Bubble size prediction in co-flowing streams

In this paper, the size of bubbles formed through the breakup of a gaseous jet in a co-axial microfluidic device is derived. The gaseous jet surrounded by a co-flowing liquid stream breaks up into monodisperse microbubbles and the size of the bubbles is determined by the radius of the inner gas jet and the bubble formation frequency. We obtain the radius of the gas jet by solving the Navier-Stokes equations for low Reynolds number flows and by minimization of the dissipation energy. The prediction of the bubble size is based on the system's control parameters only, i.e. the inner gas flow rate $Q_i$, the outer liquid flow rate $Q_o$, and the tube radius $R$. For a very low gas-to-liquid flow rate ratio ($Q_i / Q_o \rightarrow 0$) the bubble radius scales as $r_b / R \propto \sqrt{Q_i / Q_o}$, independently of the inner to outer viscosity ratio $\eta_i/\eta_o$ and of the type of the velocity profile in the gas, which can be either flat or parabolic, depending on whether high-molecular-weight surfactants cover the gas-liquid interface or not. However, in the case in which the gas velocity profiles are parabolic and the viscosity ratio is sufficiently low, i.e. $\eta_i/\eta_o \ll 1$, the bubble diameter scales as $r_b \propto (Q_i/Q_o)^\beta$, with $\beta$ smaller than 1/2.