Two-dimensional convection–diffusion in multipolar flows with applications in microfluidics and groundwater flow

Abstract

Advection–diffusion in two-dimensional plane flows plays a key role in numerous transport problems in physics, including groundwater flow, micro-scale sensing, heat dissipation, and, in general, microfluidics. However, transport profiles are usually only known in a purely convective approximation or for the simplest geometries, such as for quasi one-dimensional planar microchannels. This situation greatly limits the use of these models as design tools for fully 2D planar flows. We present a complete analysis of the problem of convection–diffusion in low Reynolds number 2D flows with distributions of singularities, such as those found in open-space microfluidics and in groundwater flows. Using Boussinesq transformations and solving the problem in streamline coordinates, we obtain concentration profiles in flows with complex arrangements of sources and sinks for both high and low Peclet numbers. These yield the complete analytical concentration profile at every point in applications such as microfluidic probes, groundwater heat pumps, or diffusive flows in porous media, which previously relied on material surface tracking, local lump models, or numerical analysis. Using conformal transforms, we generate families of symmetrical solutions from simple ones and provide a general methodology that can be used to analyze any arrangement of source and sinks. The solutions obtained include explicit dependence on the various parameters of the problems, such as Pe, the spacing of the apertures, and their relative injection and aspiration rates. We then show how these same models can be used to model diffusion in confined geometries, such as channel junctions and chambers, and give examples for classic microfluidic devices such as T-mixers and hydrodynamic focusing. The high Pe models can model problems with Pe as low as 1 with a maximum error committed of under 10%, and this error decreases approximately as Pe−1.5.