Homogenization and Multiscale Analysis of Electro-Diffusive Transport in Complex Media
Homogenisierung und Mehrskalenanalyse des elektro-diffusiven Transports in komplexen Medien
Please always quote using this URN: urn:nbn:de:bvb:29-opus4-219182
- Electro-diffusive and reactive-diffusive transports in complex media are fundamental processes
that occur naturally in the physical world and also in numerous applications from engineering
sciences. However, the heterogeneity and complexity of such systems give rise to mathematical
models involving multiple scales in space and time. Hence, appropriate methods of
homogenization and asymptotic analysis are required to derive homogenized or effective
descriptions which connect the different scales and are accessible for numerical simulations. A
special focus of this thesis is on transport processes in two bulk regions, which are connected
by a thin layer perforated by channels. SuchElectro-diffusive and reactive-diffusive transports in complex media are fundamental processes
that occur naturally in the physical world and also in numerous applications from engineering
sciences. However, the heterogeneity and complexity of such systems give rise to mathematical
models involving multiple scales in space and time. Hence, appropriate methods of
homogenization and asymptotic analysis are required to derive homogenized or effective
descriptions which connect the different scales and are accessible for numerical simulations. A
special focus of this thesis is on transport processes in two bulk regions, which are connected
by a thin layer perforated by channels. Such structures appear, e.g., in the transport of
ions between extra-cellular and intra-cellular regions through the ion channels of cellular
membranes. Here, in addition to the microstructure of the membrane, its thickness, which
is much smaller than the dimension of the bulk regions consisting of extra-cellular and
intra-cellular spaces, increases the complexity of the models formulated at the microscopic
scale. Thus, homogenization and dimension reduction have to be performed simultaneously
to obtain effective approximations in the limit when the microscopic scale parameter tends to
zero and the thin layer reduces to an interface between the two bulk regions.
Motivated by electro-diffusive processes occurring in ion transport, in this thesis a nonlinear
drift-diffusion system for multiple charged species in a porous medium with periodic
microstructure in 2D and 3D is considered. The system consists of transport equations for
the concentrations of the ion species and Poisson’s equation for the electric potential. The
diffusion terms depend nonlinearly on the concentrations, giving rise to a system of quasilinear
partial differential equations. This system approximates the classical Poisson-Nernst-Planck
system. We rigorously derive an effective (homogenized) model in the limit when the scale
parameter $\epsilon$ tends to zero. This is based on uniform a priori estimates for the solutions of
the microscopic model. The crucial result is the uniform $L^\infty$-estimate for the concentration
in space and time. We obtain this result by exploiting the fact that the system admits a
nonnegative energy functional, which decreases in time along the solutions of the system. By
using two-scale convergence properties of the microscopic solutions, effective models are
derived for different scalings of the microscopic model.
Furthermore, diffusive-reactive transport is considered in a domain consisting of two bulk
regions connected via a large number of small channels periodically distributed within a
thin layer. The height and the thickness of the channels are both of order $\epsilon> 0$. We extend
the classical two-scale convergence method to the setting of thin channels and prove related
compactness results. This method allows the derivation of effective and dimension-reduced
models, especially it enables the derivation of effective interface laws which hold in the
singular limit when the thickness of the perforated layer separating the bulk regions tends to
zero. We consider different scalings of the microscopic model, which allow different sizes
of diffusion in the channels compared to the bulk domains. However, all the scalings lead
to the same homogenized model involving an ordinary differential equation for the limit
concentration on the interface. This is different from previous results, where a thin layer
with a heterogeneous structure was considered and a reaction-diffusion equation for the limit
concentration on the interface in the form of a dynamic Wentzell interface condition was
obtained in the case of critical scaling. However, in our case, the channel geometry prevents
the appearance of such a reaction-diffusion equation on the interface.…
Author: | Apratim BhattacharyaORCiD |
---|---|
Persistent identifiers - URN: | urn:nbn:de:bvb:29-opus4-219182 |
Referee: | Maria Neuss-Radu, Adrian Muntean |
Advisor: | Maria Neuss-Radu |
Document Type: | Doctoral Thesis |
Language: | English |
Year of publication: | 2023 |
Date of online publication (Embargo Date): | 2023/03/16 |
Publishing Institution: | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) |
Granting institution: | Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Naturwissenschaftliche Fakultät |
Acceptance date of the thesis: | 2023/03/08 |
Release Date: | 2023/03/20 |
Tag: | Dimension reduction; Effective transmission conditions; Homogenization; Multiple charged species; Nonlinear electro-diffusion; Poisson-Nernst-Planck equations; Thin channels; Two-scale convergence |
Length/size: | 73 + ix pages |
Institutes: | Naturwissenschaftliche Fakultät |
Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
open_access (DINI-Set): | open_access |
Licence (German): | ![]() |