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

Christiane Richter1, Frederik Kotz1, Stefan Giselbrecht2

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Biomedical Microdevices
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Summary

Fluid mechanics in microfluidics simplifies analysis by neglecting factors like gravity. This study presents a spreadsheet method to solve the Navier-Stokes equation for complex microchannel shapes, avoiding complex software.

Keywords:
Finite differents methodMicrosoft ExcelNavier–StokesNumericsVelocity profile

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Area of Science:

  • Fluid Mechanics
  • Microfluidics
  • Computational Science

Background:

  • Macroscopic fluid mechanics involves complex factors like non-laminar flow and gravity, often simplified in microfluidic systems.
  • Analytical solutions for pressure-driven Poiseuille flow in microchannels are limited to highly symmetric cross-sections (e.g., rectangular, elliptical, circular).
  • Non-symmetric microchannel cross-sections typically require complex numerical methods to solve the Navier-Stokes equation.

Purpose of the Study:

  • To present a simplified numerical approach for solving the Navier-Stokes equation for pressure-driven flow in arbitrary microchannel cross-sections.
  • To demonstrate the utility of a basic spreadsheet tool for microfluidic analysis.
  • To provide an accessible method for calculating velocity profiles and flow rates in complex microchannels.

Main Methods:

  • Implementation of a simple numerical scheme within a spreadsheet program (Microsoft Excel).
  • Application of the numerical scheme to solve the Navier-Stokes equation for pressure-driven flow.
  • Analysis of fluid flow characteristics for various microchannel cross-sections.

Main Results:

  • The spreadsheet tool successfully solves the Navier-Stokes equation for arbitrary microchannel cross-sections.
  • The method allows for the calculation of velocity profiles and flow rates without resorting to complex computational fluid dynamics software.
  • Demonstrates a practical and accessible approach to microfluidic channel design and analysis.

Conclusions:

  • A simple spreadsheet-based numerical method can effectively analyze fluid mechanics in microfluidic systems with complex channel geometries.
  • This approach democratizes the analysis of microfluidic flow, making it accessible beyond specialized computational fluid dynamics expertise.
  • The findings facilitate the design and optimization of microfluidic devices with non-standard channel shapes.