AMaSiS 2018 Workshop: Abstracts

Poster Two-scale convergence of the generalized Poisson–Nernst–Planck problem in a two-phase medium

Anna V. Zubkova${}^{\mathbf{\text{(1)}}}$ and Victor A. Kovtunenko${}^{\mathbf{\text{(1,2)}}}$

(1) University of Graz, Institute for Mathematics and Scientific Computing

(2) Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk

We study a generalized Poisson–Nernst–Planck (PNP) problem formulated in a two-phase domain composed of periodic cells. The PNP problem describes cross-diffusion of multiple charged species, which are expressed in terms of species concentrations and an overall electrostatic potential.

The variational formulation of the PNP problem: Find discontinuous over the interface functions $(c_1^{\epsilon },\mathrm{\dots },c_n^{\epsilon })$ and ${\phi }^{\epsilon }$ such that for $i=1,\mathrm{\dots },n$:

$$\begin{array}{c}{\int }_0^T\left\{{\int }_{Q_{\epsilon }\cup {\omega }_{\epsilon }}\left[\frac{\partial c_i^{\epsilon }}{\partial t}{\overline{c}}_i+\sum _{j=1}^n{(\nabla c_j^{\epsilon })}^{\top }{D}_{\epsilon }^{ij}\nabla {\overline{c}}_i\right]𝑑x+\sum _{j=1}^n{\int }_{Q_{\epsilon }}{\epsilon }^{\kappa }{\mathrm{{\rm Y}}}_j({\text{𝐜}}^{\epsilon }){(\nabla {\phi }^{\epsilon })}^{\top }{D}_{\epsilon }^{ij}\nabla {\overline{c}}_{i}dx\right\}𝑑t\hfill \\ \hfill ={\int }_0^T{\int }_{\partial {\omega }_{\epsilon }}{\epsilon }^{1+\gamma }{g}_i({\widehat{\text{𝐜}}}^{\epsilon },{\widehat{\phi }}^{\epsilon })[[{\overline{c}}_i]]𝑑S_x𝑑t,\end{array}$$ (1a)

$$\begin{array}{c}{\int }_{Q_{\epsilon }\cup {\omega }_{\epsilon }}{(\nabla {\phi }^{\epsilon })}^{\top }{A}^{\epsilon }\nabla \overline{\phi }dx-{\int }_{Q_{\epsilon }}{\mathrm{{\rm Y}}}_0({\text{𝐜}}^{\epsilon })\overline{\phi }𝑑x+{\int }_{\partial {\omega }_{\epsilon }}\frac{\alpha }{\epsilon }[[{\phi }^{\epsilon }]][[\overline{\phi }]]𝑑S_x={\int }_{\partial {\omega }_{\epsilon }}{g}^{\epsilon }[[\overline{\phi }]]𝑑S_x,\hfill \end{array}$$ (1b)

with nonlinear terms ${\mathrm{{\rm Y}}}_j({\text{𝐜}}^{\epsilon })$, $j=0,1,\mathrm{\dots },n$.

We examine the nonlinear inhomogeneous Neumann boundary conditions for concentrations which depend on the variables, hence do not satisfy any periodicity assumption. Nonlinearity describes electro-chemical reactions on the boundary.

We aim at the two-scale convergence as $\epsilon \to 0$ of the model applying the periodic unfolding technique defined in the two-phase domain with an interface.

Acknowledgments: The authors are supported by the Austrian Science Fund (FWF) Project P26147-N26: “Object identification problems: numerical analysis” (PION). The authors thank the Austrian Academy of Sciences (OeAW) and IGDK1754 for partial support.