A hybrid numerical approach for predicting mixing length and mixing time in microfluidic junctions from moderate to arbitrarily large values of the Péclet number

We investigate numerically the homogenization process of a diffusive species in a mixing channel of arbitrary length downstream a microfluidic cross-junction. The channel length, λα, necessary to achieve a prescribed level of mixedness, α, is targeted as primary quantity of interest, and its dependence on the Reynolds number, Re, on the flow ratio between the impinging currents, R, and on the Schmidt number of the solute, Sc, is analyzed. The accurate numerical solution of the mass transport equations up to values Pe=ReSc≃106 of the Péclet number is here made possible by a hybrid numerical approach. This approach combines a recently proposed Monte Carlo method, enforced near the impinging zone, with a pseudo-transient 2D formulation of mass transport in the mixing channel, where purely axial (Poiseuille) flow settles in. At values of the flow ratio significantly different from unity (e.g. those used in flash nanoprecipitation) a non-trivial dependence of λα on Re is found at fixed Sc and R. This result is interpreted based on the spectral (eigenvalue/eigenfunction) structure of the 2D generalized Sturm-Liouville pure diffusion problem defined onto the cross-section of the mixing channel. Hinging on this interpretation, we show that for Sc⩾103 the dependence of λα on Sc at fixed Re and R can be singled out and theoretically predicted. By this property, universal ready-to-use curves yielding the mixing length for each assigned geometry can be constructed, which could be used to correlate the mixing time with other phenomena (chemical reactions, phase changes) occurring alongside the mass transport.