  We present a complete analysis of the problem of convection-diffusion in low
Re, 2-dimensional 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 Pe. These yield the complete analytical concentration
profile at every point in applications that previously relied on material
surface tracking, local lump models or numerical analysis such as microfluidic
probes, groundwater heat pumps, or diffusive flows in porous media. 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 that contain the
explicit dependence on the various parameters of the problems, such as Pe, the
spacing of the apertures and their relative injection and aspiration rates. In
particular, we show that the high Pe models can model problems with Pe as low
as 1 with a maximum error committed of under $10\%$, and that this error
decreases approximately as $Pe^{-1.5}$.
