Characterizing relaxation timescales and overall steady-state efficiency of continuous inflow-outflow micromixers

Most of the efforts for developing a consistent theory of mixing have mainly considered two classes of problems, namely mixing in closed bounded flows and dispersion in infinitely extended domains. However, micorfluidic devices are typically bounded inflow-outflow systems, and a tailored approach to address quantitatively their mixing properties must be considered. Based on the spectral (eigenvalue-eigenfunction) structure of the advection-diffusion operator, we develop a quantitative framework for describing both transient and steady-state mixing in open flow devices. We show that the relaxation timescale for reaching the steady state condition can be conveniently quantified by the Frobenius eigenvalue of the advection-diffusion operator associated with the assigned boundary value problem. The steady-state homogenization process along the mean streamwise direction of the mixer can instead be approached by considering the spectral properties of the linear operator that maps the inlet profile into the corresponding profile at the generic mixer cross-section. Concrete examples of physically realizable flows, ranging from a simple 2d channel flows to a three-dimensional helical flow giving rise to complex kinematic structures are thoroughly analyzed and used as benchmark test cases to illustrate the spectral characterization of mixing flow devices.