Bubbling regime in planar co-flowing air-water sheets

We study by means of experiments and numerical simulations the periodic breakup of an air sheet surrounded by a co-flowing water sheet, for a fixed liquid- to-gas thickness ratio, $h = h_ {w,0}^*/h_{a,0}^*\simeq 5.27$, as a function of their control parameters, namely the Weber number, $We=\rho_w\,u_{w,0}^{*2} \,h_{a,0}^*/\sigma$, and the water-to-air velocity ratio, $\Lambda=u_{w,0}^*/\bar u_{a,0}^*$, where $h_{a,0}^*$ and $h_ {w,0}^*$ are the half-thicknesses of the air and water sheets at the exit. The bubble formation process is divided into two stages governed by different physical phenomena: the formation of an incipient neck, which moves downstream at the water velocity, and the collapse of the neck. Both stages are characterized and a simple theoretical model, which depends on the control parameters, is proposed for each one in order to provide the time scale for the bubbling process, finding a good agreement with the experiments and simulations. Finally, the experimental measurements of the bubbling frequency and bubble size are compared with those obtained numerically within the $We$- $\Lambda$ parameter space studied.