Long time dynamics of nonequilibrium electroconvection

Lee, Fizay-Noah

doi:10.1090/tran/9171

Long time dynamics of nonequilibrium electroconvection
HTML articles powered by AMS MathViewer

by Fizay-Noah Lee

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9171

Published electronically: April 24, 2024

PDF | Request permission

Abstract:

The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove (1) the existence of a compact global attractor with finite fractal (box-counting) dimension and (2) space-time averaged electroneutrality $\rho \approx 0$ in the singular limit of Debye length going to zero, $\epsilon \to 0$.

References

S. Agmon and L. Nirenberg, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math. 20 (1967), 207–229. MR 204829, DOI 10.1002/cpa.3160200106
Elie Abdo and Mihaela Ignatova, Long time finite dimensionality in charged fluids, Nonlinearity 34 (2021), no. 9, 6173–6209. MR 4295548, DOI 10.1088/1361-6544/ac13bf
Piotr Biler, Waldemar Hebisch, and Tadeusz Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Anal. 23 (1994), no. 9, 1189–1209. MR 1305769, DOI 10.1016/0362-546X(94)90101-5
Dieter Bothe, André Fischer, and Jürgen Saal, Global well-posedness and stability of electrokinetic flows, SIAM J. Math. Anal. 46 (2014), no. 2, 1263–1316. MR 3180849, DOI 10.1137/120880926
Y. S. Choi and Roger Lui, Multi-dimensional electrochemistry model, Arch. Rational Mech. Anal. 130 (1995), no. 4, 315–342. MR 1346361, DOI 10.1007/BF00375143
P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for $2$D Navier-Stokes equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 1–27. MR 768102, DOI 10.1002/cpa.3160380102
Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259
Peter Constantin and Mihaela Ignatova, On the Nernst-Planck-Navier-Stokes system, Arch. Ration. Mech. Anal. 232 (2019), no. 3, 1379–1428. MR 3928752, DOI 10.1007/s00205-018-01345-6
Peter Constantin, Mihaela Ignatova, and Fizay-Noah Lee, Interior electroneutrality in Nernst-Planck-Navier-Stokes systems, Arch. Ration. Mech. Anal. 242 (2021), no. 2, 1091–1118. MR 4331022, DOI 10.1007/s00205-021-01700-0
Peter Constantin, Mihaela Ignatova, and Fizay-Noah Lee, Nernst-Planck-Navier-Stokes systems near equilibrium, Pure Appl. Funct. Anal. 7 (2022), no. 1, 175–196. MR 4396256
Peter Constantin, Mihaela Ignatova, and Fizay-Noah Lee, Nernst-Planck-Navier-Stokes systems far from equilibrium, Arch. Ration. Mech. Anal. 240 (2021), no. 2, 1147–1168. MR 4244828, DOI 10.1007/s00205-021-01630-x
Peter Constantin, Mihaela Ignatova, and Fizay-Noah Lee, Existence and stability of nonequilibrium steady states of Nernst-Planck-Navier-Stokes systems, Phys. D 442 (2022), Paper No. 133536, 18. MR 4496950, DOI 10.1016/j.physd.2022.133536
Peter Constantin, Andrei Tarfulea, and Vlad Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys. 335 (2015), no. 1, 93–141. MR 3314501, DOI 10.1007/s00220-014-2129-3
S. M. Davidson, M. Wissling, and A. Mani, On the dynamical regimes of pattern-accelerated electroconvection, Sci. Rep. 6 (2016), 22505, \PrintDOI{19.1039/srep22505}.
Chia-Yu Hsieh and Yong Yu, Debye layer in Poisson-Boltzmann model with isolated singularities, Arch. Ration. Mech. Anal. 236 (2020), no. 1, 289–327. MR 4072215, DOI 10.1007/s00205-019-01466-6
S. Kang and R. Kawk, Pattern formation of three-dimensional electroconvection on a charge selective surface, Phys. Rev. Lett 124 (2020), 154502, https://doi.org/10.1103/PhysRevLett.124.154502.
Fizay-Noah Lee, Global regularity for Nernst-Planck-Navier-Stokes systems with mixed boundary conditions, Nonlinearity 36 (2023), no. 1, 255–286. MR 4521944
Michael S. Mock, Analysis of mathematical models of semiconductor devices, Advances in Numerical Computation Series, vol. 3, Boole Press, Dún Laoghaire, 1983. MR 697094
V. S. Pham, Z. Li, K. M. Lim, J. K. White, and J. Han, Direct numerical simulation of electroconvective instability and hysteretic current-voltage response of a permselective membrane, Phys. Rev. E 86 (2012), 046310, https://doi.org/10.1103/PhysRevE.86.046310.
R. Probstein, Physicochemical hydrodynamics: an introduction, 2nd ed., Wiley-Interscience, 2003.
Isaak Rubinstein, Electro-diffusion of ions, SIAM Studies in Applied Mathematics, vol. 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1075016, DOI 10.1137/1.9781611970814
S. M. Rubinstein, G. Manukyan, A. Staicu, I. Rubinstein, B. Zaltzman, R.G.H. Lammertink, F. Mugele, M. Wessling, Direct observation of a nonequilibrium electro-osmotic instability, Phys. Rev. Lett. 101 (2008), 236101–236105.
I. Rubinstein and L. A. Segel, Breakdown of a stationary solution to the Nernst-Planck-Poisson equations, J. Chem. Soc., Faraday Trans. 2 75 (1979), 936–940.
I. Rubinstein and B. Zaltzman, Electro-osmotically induced convection at a permselective membrane, Phys. Rev. E 62 (2000), 2238–2251.
R. Ryham, Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics, arXiv:0910.4973v1, 2009.
Markus Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci. 19 (2009), no. 6, 993–1015. MR 2535842, DOI 10.1142/S0218202509003693
B. Zaltzman and I. Rubinstein, Electro-osmotic slip and electroconvective instability, J. Fluid Mech. 579 (2007), 173–226. MR 2325045, DOI 10.1017/S0022112007004880