DFG Research Center MATHEON, Project D27
Numerical methods for coupled
micro- and nanoflows with strong electrostatic forces
(Domain of Expertise: Electronic devices)
Project members
Contents
Backgrounds
Research Program for next funding period
References
Cooperation
Backgrounds
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
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].
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.
References
[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.
Cooperation
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.
