Buoyancy-driven drop generation via microchannel revisited

Abstract

Droplets formed under the buoyancy force in a quiescent continuous phase provides a facile yet interesting system for fundamental studies on the drop-rupture mechanism, which was a field of vast research interest prior to the arrival of the microfluidic techniques. The formation of macro-drops via a microchannel in a buoyancy-driven system is revisited for three low-viscosity hydrocarbon oils using a wide range of surfactant concentrations. The dripping-to-jetting transition was found to occur at a Bond number of ~0.85, reflecting the significance of buoyancy force even in the jetting regime. The pinch-off time and the satellite drops were found to have negligible impact on the final drop volume throughout the dripping regime. A modified force balance model is presented which includes the dynamic interfacial tension based on the surface-average age of the drop. The model clearly predicts a region where the drop size increases initially with dispersed phase flow rate (Q) due to increasing dynamic interfacial tension, followed by a region of constant drop size with Q, where the increase in interfacial force is compensated by an equal increase in the kinetic force. Polydispersity in the drops formed under the jetting regime was found to vary in accordance with the polydispersity in the break-up length of the drops. Interestingly, highly monodisperse drops were not limited to dripping regime and formed under the jetting regime at a higher frequency too, but only under certain flow rate and interfacial tension conditions.

Appendix: Derivation of surface-average age (t av) of a growing spherical drop

We assume the drop is always spherical while growing at the tip of the channel. Q is the constant volumetric flow rate feeding the drop; A is the surface area of the drop having radius r at any time t.

The interface created at the beginning of the drop formation has the longest age, which is equivalent to the time of drop formation t _f. Similarly, the surface generated at the point of drop detachment has the shortest (zero) surface age. Thus, the surface-average age of a drop, which is eventually ruptured at t _f with volume V _f, area A _f and radius r _f, can be expressed by,

$$\begin{aligned} t_{\text{av}} & = \frac{1}{{A_{\text{f}} }}\mathop \int \limits_{0}^{{A_{\text{f}} }} (t_{\text{f}} - t) dA \\ t_{\text{av}} & = t_{\text{f}} - \frac{1}{{QA_{f} }}\mathop \int \limits_{0}^{{r_{\text{f}} }} \frac{{4\pi r^{3} }}{3} 8\pi rdr \\ t_{\text{av}} & = t_{\text{f}} - \frac{1}{{QA_{\text{f}} }}\left( {\frac{{2V_{\text{f}} A_{\text{f}} }}{5}} \right) \\ t_{\text{av}} & = t_{\text{f}} - \frac{2}{5}t_{\text{f}} = 0.6t_{\text{f}} \\ \end{aligned}$$

Note that the surface-average age of the drop is independent of volumetric flow rate.