Classical Diffusion Research Articles

Optimal convergence rate of the explicit Euler method for convection–diffusion equations

Revisiting the explicit Euler method of the classical diffusion equation, a new difference scheme with the optimal convergence rate four is achieved under the condition of the specific step-ratio r=1/6. Applying the corrected idea to the convection–diffusion equation, a new corrected numerical scheme is obtained which owns a similar fourth-order optimal convergence rate. Rigorous numerical analysis is carried out by the maximum principle. Compared with the standard difference schemes, the new proposed difference schemes have obvious advantage in accuracy. Extensive numerical examples with and without exact solutions confirm our theoretical results. Moreover, extending our technique to nonlinear problems such as the Fisher equation and viscous Burgers’ equation is available.

ADI Methods for Three-dimensional Fractional Diffusions

ADI methods can be generalized to solve numerically multidimensional fractional diffusion equations, which describe fluid flows through porous media better than classical diffusion equations. A new, unconditionally stable, second-order and well balanced in space, third-order in time ADI scheme has been constructed and its convergence accelerated by an extrapolation technique coupled with the PageRank algorithm.

Three-dimensional sintering morphology simulation of silica fibrous insulator and its quantitative description

Silica fibrous insulator possessing excellent thermal properties is widely used in the aerospace engineering field. A finite element model is developed for simulating the solid-state sintering process in the preparation process of the fibrous insulator. The model is capable of simulating initial surfaces with complex geometric features. The model is validated by using the classic double-sphere diffusion model and comparing the simulation results with the experimental results in the reference. Results suggest sintered morphology is strongly controlled by the highest holding temperature and has little relationship with the heating rate. The detailed simulation parameters for sintering fibrous silica are determined by the experiment of sintering parallel fibers. Comparison between the simulation results of planar simplification model, double-sphere model, and double-cylinder model reveals the significance of neighborhood diffusion on the sintering of parallel fibers. We found the morphological geometric feature of the sintered fibrous body is consistent and accordingly the Feature Neck Ratio (FNR) is defined to quantitatively describe the degree of sintering and the morphology feature of the sintered fibers. From the perspective of the sintering morphology, when FNR reaches about 0.7–1 no matter how the sintering route is, the fibers are considered to be fully sintered together.

Cavity Nucleation and Growth in Nickel-Based Alloys during Creep

The number of fossil fueled power plants in electricity generation is still rising, making improvements to their efficiency essential. The development of new materials to withstand the higher service temperatures and pressures of newer, more efficient power plants is greatly aided by physics-based models, which can simulate the microstructural processes leading to their eventual failure. In this work, such a model is developed from classical nucleation theory and diffusion driven growth from vacancy condensation. This model predicts the shape and distribution of cavities which nucleate almost exclusively at grain boundaries during high temperature creep. Cavity radii, number density and phase fraction are validated quantitively against specimens of nickel-based alloys (617 and 625) tested at 700 °C and stresses between 160 and 185 MPa. The model’s results agree well with the experimental results. However, they fail to represent the complex interlinking of cavities which occurs in tertiary creep.

Carrier Diffusion in GaN : A Cathodoluminescence Study. II. Ambipolar versus Exciton Diffusion

We determine the diffusion length of excess carriers in $\mathrm{Ga}\mathrm{N}$ by spatially resolved cathodoluminescence spectroscopy utilizing a single quantum well as a carrier collector or carrier sink. Monochromatic intensity profiles across the quantum well are recorded for temperatures between 10 and 300 K. A classical diffusion model accounts for the profiles acquired between 120 and 300 K, while for temperatures lower than 120 K, a quantum capture process has to be taken into account in addition. Combining the diffusion length extracted from these profiles and the effective carrier lifetime measured by time-resolved photoluminescence experiments, we deduce the carrier diffusivity as a function of temperature. The experimental values are found to be close to theoretical ones for the ambipolar diffusivity of free carriers limited only by intrinsic phonon scattering. This agreement is shown to be fortuitous. The high diffusivity at low temperatures instead originates from an increasing participation of excitons in the diffusion process.

On the application of the analytical discrete ordinates method to the solution of nonclassical transport problems in slab geometry

In this work we investigate the use of the Analytical Discrete Ordinates (ADO) method when solving the spectral approximation of the nonclassical transport equation. The spectral approximation is a recently developed method based on the representation of the nonclassical angular flux as a series of Laguerre polynomials. This representation generates, as outcome, a system of equations that have the form of classical transport equations and can therefore be solved by current deterministic algorithms. Thus, the investigation of efficient approaches to solve the nonclassical transport equation is of interest and shall be pursued. This is the case of the ADO method which has been successfully used to solve a wide class of problems in the general area of particle transport. Numerical results are presented for two nonclassical test problems in slab geometry. These nonclassical transport problems are chosen in such way that their solution exactly reproduces the solution of the classical diffusion problem. Very accurate results are obtained for both test problems. However, the use of high precision arithmetic is sometimes required as illustrated in the second test problem. Limitations of the spectral approximation are also analyzed and discussed.

