Tracer dispersion in inertial flow of an ideal gas through spatially-periodic channels: a new class of problems in Brenner’s macrotransport paradigm

Predicting solute dispersion in channels possessing a small-scale periodic structure is a key step for the optimal design of a number of microfluidics-based processes. Because of the intrinsic multiscale nature of these processes, direct numerical approaches to solving the time-dependent advection-diffusion equation are practically unfeasible. In incompressible spatially periodic flows, Brenner’s macrotransport paradigm provides a valuable alternative to the direct full-scale approach in that it allows to predict the axial dispersion coefficient of the solutes solely by solving a steady-state advection-diffusion equation defined on the minimal periodic cell of the channel. However, the same strategy cannot be straightforwardly enforced in the case of gas flow under large pressure drops, since the velocity field is not spatially-periodic and the diffusion coefficient is position-dependent. Based on a recent result providing a factorization of gas velocity, density and pressure in terms of large- and small-scale components, we here extend Brenner’s approach to compressible flows through periodic channels and propose a generalized version of the 1D Brenner’s effective transport equation, where the transport coefficients can be strongly dependent on the large-scale axial coordinate. The approach is validated by comparison with statistical averaging of a Langevin-type equation mimicking the full-scale advective-diffusive process, using the compressible Poiseuille flow through a cylindrical capillary as benchmark. The application of the extended macrotransport framework to gas flow through fully 3d periodic geometries defines a new class of macrotransport problems, where the lack of axial symmetry entails a strong impact of flow inertia on axial dispersion.