Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries

We adress the question of the stability of a confined coflowing jet at low Reynolds number in various geometries. Our study is motivated by recent experiments in microfluidic devices. When immiscible fluids flow in microchannels, either monodisperse droplets or parallel flows are obtained depending upon the flow rate of the aqueous phase and the oil phase. In these experiments, the confining and the shape of the geometry play a fundamental role. In a previous paper [Guillot, Phys. Rev. Lett 99, 104502 (2007)], we analyzed the stability of the jet in the framework of the lubrication approximation at low Reynolds number in a cylindrical geometry, and we related the transition between the droplets regime and the jet regime to the absolute-convective transition of the Rayleigh plateau instability. In this work, the effect of the channel geometry and the jet position within the microfluidic device are discussed. New flow patterns are pointed out. Bidimensional jets are encountered in square and rectangular geometry. Contrary to jets occuring in circular geometry, these two-dimensional jets are absolutely stable. Focusing on situations where the inner fluid is more viscous than the outer one, we evidence a range of parameters where droplets are produced through a blocking and pinching mechanism. In this particular case, the flow is unstable, the growing perturbations are convected upstream. This induces the clogging of the channel by the internal phase and its pinching by the external one. In a future presentation we will give a comparison between this model and experimental data.