Fig. 5: Modeling of the out-of-phase synchronization mode.

a Droplet-breakup geometry of the out-of-phase mode. We define the smallest gap width between the interfaces as w_g,1, which affects the filling stage of the dispersed phase 1. The distance between the dispersed phase 2 and the wall of the main channel is w_c – b₂. Considering the similarity between these two lengths, we assume w_g,1 = w_c – ξ·b₂ with a correction factor ξ. b Experimentally measured protrusion heights (normalized), bw_c⁻¹, as a function of dimensionless time (t^* = t · \(\frac{{Q_{\mathrm{w}}}}{{w_{\mathrm{c}}w_{\mathrm{d}}h}}\)). The dots on the curves indicate the end of the filling stage in each cycle and the corresponding b_fill. c Model calculation showing the stability of the out-of-phase state. The evolution of the protrusion heights shows how the initial state, arbitrarily chosen, converges to the out-of-phase state of a constant synchronization parameter α. The experimental parameters from the data in (b) are adopted for the model calculation.