Universal size and shape of viscous capillary jets: application to gas-focused microjets

The size and shape of capillary microjets are analysed theoretically and experimentally. We focus on the particular case of gas-focused viscous microjets, which are shaped by both the pressure drop in the axial direction occurring in front of the discharge orifice, and the tangential viscous stress caused by the difference between the velocities of the co-flowing gas stream and liquid jet behind the orifice. The momentum equation obtained from the slender approximation reveals that the momentum injected into the jet in these two regions is proportional to the ratio of the pressure drop to the orifice diameter. Thus, the liquid-driving forces can be reduced to a single term in the momentum equation. Besides, the size and shape of gas-focused microjets were experimentally measured. The experiments indicated that the Weber number has a minor influence on the jet diameter for steady, stable jets, while both the axial coordinate and the Reynolds number affect its size significantly. When the experimental results are expressed in terms of conveniently scaled variables, one obtains a remarkable collapse of all measured jet diameters into a single curve. The curve matches a universal self-similar solution of the momentum equation for a constant driving force, first calculated by Clarke (Mathematika, vol. 12, 1966, p. 51) and not yet exploited in the field of steady tip-streaming flows, such as flow focusing and electrospray. This result shows that the driving force or motor mentioned above attains a rather homogeneous value at the region where the gas-focused microjet develops. The approach used in this work can also be applied to study other varied microjet generation means (e.g. co-flowing, electrospray and electrospinning).