Instability analysis and geometric perturbation theory to a mutual beneficial interaction between species with a higher order operator

<abstract><p>The higher order diffusion can be understood as a generalization to the classical fickian diffusion. To account for such generalization, the Landau-Ginzburg free energy concept is applied leading to a fourth order spatial operator. This kind of diffusion induces a set of instabilities in the proximity of the critical points raising difficulties to study the convergence of Travelling Waves (TW) solutions. This paper aims at introducing a system of two species driven by a mutual interaction towards prospering and with a logistic term in their respective reactions. Previous to any analytical finding of TW solutions, the instabilities of such solutions are studied. Afterwards, the Geometric Perturbation Theory is applied to provide means to search for a linearized hyperbolic manifold in the proximity of the equilibrium points. The homotopy graphs for each of the flows to the hyperbolic manifolds are provided, so that analytical solutions can be obtained in the proximity of the critical points. Additionally, the set of eigenvalues in the homotopy graphs tend to cluster and synchronize for increasing values of the TW-speed.</p></abstract>

Substantiating the Diffusion Model of Innovation Implementation and its Application to Vaccine Propagation.

The paper examines prerequisites and assumptions of the classical Bass innovation diffusion model with the aim of applying it in modeling of relevant stochastic processes related to the pandemic. The Bass model has proven its versatility and applicability to various environments. A thorough mathematical substantiation of the model properties is presented based on theories of evolutionary equations and stochastic processes for its further development, as well as search for uncertainty parameters and observable variables. The paper provides realistic estimation results of the Bass model parameters for vaccination in Ukraine and Belarus on weekly data of the first half of 2021. Similar studies are suggested for other countries, as well as regions and districts of Ukraine.

Numerical Calculation of Diffusion Depth and Skin Depth Based on a GEMTIP Model in Electromagnetic Sounding Methods

Electromagnetic (EM) sounding methods in the time and frequency domains are widely used in polymetallic ore exploration and can measure both EM induction and induced polarization (IP) responses simultaneously. Depth inversion is a crucial factor affecting the accuracy of data interpretation. The effect of IP on EM wave propagation has been ignored in traditional studies of diffusion depth and skin depth. This article is based on the generalized effective-medium theory of the IP effect (GEMTIP) model to represent the IP effect. The frequency domain solution based on GEMTIP is obtained. To address the problem whereby the time-domain solution cannot be solved by inverse Laplace transform, rational approximation and linear programming methods are used to simulate the variation of pulse field source with time and distance. We analyze the influences of geometrical parameters and physical properties of rocks and ores with the IP effect on the depth. Finally, we consider the influence of intrinsic chargeability and improve the classic diffusion depth formula. The formulae for estimating the time-domain diffusion depth of the IP effect under certain conditions are given and verified. These methods are of great significance to better understand the relationship between the subsurface dispersive media and the propagation of EM waves.

Simulation and 3-D Visualization of Crystallization Process Based on Cellular Automata

Crystallization is widely used for separation and purification. During this process, the different growth conditions, such as temperature and supersaturation, would result in different size and morphology, which not only affect the product quality, but also affect the subsequent process operations. Through the simulation of the crystallization process, a better understanding of crystallization process dynamics can be extracted. The population balance (PB) model has been used to simulate industrial crystallization processes, while the multi-dimensional population balance (MPB) model can be used to obtain the information of size and morphology distribution. However, the morphological distribution information provided by numerical solution of MPB is macroscopic, and the specific morphology of a crystal particle cannot be displayed at the microscopic level, so the difference among crystals morphology cannot be clearly displayed. Cellular automata (CA) is a method that simulates a process by considering simple cell changes and cell-to-cell interactions. It can be employed to conduct cross-scale simulation, i.e., both macro and micro. In this work, a CA method is introduced into an MPB model to simulate the batch cooling crystallization process from solution. The rule of crystal growth comes from classical diffusion theory, and whether the solution is transformed into a crystal is realized by the Monte Carlo method. The crystallization of potassium dihydrogen phosphate (KDP) is taken as a case-study and the results is verified with the results in the literature. Through CA simulation, not only the macroscopic crystal size and morphology distribution information can be obtained, but also the crystal morphology in microscopic can be exhibited. At the same time, the crystallization process can be visualized in 3-D environment, with size and morphology distribution being presented intuitively. In addition, the calculation time is reduced under certain accurate conditions. This result provides a theoretical reference for modeling, analysis and controlling of the crystallization process in the future.

