Despite single-phase parallel flows for which analytical solutions are available for species conservation equation in its most general form, little progress has been made in analytical modeling of cross-stream diffusion in stratified multiphase flows. The main reason is that solute concentration and fluid properties are discontinuous at the interface of phases. In the present study, a new method of solution is presented for the analytical treatment of solute transport in stratified multiphase flows. The solution methodology starts with developing separate species conservation equations for the phases, which, upon non-dimensionalization, are replaced with a single equation for which series solutions are obtained utilizing the variational calculus. Using the method proposed, analytical solutions are obtained for mass transport in microfluidic two-phase extraction by taking the influences of the non-uniform velocity and the axial diffusion effects into account. The inclusion of the non-uniform velocity effects, which leads to 3D solutions, enables us to capture the heterogeneous transport of solutes, a phenomenon that is ignored by the available simple 2D solutions. The results indicate that axial diffusion in stratified multiphase flows is significantly more important than in single-phase flow and is not characterized solely by the Péclet number. It is also found that the only parameter controlling the length required for a complete extraction is the solvent-to-solution viscosity ratio.

尽管单相平行流具有最一般形式的物种保护方程式可用的解析解，但在分层多相流中的横流扩散的解析模型方面进展甚微。主要原因是在相界面处溶质浓度和流体性质不连续。在本研究中，提出了一种用于分层多相流中溶质运移分析处理的新方法。该解决方案方法学始于为各相开发单独的物种守恒方程，在无量纲化时，将其替换为一个单个方程，利用变分演算可为其获得级数解。使用建议的方法，通过考虑非均匀速度和轴向扩散效应的影响，获得了微流体两相萃取中传质的解析解。包含非均匀速度效应的3D解决方案使我们能够捕获溶质的异质传输，这种现象被可用的简单2D解决方案所忽略。结果表明，分层多相流中的轴向扩散比单相流中的轴向扩散要重要得多，并且不能仅通过佩克利数来表征。还发现控制完全萃取所需长度的唯一参数是溶剂与溶液的粘度比。包含非均匀速度效应的3D解决方案使我们能够捕获溶质的异质传输，这种现象被可用的简单2D解决方案所忽略。结果表明，分层多相流中的轴向扩散比单相流中的轴向扩散要重要得多，并且不能仅通过佩克利数来表征。还发现控制完全萃取所需长度的唯一参数是溶剂与溶液的粘度比。包含非均匀速度效应的3D解决方案使我们能够捕获溶质的异质传输，这种现象被可用的简单2D解决方案所忽略。结果表明，分层多相流中的轴向扩散比单相流中的轴向扩散要重要得多，并且不能仅通过佩克利数来表征。还发现控制完全萃取所需长度的唯一参数是溶剂与溶液的粘度比。结果表明，分层多相流中的轴向扩散比单相流中的轴向扩散要重要得多，并且不能仅通过佩克利数来表征。还发现控制完全萃取所需长度的唯一参数是溶剂与溶液的粘度比。结果表明，分层多相流中的轴向扩散比单相流中的轴向扩散要重要得多，并且不能仅通过佩克利数来表征。还发现控制完全萃取所需长度的唯一参数是溶剂与溶液的粘度比。