Continuum description of flows in micro- and nano-systems requires ad hoc addition of effects such as slip at walls, surface diffusion, Knudsen diffusion and others. While all these effects are derived from various phenomenological formulations, a sound theoretical ground unifying these effects and observations is still lacking. In this paper, adopting the definition and existence of various type of flow velocities beyond that of the standard mass velocity, we suggest derivation of model boundary conditions that may systematically justify various diffusion process occurring in micro- and nano-flows where the classical continuum model breaks down. Using these boundary conditions in conjunction with the classical continuum flow equations we present a unified derivation of various expressions of mass flow rates and flow profiles in micro- and nano-channels that fit experimental data and provide new insights into these flow profiles. The methodology is consistent with recasting the Navier–Stokes equations and appears justified for both gas and liquid flows. We conclude that these diffusion type of boundary conditions may be more appropriate to use in simulating flows in micro- and nano-systems and may also be adapted as boundary condition models in other interfacial flow modelling.

Over the past two decades, several researchers have presented experimental data from pressure-driven water flow through carbon nanotubes quoting mass flow rates which are four to five orders of magnitude higher than those predicted by the Navier-Stokes equations with no-slip condition. The current work examines the development of an OpenFOAM solver for creeping flows that better accounts for some micro-and nano-scale diffusion processes. It is based on the observation that a change of velocity variable within the classical Navier-Stokes equations leads to a form of flow model with additional diffusive terms which become apparent at the micro-and nano-scale. Numerical simulations from the new solver compare well with associated analytical solutions that match the experimental flow enhancement observed in cylindrical tubes. This lays the foundations for further investigations of liquid flows in more complex nano-sized geometries, such as those obtained from pore-scale imaging.