Dynamics of Moments and Stationary States for GKSL Equations of Classical Diffusion Type

Dynamics of Moments and Stationary States for GKSL Equations of Classical Diffusion Type

Convolution integral for fractional diffusion equation

A modified version of the classical diffusion equation, called the fractional diffusion equation (FDE), has been developed to describe anomalous fluid flow through porous media. Since the FDE does not contain the rate as the input of the well-reservoir system, in this paper, we re-develop the FDE containing both rate (input) and pressure (output). Based on the new FDE, we show that the conventional convolution integral can still be used to relate the rate and pressure in reservoirs with anomalous behavior. A practical application of the convolution integral and how to use it in practice is presented using a synthetic example.

Macroscopic limit of a kinetic model describing the switch in T cell migration modes via binary interactions

Experimental results on the immune response to cancer indicate that activation of cytotoxic T lymphocytes (CTLs) through interactions with dendritic cells (DCs) can trigger a change in CTL migration patterns. In particular, while CTLs in the pre-activation state move in a non-local search pattern, the search pattern of activated CTLs is more localised. In this paper, we develop a kinetic model for such a switch in CTL migration modes. The model is formulated as a coupled system of balance equations for the one-particle distribution functions of CTLs in the pre-activation state, activated CTLs and DCs. CTL activation is modelled via binary interactions between CTLs in the pre-activation state and DCs. Moreover, cell motion is represented as a velocity-jump process, with the running time of CTLs in the pre-activation state following a long-tailed distribution, which is consistent with a Lévy walk, and the running time of activated CTLs following a Poisson distribution, which corresponds to Brownian motion. We formally show that the macroscopic limit of the model comprises a coupled system of balance equations for the cell densities, whereby activated CTL movement is described via a classical diffusion term, whilst a fractional diffusion term describes the movement of CTLs in the pre-activation state. The modelling approach presented here and its possible generalisations are expected to find applications in the study of the immune response to cancer and in other biological contexts in which switch from non-local to localised migration patterns occurs.

Electron vortexes in two-dimensional steady photoplasma

This paper reports the new phenomenon – the presence of electron vortexes in steadily generated two-dimensional photoplasma. A fluid model of a two-chamber photoplasma cell with sodium vapor and Argon as a buffer gas has been conducted in this study. The excitation source is in the first (driver) chamber. The transport processes of the generated plasma from the first into the second (diffusion) chamber create an electromotive force (EMF), which is the potential difference between them. It is shown that, in the two-chamber photoplasma, along with the flows of classical ambipolar diffusion, vortex currents of electrons are also formed. The change in temperature and electron density caused by them could affect the potential difference between the two chambers. The dependence of the current vortices and, consequently, the generated EMF on the buffer gas pressure is shown.

Projecting seismicity induced by complex alterations of underground stresses with applications to geothermal systems

Seismicity associated with subsurface operations is a major societal concern. It is therefore critical to improve predictions of the induced seismic hazard. Current statistical approaches account for the physics of pore pressure increase only. Here, we present a novel mathematical model that generalises adopted statistics for use in arbitrary injection/production protocols and applies to arbitrary physical processes. In our model, seismicity is driven by a normalised integral over the spatial reservoir volume of induced variations in frictional Coulomb stress, which—combined with the seismogenic index—provides a dimensionless proxy of the induced seismic hazard. Our model incorporates the classical pressure diffusion based and poroelastic seismogenic index models as special cases. Applying our approach to modeling geothermal systems, we find that seismicity rates are sensitive to imposed fluid-pressure rates, temperature variations, and tectonic conditions. We further demonstrate that a controlled injection protocol can decrease the induced seismic risk and that thermo-poroelastic stress transfer results in a larger spatial seismic footprint and in higher-magnitude events than does direct pore pressure impact for the same amount of injected volume and hydraulic energy. Our results, validated against field observations, showcase the relevance of the novel approach to forecast seismic hazards induced by subsurface activities.

