Inertia-driven enhancement of mixing efficiency in microfluidic cross-junctions: a combined Eulerian/Lagrangian approach

Mixing of a diffusing species entrained in a three-dimensional microfluidic flow-focusing cross-junction is numerically investigated at low Reynolds numbers, \(1 \le Re \le 150\), for a value of the Schmidt number representative of a small solute molecule in water, \(Sc = 10^3\). Accurate three-dimensional simulations of the steady-state incompressible Navier–Stokes equations confirm recent results reported in the literature highlighting the occurrence of different qualitative structures of the flow geometry, whose range of existence depends on Re and on the ratio, R, between the volumetric flowrates of the impinging currents. At low values of R and increasing Re, the flux tube enclosing the solute-rich stream undergoes a topological transition, from the classical flow-focused structure to a multi-branched shape. We here show that this transition causes a nonmonotonic behavior of mixing efficiency with Re at constant flow ratio. The increase in efficiency is the consequence of a progressive compression of the cross-sectional diffusional lengthscale, which provides the mechanism sustaining the transversal Fickian flux even when the Peclet number, \({Pe=Re \, Sc}\), characterizing mass transport, becomes higher due to the increase in Re. The quantitative assessment of mixing efficiency at the considerably high values of the Peclet number considered (\(10^3 \le Pe \le 1.5 \times 10^5\)) is here made possible by a novel method of reconstruction of steady-state cross-sectional concentration maps from velocity-weighted ensemble statistics of noisy trajectories, which does away with the severe numerical diffusion shortcomings associated with classical Eulerian approaches to mass transport in complex 3d flows.