Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

On electro-kinetic fluids: One dimensional configurations

Pages: 357 - 371, Volume 6, Issue 2, March 2006      doi:10.3934/dcdsb.2006.6.357

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R. Ryham - Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States (email)
Chun Liu - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)
Zhi-Qiang Wang - Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, United States (email)

Abstract: Electro-kinetic fluids can be modeled by hydrodynamic systems describing the coupling between fluids and electric charges. The system consists of a momentum equation together with transport equations of charges. In the dynamics, the special coupling between the Lorentz force in the velocity equation and the material transport in the charge equation gives an energy dissipation law. In stationary situations, the system reduces to a Poisson-Boltzmann type of equation. In particular, under the no flux boundary conditions, the conservation of the total charge densities gives nonlocal integral terms in the equation. In this paper, we analyze the qualitative properties of solutions to such an equation, especially when the Debye constant $\epsilon$ approaches zero. Explicit properties can be derived for the one dimensional case while some may be generalized to higher dimensions. We also present some numerical simulation results of the system.

Keywords:  electrokinetics, Poisson-Boltzmann, Debye length, Nernst-Plank.
Mathematics Subject Classification:  Primary: 35Q35 35Q99; Secondary: 35J60.

Received: March 2005;      Revised: September 2005;      Available Online: December 2005.