respect to the modulation on the surface tension coefficient for $\omega =0.7$. Insets show the two cases where the pinch-off characteristic shifts from (a) forward pinchoff to (b) rear pinchoff.

Figure 8

Growth of the stress perturbations for different surface tension sensitivities: (a) ${\mathrm{\Pi }}_{\gamma }{\varepsilon }_T=0.0025$, (b) ${\mathrm{\Pi }}_{\gamma }{\varepsilon }_T=0.025$, and (c) ${\mathrm{\Pi }}_{\gamma }{\varepsilon }_T=0.25$. The modulation frequency, $\omega$, is fixed at 0.7.

Figure 9

Free surface profiles from our 1D model with the regularization scheme. Our results are in near perfect agreement with the 1D- and 2D-axisymmetric simulations shown in Fig. 11 of McIlroy and Harlen [44]. The model parameters are $\text{We}=338, \text{Oh}=0.122, k{R}_0=0.7$. The amplitude of velocity perturbations are (a) ${\varepsilon }_v=0.1$ and (b) ${\varepsilon }_v=0.15$. Shift from inverted pinching back to downstream pinching is reproduced with our code as well.

Figure 10

Filament shapes from the case described in Sec. 3.1 of Ref. [24].