Numerics made easy: solving the Navier–Stokes equation for arbitrary channel cross-sections using Microsoft Excel


The fluid mechanics of microfluidics is distinctively simpler than the fluid mechanics of macroscopic systems. In macroscopic systems effects such as non-laminar flow, convection, gravity etc. need to be accounted for all of which can usually be neglected in microfluidic systems. Still, there exists only a very limited selection of channel cross-sections for which the Navier–Stokes equation for pressure-driven Poiseuille flow can be solved analytically. From these equations, velocity profiles as well as flow rates can be calculated. However, whenever a cross-section is not highly symmetric (rectangular, elliptical or circular) the Navier–Stokes equation can usually not be solved analytically. In all of these cases, numerical methods are required. However, in many instances it is not necessary to turn to complex numerical solver packages for deriving, e.g., the velocity profile of a more complex microfluidic channel cross-section. In this paper, a simple spreadsheet analysis tool (here: Microsoft Excel) will be used to implement a simple numerical scheme which allows solving the Navier–Stokes equation for arbitrary channel cross-sections.

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  1. S. Khodaparast, N. Borhani, J.R. Thome, Sudden expansions in circular microchannels: flow dynamics and pressure drop. Microfluid. Nanofluid. 17, 561–572 (2014). doi:10.1007/s10404-013-1321-7

  2. J. Kim, J. Lee, C. Wu, S. Nam, D. Di Carlo, W. Lee, Inertial focusing in non-rectangular cross-section microchannels and manipulation of accessible focusing positions. Lab Chip 16, 992–1001 (2016)

  3. R. Truckenmüller et al., Thermoforming of film‐based biomedical microdevices. Adv. Mater. 23, 1311–1329 (2011)

  4. A. Waldbaur, B. Carneiro, P. Hettich, E. Wilhelm, B. Rapp, Computer-aided microfluidics (CAMF): from digital 3D-CAD models to physical structures within a day. Microfluid. Nanofluid. 15, 625–635 (2013). doi:10.1007/s10404-013-1177-x

  5. E. Wilhelm, C. Neumann, T. Duttenhofer, L. Pires, B.E. Rapp, Connecting microfluidic chips using a chemically inert, reversible, multichannel chip-to-world-interface. Lab Chip 13, 4343–4351 (2013). doi:10.1039/c3lc50861g

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This work was funded by the Bundesministerium für Bildung und Forschung (BMBF), funding code 03X5527 “Fluoropor”

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Correspondence to Bastian E. Rapp.

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Richter, C., Kotz, F., Giselbrecht, S. et al. Numerics made easy: solving the Navier–Stokes equation for arbitrary channel cross-sections using Microsoft Excel. Biomed Microdevices 18, 52 (2016) doi:10.1007/s10544-016-0070-2

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  • Numerics
  • Navier–Stokes
  • Microsoft Excel
  • Finite differents method
  • Velocity profile