DFG Research Center MATHEON, Project D27

Numerical methods for coupled micro- and nanoflows with strong electrostatic forces (Domain of Expertise: Electronic devices)

by Volker John

Project members


Research Program for next funding period


In an electrolyte, the charge distribution of n ionic species with molar concentrations ci influences the electrostatic potential phi in a self-consistent manner. On the one hand, the species transport is driven by convection due to the fluid velocity u, by migration due to the electric field and by diffusion. On the other hand, moving ions exert a force driving the flow. Assuming dilute solutions, absence of reactions, and ignoring inertia forces, these interactions are modeled by the Nernst-Planck-Poisson-Stokes system

Nernst-Planck-Poisson-Stokes system

In (1), Ni is the molar flux, zi the charge number, Di the diffusion coeffcient, and mui the mobility of species i. Further, rho is the space charge, F is the Faraday constant, epsilon is the dielectric, nu is the kinematic viscosity of the fluid, and d is a constant of order 1 relating the electro-osmotic body force to the force exerted by the space charge. System (1) has to be equipped with appropriate boundary conditions. The electro-osmotic force on the fluid occurs only in boundary layer regions with charge separation, where epsilon / F  ~ 10-16 mol / (V m) cannot assumed to be comparably small. In the bulk of the fluid, the electroneutrality condition holds. In small scale devices, this effect is used to induce a controlled movement of the fluid, as in electro-osmotic micropumps used in lab-on-a chip technologies [LS04].

Likewise, it becomes relevant in biological systems [DLG07] and bio-electronic devices [JCLS08]. In the case of large derivatives of the electrostatic potential perpendicular to a permselective membrane, this force may lead to a destabilization of the boundary layer as observed in electro-dialysis devices [ZR07].

Schematic description of electro dialysis

In the pore space of electrodes and the voids of polymer membranes, this effect influences the performance of electrochemical storage devices. The system is used as well to describe the movement of charged particles in printing devices. The solution of (1) in the boundary layers of the electrostatic potential is of importance in applications. Because of the large gradient of phi and the typical values of the parameters given above, each equation of (1) has a locally large term. The presence of these huge terms in the system affects its stability in a sophisticatedway. Note that for zero fluid velocity, (1) becomes the van Roosbroeck equations of charge transport in semiconductors, as considered in MATHEON project D22 [GG09].

Research program

The project aims at the development of stable and accurate discretizations for system (1) in two and three dimensions. The investigated discretizations should be assessed in particular at the electro-dialysis membrane which was studied in [ZR07].  All algorithms will be implemented on the basis of existing software of the Research Group 3 of the WIAS. The simulation of the electro-dialysis membrane are considered as a proof of concept of the studied methods. We have chosen to use this example because it comes on the one hand  from applications and on the other hand a lot of details to compare with are known.


[ACF+11] M. Augustin, A. Caiazzo, A. Fiebach, J. Fuhrmann, V. John, A. Linke, and R. Umla. An assessment of discretizations for convection-dominated convection-diffusion equations. Comput. Methods Appl. Mech. Engrg., 200:3395-3409, 2011.

[CELR11] Michael A. Case, Vincent J. Ervin, Alexander Linke, and Leo G. Rebholz. A connection between Scott-Vogelius and grad-div stabilized Taylor-Hood FE approximations of the Navier-Stokes equations. SIAM J. Numer. Anal., 49(4):1461-1481, 2011.

[DLG07] G. De Luca and M.I. Glavinovic. Glutamate, water and ion transport through a charged nanosize pore. Biochim. Biophys. Acta, Biomembr., 1768(2):264-279, 2007.

[EFL11] R. Eymard, J. Fuhrmann, and A. Linke. Mac schemes on triangular delaunay meshes. Preprint 1654, WIAS, 2011.

[EGHL12] R. Eymard, Th. Gallouet, R. Herbin, and A. Linke. Convergence of a finite volume scheme for the biharmonic problem. Math. Comp., 2012 (electronic).

