American Institute of Mathematical Sciences

Advanced
March  2006, 6(2): 357-371. doi: 10.3934/dcdsb.2006.6.357

On electro-kinetic fluids: One dimensional configurations

R. Ryham 1, , Chun Liu 2, and  Zhi-Qiang Wang 3,

1. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802
2. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
3. 

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, United States

Received  March 2005 Revised  September 2005 Published  December 2005

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: Nernst-Plank., electrokinetics, Poisson-Boltzmann, Debye length.
Mathematics Subject Classification: Primary: 35Q35 35Q99; Secondary: 35J6.
Citation: R. Ryham, Chun Liu, Zhi-Qiang Wang. On electro-kinetic fluids: One dimensional configurations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 357-371. doi: 10.3934/dcdsb.2006.6.357
[1]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939
[2]

Chiun-Chang Lee. Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3251-3276. doi: 10.3934/dcds.2016.36.3251
[3]

L. Bedin, Mark Thompson. Existence theory for a Poisson-Nernst-Planck model of electrophoresis. Communications on Pure & Applied Analysis, 2013, 12 (1) : 157-206. doi: 10.3934/cpaa.2013.12.157
[4]

Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269
[5]

Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic & Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036
[6]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078
[7]

Jonathan Zinsl. Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2915-2930. doi: 10.3934/dcds.2016.36.2915
[8]

Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007
[9]

Yusheng Jia, Weishi Liu, Mingji Zhang. Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1775-1802. doi: 10.3934/dcdsb.2016022
[10]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457
[11]

Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13
[12]

N. Ben Abdallah, M. Lazhar Tayeb. Diffusion approximation for the one dimensional Boltzmann-Poisson system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1129-1142. doi: 10.3934/dcdsb.2004.4.1129
[13]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361
[14]

Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253
[15]

Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg, Mingji Zhang. Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1623-1643. doi: 10.3934/dcdsb.2018064
[16]

Alexander Arbieto, Carlos Matheus. On the periodic Schrödinger-Debye equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 699-713. doi: 10.3934/cpaa.2008.7.699
[17]

F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[18]

Marcelo Nogueira, Mahendra Panthee. On the Schrödinger-Debye system in compact Riemannian manifolds. Communications on Pure & Applied Analysis, 2020, 19 (1) : 425-453. doi: 10.3934/cpaa.2020022
[19]

Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339
[20]

Feng Luo. Geodesic length functions and Teichmuller spaces. Electronic Research Announcements, 1996, 2: 34-41.

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences