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We study a reaction-diffusion system within a long channel in the regime in which the projected Fick-Jacobs-Zwanzig operator for confined diffusion can be used. We found that under this approximation, Turing instability conditions can be modified due to the channel geometry. The dispersion relation, range of unstable modes where pattern formation occurs, and spatial structure of the patterns itself change as functions of the geometric parameters of the channel. This occurs for the three channels analyzed, for which the values of the projected operators can be found analytically. For the reaction term, we use the well-known Schnakenberg kinetics. [ABSTRACT FROM AUTHOR]

Highlights • We propose a model to the adsorption-diffusion process in fractal porous materials. • We interpret experimental data for three solutes adsorbed on activated carbon and find their fractal dimensions. • Fractal dimensions significantly affect the adsorption-diffusion dynamics. • We find the fractal dimensions do not depend on the adsorbed solute. Abstract In this work, we present a mathematical model to describe the adsorption-diffusion process on fractal porous materials. This model is based on the fractal continuum approach and considers the scale-invariant properties of the surface and volume of adsorbent particles, which are well-represented by their fractal dimensions. The method of lines was used to solve the nonlinear fractal model, and the numerical predictions were compared with experimental data to determine the fractal dimensions through an optimization algorithm. The intraparticle mass flux and the mean square displacement dynamics as a function of fractal dimensions were analyzed. The results suggest that they can be potentially used to characterize the intraparticle mass transport processes. The fractal model demonstrated to be able to predict adsorption-diffusion experiments and jointly can be used to estimate fractal parameters of porous adsorbents. [ABSTRACT FROM AUTHOR]

In this work, we show that under specific anomalous diffusion conditions, chemical systems can produce well-ordered self-similar concentration patterns through diffusion-driven instability. We also find spiral patterns and patterns with mixtures of rotational symmetries. The type of anomalous diffusion discussed in this work, either subdiffusion or superdiffusion, is a consequence of the medium heterogeneity, and it is modeled through a space-dependent diffusion coefficient with a power-law functional form. [ABSTRACT FROM AUTHOR]