Homogenization and Multiscale Analysis of Electro-Diffusive Transport in Complex Media

Document Type
Doctoral Thesis
Issue Date
Issue Year
Bhattacharya, Apratim

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. 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 tends to zero. This is based on uniform a priori estimates for the solutions of the microscopic model. The crucial result is the uniform -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 . 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.

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