2D Convection-Diffusion in Multipolar Flows

TL;DR

This paper provides a comprehensive analytical framework for 2D convection-diffusion in low Reynolds number flows with singularities, applicable to microfluidics and groundwater, enabling precise concentration profiling across complex source-sink arrangements.

Contribution

It introduces a novel analytical methodology using Boussinesq transformations and conformal mapping to solve convection-diffusion problems with arbitrary source-sink configurations, including high Pe regimes.

Findings

Analytical concentration profiles are derived for complex source-sink arrangements.

High Pe models are accurate for Pe as low as 1, with errors under 10%.

Error decreases approximately as Pe^{-1.5}.

Abstract

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 $Pe^{-1.5}$.