[FLL11] J. Fuhrmann, H. Langmach, and A. Linke. A numerical method for mass conservative coupling between fluid flow and solute transport. Appl. Numer. Math., 61(4):530-553, 2011.

[FLLB09] J. Fuhrmann, A. Linke, H. Langmach, and H. Baltruschat. Numerical calculation of the limiting current for a cylindrical thin layer  flow cell. Electroch. Acta, 55(2):430-438, 2009.

[GG09] A. Glitzky and K. Gärtner. Energy estimates for continuous and discretized electro-reaction-diffusion systems. Nonlinear Anal., 70(2):788-805, 2009.

[GLRW11] Keith Galvin, Alexander Linke, Leo Rebholz, and Nicholas Wilson. Stabilizing poor mass conservation in incompressible  flow problems with large irrotational forcing and application to thermal convection. Computer Methods in Applied Mechanics and Engineering, Vol. 237-240, pp. 166-176, 2012.

[JCLS08] J.W. Jerome, B. Chini, M. Longaretti, and R. Sacco. Computational modeling and simulation of complex systems in bio-electronics. J. Comput. Electron., 7(1):10-13, 2008.

[JK05] V. John and S. Kaya. A finite element variational multiscale method for the Navier-Stokes equations. SIAM J. Sci. Comp., 26:1485-1503, 2005.

[JK07] V. John and P. Knobloch. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations: Part I - a review. Comp. Meth. Appl. Mech. Engrg., 196:2197-2215, 2007.

[JK08a] V. John and S. Kaya. Finite element error analysis of a variational multiscale method for the Navier-Stokes equations. Adv. Comp. Math., 28:43-61, 2008.

[JK08b] Volker John and Petr Knobloch. On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. II. Analysis for P1 and Q1 finite elements. Comput. Methods Appl. Mech. Engrg., 197(21-24):1997-2014, 2008.

[JK10] V. John and A. Kindl. A variational multiscale method for turbulent  flow simulation with adaptive large scale space. J. Comput. Phys., 229:301-312, 2010.

[JN11] V. John and J. Novo. Error analysis of the supg finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal., 49:1149-1176, 2011.

[Joh04] V. John. Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and Numerical Results for a Class of LES Models, volume 34 of Lecture Notes in Computational Science and Engineering. Springer-Verlag Berlin, Heidelberg, New York, 2004.

[JR10] V. John and M. Roland. On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems. Int. J. Numer. Meth. Engng., 82:1450-1474, 2010.

[JRM+09] V. John, M. Roland, T. Mitkova, K. Sundmacher, L. Tobiska, and A. Voigt. Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Engrg. Sci., 64:733-741, 2009.

[Lin08] A. Linke. Divergence-free mixed finite elements for the incompressible Navier-Stokes Equation. PhD thesis, University of Erlangen, 2008.

[Lin09] A. Linke. Collision in a cross-shaped domain - a steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD. Comput. Methods Appl. Mech. Engrg., 198(41{44):3278-3286, 2009.

[LS04] D.J. Laser and J.G. Santiago. A review of micropumps. J. Micromech. Microeng., 14:R35, 2004.

[ZR07] B. Zaltzman and I. Rubinstein. Electro-osmotic slip and electroconvective instability. J. Fluid Mech., 579:173-226, 2007.


Internal MATHEON cooperation
Phenomena which can be modeled by systems of type (1) are studied in project C10. After having available a stable and accurate method, the investigations in C10 will be supplemented with numerical simulations. The numerical analysis of the methods will be performed in collaboration with project D22.

External cooperation
On the numerical analysis of finite volume schemes there existsa close collaboration with R. Eymard (Universite de Marne-la-Vallee, France).

Transfer of results
The successful development of stable and accurate numerical schemes will allow the cooperation with groups developing nano fluidic devices, e.g., based on joint funding applications.