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Diffusion equations are derived for an isothermal lattice Boltzmann model with two components. The first-order upwind finite difference scheme is used to solve the evolution equations for the distribution functions. When using this scheme, the numerical diffusivity, which is a spurious diffusivity in addition to the physical diffusivity, is proportional to the lattice spacing and significantly exceeds the physical value of the diffusivity if the number of lattice nodes per unit length is too small. Flux limiter schemes are introduced to overcome this problem. Empirical analysis of the results of flux limiter schemes shows that the numerical diffusivity is very small and depends quadratically on the lattice spacing.

We reinvestigate the model of Bonabeau et al.¹ of self-organizing social hierarchies by including a distribution of attractive sites. Agents move randomly except in the case where an attractive site is located in its neighborhood. We find that the transition between an egalitarian society at low population density and a hierarchical one at high population density strongly depends on the distribution and percolation of the valuable sites. We also show how agent diffusivity is closely related to social hierarchy.

Importance of studying Brinkman–Navier–Stokes flow under the influence of electric and magnetic fields lies in its relevance to fundamental physical phenomena, its applications in various fields of science and engineering and its potential for technological advancements. The study of Brinkman–Navier–Stokes flow under the influence of electric and magnetic fields, along with additional factors such as Joule heating, fractional derivatives and convection, represents a multifaceted and challenging problem in fluid dynamics and applied mathematics. In this study, we explore the combined effects of Joule heating due to electric current passing through the fluid and fractional derivatives that describe nonlocal behaviors. The fractional formulation leads to a relaxation mechanism that exhibits delay and recollection of the fluid motion. The organization of direct temperature and concentration changes in MHD flow is made possible by this formalization. A finite difference/finite element method is used to calculate the flow dynamics issue and fractionally linked fields. The physical factors explain how the topic under examination is relevant.

Gas diffusion in dry porous media has been a hot topic in several areas of technology for many years. In this paper, a diffusivity model for gas diffusion in dry porous media is developed based on fractal theory and Fick’s law, which incorporates the effects of converging–diverging pores and tortuous characteristics of capillaries as well as Knudsen diffusion. The effective gas diffusivity model is expressed as a function of the fluctuation amplitude of the capillary cross-section size variations, the porosity, the pore area fractal dimension and the tortuosity fractal dimension. The results show that the relative diffusivity decreases with the increase of the fluctuation amplitude and increases with the increase of pore area fractal dimension. To verify the validity of the present model, the relative diffusivity from the proposed fractal model is compared with the existing experimental data as well as two available models of Bruggeman and Shou. Our proposed diffusivity model with pore converging–diverging effect included is in good agreement with reported experimental data.

In this study, the optimization of the fractal-like architecture of porous fibrous materials related to permeability, diffusivity, and thermal conductivity was analyzed by applying the established theoretical models. It was observed that the ratio of dimensionless permeability over dimensionless effective diffusivity decreased with the decrease of porosity and tortuosity fractal dimension, respectively, which implied that lower porosity and tortuosity fractal dimension were beneficial to wind/water resistant fabric, as it reduced the ratio of dimensionless permeability over dimensionless effective diffusivity. Besides, it was found that the ratio of the dimensionless total effective thermal conductivity over dimensionless effective diffusivity increased with tortuosity fractal dimension, which implied lower tortuosity fractal dimension was beneficial to clothing insulation, as it reduced the ratio of dimensionless total effective thermal conductivity over dimensionless effective diffusivity. The optimization results indicated that fabrics with more aligned fibers were preferred for protective clothing, as the low tortuosity fractal dimension implied fibers in the fibrous materials should be more aligned.

The understanding of the diffusion process and mechanisms of harmful species (e.g. chlorides) in porous cementitious materials is important to control and improve the material durability under harsh environments. In this paper, fractal analysis on the pore structure of porous cementitious materials was conducted and involved in a diffusion model. Macro material geometric parameters were considered in the model to avoid the difficulties in the measurements of microscopic pore parameters. The deformations of porous cementitious materials under the uniaxial elastic loads were considered to correct the diffusion model. The stress-affected diffusivity was displayed in an elegant expression involving some macro material parameters (e.g. total porosity, elastic modulus of solid skeleton, Poisson ratio). Results show that the effective diffusivity is greatly influenced by the porosity and stress ratio. The uniaxial elastic loads decrease the pore areas but increase the lengths of the pore channels for mass diffusion, which eventually causes the decrease of the effective diffusivity. The plots of the relative diffusivity against the stress ratio follow linear forms. The developed fractal diffusion model may help better understand the diffusion process in complex porous cementitious materials under elastic loads. Going beyond this, the fractal diffusion model may provide a new tool to predict the diffusivity of porous building materials under complex mechanical and environmental loads.

The present work studies the optical nonlinearity exhibited by the material (for Continuous Wave (CW) laser or long pulse) due to the change in thermal properties of the material on illumination. Thermal lens (TL) technique has been used to measure the refractive index change due to the formation of TL along with other thermo-optic properties of the material in solution. A CW Ar-ion laser has been used as light source and the laser beam was chopped at 25Hz frequency to obtain 12ms pulse to observe the formation of the TL within the sample. The $n_2$ value have been calculated by the TL technique for Benzene, Toluene and Dimethylaniline (DMA) in toluene and Benzene. The $n_2$ value is found to be in the order of 10${}^{-6}$ to 10${}^{-7}$cm²W${}^{-1}$.

An analysis of the equations used for modeling thermal arsenic diffusion in silicon has been carried out. It was shown that for arsenic diffusion governed by the vacancy-impurity pairs and the pairs formed due to interaction of impurity atoms with silicon self-interstitials in a neutral charge state, the doping process can be described by the Fick’s second law equation with a single effective diffusion coefficient which takes into account two impurity flows arising due to interaction of arsenic atoms with vacancies and silicon self-interstitials, respectively. Arsenic concentration profiles calculated with the use of the effective diffusivity agree well with experimental data if the maximal impurity concentration is near the intrinsic carrier concentration. On the other hand, for higher impurity concentrations a certain deviation in the local regions of arsenic distribution is observed. The difference from the experiment can occur due to the incorrect use of effective diffusivity for the description of two different impurity flows or due to the formation of nonuniform distributions of neutral vacancies and neutral self-interstitials in heavily doped silicon layers. We also suppose that the migration of nonequilibrium arsenic interstitial atoms makes a significant contribution to the formation of a low concentration region on thermal arsenic diffusion.