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.

中文翻译：

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多极流中的二维对流扩散在微流体和地下水流中的应用

二维平面流中的平流扩散在物理学中的许多传输问题中起着关键作用，包括地下水流、微尺度传感、散热，以及一般的微流体。然而，传输剖面通常仅在纯对流近似或最简单的几何形状中已知，例如对于准一维平面微通道。这种情况极大地限制了这些模型作为全二维平面流设计工具的使用。我们对具有奇点分布的低雷诺数二维流中的对流扩散问题进行了完整分析，例如在开放空间微流体和地下水流中发现的那些。使用 Boussinesq 变换并解决流线坐标中的问题，我们获得了高和低佩克莱特数的源和汇的复杂排列的流动中的浓度分布。这些在微流体探针、地下水热泵或多孔介质中的扩散流等应用中的每个点产生完整的分析浓度分布，这些应用以前依赖于材料表面跟踪、局部块模型或数值分析。使用保形变换，我们从简单的解决方案生成对称解决方案族，并提供可用于分析源和汇的任何排列的通用方法。获得的解决方案包括对问题的各种参数的明确依赖，例如 Pe、孔的间距以及它们的相对注射和吸入速率。然后，我们展示了如何使用这些相同的模型来模拟受限几何结构中的扩散，例如通道连接和腔室，并举例说明经典微流体设备，例如 T 型混合器和流体动力学聚焦。高 Pe 模型可以对 Pe 低至 1 的问题进行建模，最大误差低于 10%，并且该误差大约降低为 Pe-1.5。