Traditional Diffusion Equation Research Articles

MATHEMATICAL MODEL FOR NANOFIBERS’ DIFFUSION IN A NANOFLUID AND NUMERICAL SIMULATION

The diffusion process of the nanofibers in a nanofluid displays a combination of diffusion and wave characteristics. The wave-like behavior is attributed to the large aspect ratio of the nanofiber, which can be modeled as a bead-spring chain system, exhibiting wave-like properties. In order to describe this mixed diffusion-wave phenomenon, a fractional diffusion-wave equation is proposed, wherein multiple time fractional derivatives are employed. By appropriately regulating the fractional time terms, the model can be transformed into either the traditional diffusion equation or the traditional wave equation, as required. A numerical schedule is developed through the implementation of suitable time and space discretization, and a stability analysis is conducted. The numerical results substantiate the reliability of the numerical schedule and its applicability to other fractional differential equations.

A unified model of suspension concentration distribution in sediment mixed turbulent flows using generalized fractional advection-diffusion equation

The fractional operator in a space fractional advection-diffusion equation (FADE) plays a significant role in the mixing and vertical movement of sediment particles in a sediment-laden turbulent flow under non-local effects. Turbulent flow exhibits non-local mixing properties, which leads to the non-Fickian diffusion process that cannot be captured by the traditional diffusion equation. In this work, we present a generalized FADE that includes the generalized fractional differential operator in the Caputo sense. The full analytical solution is proposed utilizing the general Laplace transformation method. This generalized solution contains weight and scale functions and includes the effects of non-locality. It has been shown that several existing famous models of suspension concentration distribution for sediment particles (including both type-I and type-II distributions) in turbulent flows can be obtained from the proposed generalized solution with proper choices of the scale and weight functions in particular. Here a total of fourteen different types of concentration distribution equations including type-I and type-II profiles are derived from the general solution. Further possible generalizations of the model are also discussed which are more useful for practical applications. It is found that the several existing sediment distribution models are equivalent up to choices of weight and scale functions. Further, we found that the scale function could be physically related to the characteristic Lagrangian length of sediment mixing. The choice of the scale and weight function for both the type-I and type-II profiles are discussed and analyzed. Finally, the model is validated with experimental data as well as field data from the Missouri River, Mississippi River, and Rio Grande conveyance channels, and in each case, satisfactory agreements are obtained. These suggest the broader applicability of the present study.

Modelling Non-Fickian Behavior in the Cell Walls of Wood Using a Fractional-In-Space Diffusion Equation

Recent studies of diffusion processes in highly heterogeneous, fractal-like media highlight that the traditional diffusion equation may not adequately describe the movement of water in the pore network because of the evidently anomalous transport phenomena. It is postulated here that the diffusion of moisture in the cell walls of wood is anomalous (non-Fickian), and this motivates the use of a moisture potential defined in terms of a fractional-in-space operator involving the Laplacian raised to a fractional index. A coupled anomalous transport model for temperature and moisture content is then derived to simulate the absorption of water in the cell walls of wood. This model is discretized in space using the finite volume method to produce a large system of ordinary differential equations, which is advanced in time using a second-order exponential Euler method. A novelty of our computational strategy is the way in which the flux boundary condition involving the moisture potential is implemented to ensure an accurate global mass balance is achieved at each time step. The model is validated against experimental data available for beech, where good agreement is observed.

Diffusion in inhomogeneous and anisotropic media

Despite common practice, inhomogeneous and/or anisotropic diffusion cannot be considered without regarding the microscopic details breaking the translational and/or angular symmetry. The macroscopic diffusion equation and the stationary solution are determined by the microscopic model and depend in general on all the microscopic parameters and not simply on the combination in the diffusion tensor. The traditional diffusion equation is only valid under special conditions and it cannot, in general, be used for anisotropic diffusion. An alternative form of the diffusion equation has a wider range of applicability. It is shown that for isotropic diffusion all variants of the diffusion equation are mathematically (but not physically) equivalent and can be transformed into each other by introduction of effective potentials. This is not the case for anisotropic diffusion where the traditional diffusion equation in most cases will give incorrect results. Two examples illustrate the differences between the two dynamic equations with respect to stationary solutions and detailed balance.