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. 2023 Mar 23;13:4728. doi: 10.1038/s41598-023-31895-z

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© The Author(s) 2023, corrected publication 2024

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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(a) Three-dimensional numerical simulation result for ethanol concentration at steady state. (b) Ethanol concentration at the yz-plane. Cut plane is placed at x = 250 µm, where the cross-junction end and long diffusive mixing section starts. Lagrangian content of the solvent over time is shown plotted in relation to the (c) flow rate ratio (FRR) and (d) solvent type. Whole data were processed by MATLAB R2019B (MathWorks, Inc., Natick, MA, USA) using Eqs. (3) and (4).