Communications in Mathematical Sciences

Volume 9 (2011)

Number 3

Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach

Pages: 685 – 710



Markus Schmuck (Chemical Engineering, Massachusetts Institute of Technology, Camridge, Massachusetts)


We formulate the basic equations modeling solid-electrolyte composites without surface reactions. From these equations we achieve by the two-scale convergence method homogenized Nernst-Planck-Poisson equations. Moreover, we extend the system by including Stokes flow. Again, the two-scale convergence allows the rigorous justification of the resulting homogenized and nonlinearly coupled overall system. So called “material tensors” naturally arise by the upscaling and replace the commonly used porosity parameter from engineering. The upscaled equations derived here capture more accurately porous structures by including the microscopic geometry in a systematic way. To the author’s best knowledge, this seems to be the first approach which derives the Stokes-Poisson-Nernst-Planck system being governed by porous materials and hence serves as a basis for additional specifications in the future.


Two-scale convergence, Stokes-Poisson-Nernst-Planck equations, porous materials, Darcy’s law, periodic homogenization

2010 Mathematics Subject Classification

35B27, 35K55, 35Q35, 35Q92, 76M50

Published 11 March 2011