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Steady advection–diffusion in polygonal microfluidic mixers

Published online by Cambridge University Press:  04 December 2025

Etienne Boulais Open the ORCID record for Etienne Boulais [Opens in a new window] ,
Aniket Udepurkar Open the ORCID record for Aniket Udepurkar [Opens in a new window] ,
Cedric Devos Open the ORCID record for Cedric Devos [Opens in a new window] ,
Martin Z. Bazant Open the ORCID record for Martin Z. Bazant [Opens in a new window] ,
Allan S. Myerson  and
Richard D. Braatz
Etienne Boulais
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Aniket Udepurkar
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Cedric Devos
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Martin Z. Bazant
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Allan S. Myerson
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Richard D. Braatz*
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
*
Corresponding author: Richard D. Braatz, braatz@mit.edu
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Abstract

We present theoretical models for flow and diffusion in microfluidic polygonal mixers of arbitrary shapes. Combining work based on Boussinesq coordinates with modern methods for the calculation of the Schwarz–Christoffel transform, we present an integrated method that yields analytical solutions for both flow and concentration profiles everywhere in microfluidic mixers with arbitrary numbers of inlets. We illustrate how the problem can be reduced to a sequence of conformal maps to a known domain, where the advection–diffusion problem can be readily solved, and map back the solution to the geometry of interest. We use the method to model a number of previously published microfluidic mixer geometries, used in lipid nanoparticle synthesis, among others. The method is also applicable to other problems described by planar transport equations in polygonal domains, for instance, in groundwater flows or electrokinetics.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Mass Transport: Coupled diffusion and flow Micro-/Nano-fluid dynamics: Microfluidics

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 1024 , 10 December 2025 , A36
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© The Author(s), 2025. Published by Cambridge University Press

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