Spectral properties and universal behaviour of advecting-diffusing scalar fields in finite-length channels

This paper analyses the relaxation towards the steady state of an advecting-diffusing field in a finite-length channel. The dominant eigenvalue, -Lambda(F), of the advection-diffusion operator provides the slowest relaxation time scale for achieving steady state in open flow devices. We focus on parallel flows and analyse how Lambda(F) depends on the velocity profile and the molecular diffusivity. As a result of the universal localization features of the eigenfunction associated with Lambda(F), we find that Lambda(F) can be predicted analytically based on the local behaviour of the velocity profile near the stagnation points. Microfluidic applications of the theory are also addressed.