Study of the wave period effect on the time-averaged suspended sediment concentration distribution under the oscillatory sheet flow condition

ABSTRACT Existing sheet flow experimental data show the measured time-averaged suspended sediment concentration (TSSC) may increase, decrease, or even remain constant if the wave period decreases. With respect to the available sheet flow experimental data, it is confirmed that the TSSC of relatively fine sand undergoes three stages, i.e. first decreases, then increases, and finally decreases again in the case that the wave period decreases from 12 s to 2 s, whereas for the relatively coarse sand, its vertical profile remains almost unchanged. The criterion for identifying the relatively fine or coarse sand was proposed in terms of the nondimensional ratio between the settling velocity and the root-mean-square flow velocity. If this ratio is less than 0.04–0.043, it is deemed as relatively fine sand. Otherwise, it is classified as relatively coarse sand. Applying the classical gradient diffusion model, sediment diffusivity is confirmed to be insensitive to the wave period. Keeping other experimental conditions uniform, TSSCs with different wave periods are proportional to each other and only affected by the reference concentration. Subsequently, a simple quantitative expression considering the wave period effect on the TSSC under the oscillatory sheet flow regime was proposed and verified with the experimental data.

On the effect of spatial fractional heat conduction in MHD boundary layer flow using Gr-Fe3O4–H2O hybrid nanofluid

In this article, the magnetohydrodynamic boundary layer flow and spatial fractional heat transport of Gr-Fe3O4–H2O hybrid nanofluid on a moving flat plate is investigated. The diffusion terms in the energy equation are modified by incorporating the spatial fractional derivatives using Fourier's law. Moreover, the effect of the joule heating, heat sink/source and viscous dissipation on the energy equation is also taken into account. The optimal collocation method as a new semi-analytical approach is introduced and the effects of physical parameters on the flow and heat transport specifications are obtained and analyzed. It is seen that changing the order of fractional derivatives from 0.92 (fractional diffusion) to 1 (classical diffusion) leads to a growth of the temperature. Furthermore, for the case λ = 0.7, the Nusselt number of Gr-Fe3O4–H2O augments by 22% when compared to Fe3O4–H2O and by 66% when compared to H2O. The results of Nusselt number and Skin friction coefficient for Prandtl number of 0.71 are validated by comparing with both the experimental and numerical studies in the existing literature, which confirms that the optimal collocation method can be successfully used to predict the spatial-fractional MHD boundary layer flow.

Maximum principle preserving finite difference scheme for 1-D nonlocal-to-local diffusion problems

In a recent paper (Du et al., 2018), a quasi-nonlocal coupling method was introduced to seamlessly bridge a nonlocal diffusion model with the classical local diffusion counterpart in a one-dimensional space. The proposed coupling framework removes interfacial inconsistency, preserves the balance of fluxes, and satisfies the maximum principle of diffusion problem. However, the numerical scheme proposed in that paper does not maintain all of these properties on a discrete level. In this paper we resolve this issue by proposing a new finite difference scheme that ensures the balance of fluxes and the discrete maximum principle. We rigorously prove these results and provide the stability and convergence analyses accordingly. In addition, we provide the Courant–Friedrichs–Lewy (CFL) condition for the new scheme and test a series of benchmark examples which confirm the theoretical findings.

Protected quantum coherence by gain and loss in a noisy quantum kicked rotor

We study the effects of non-Hermiticity on quantum coherence via a noisy quantum kicked rotor (NQKR). The random noise comes from the fluctuations in kick amplitude at each time. The non-Hermitian driving indicates the imaginary kicking potential, representing the environment-induced atom gain and loss. In the absence of gain and loss, the random noise destroys quantum coherence manifesting dynamical localization, which leads to classical diffusion. Interestingly, in the presence of non-Hermitian kicking potential, the occurrence of dynamical localization is highly sensitive to the gain and loss, manifesting the restoration of quantum coherence. Using the inverse participation ratio arguments, we numerically obtain a phase diagram of the classical diffusion and dynamical localization on the parameter plane of noise amplitude and non-Hermitian driving strength. With the help of analysis on the corresponding quasieigenstates, we achieve insight into dynamical localization, and uncover that the origin of the localization is interference between multiple quasi-eigenstates of the quantum kicked rotor. We further propose an experimental scheme to realize the NQKR in a dissipative cold atomic gas, which paves the way for future experimental investigation of an NQKR and its anomalous non-Hermitian properties.

A homogenized cerebrospinal fluid model for diffuse optical tomography in the neonatal head

Diffuse optical tomography is a non-invasive and non-irradiating medical imaging technique that is particularly suitable for cerebral monitoring of newborns since it can be used at the bedside of the patient. Here, a new model for optical tomography in the neonatal brain is presented that takes into account the presence of arachnoid trabeculae in the cerebrospinal fluid (CSF). It is known that the classical diffusion approximation (DA) for light propagation is at the limit of validity in the CSF layer due to the low values of the absorption and scattering coefficients. The new model is obtained by the DA of the homogenized radiative transfer equation and is rigorously justified. Numerical results in two and three dimensions attest for the improved sensitivity of the new model to the presence of perturbations in the brain layer.