Constant Diffusion Coefficient Research Articles

Diffusion-driven flows in a nonlinear stratified fluid layer

Diffusion-driven flow is a boundary layer flow arising from the interplay of gravity and diffusion in density-stratified fluids when a gravitational field is non-parallel to an impermeable solid boundary. This study investigates diffusion-driven flow within a nonlinearly density-stratified fluid confined between two tilted parallel walls. We introduce an asymptotic expansion inspired by the centre manifold theory, where quantities are expanded in terms of derivatives of the cross-sectional averaged stratified scalar (such as salinity or temperature). This technique provides accurate approximations for velocity, density and pressure fields. Furthermore, we derive an evolution equation describing the cross-sectional averaged stratified scalar. This equation takes the form of the traditional diffusion equation but replaces the constant diffusion coefficient with a positive-definite function dependent on the solution's derivative. Numerical simulations validate the accuracy of our approximations. Our investigation of the effective equation reveals that the density profile depends on a non-dimensional parameter denoted as $\gamma$ representing the flow strength. In the large $\gamma$ limit, the system is approximated by a diffusion process with an augmented diffusion coefficient of $1+\cot ^{2}\theta$ , where $\theta$ signifies the inclination angle of the channel domain. This parameter regime is where diffusion-driven flow exhibits its strongest mixing ability. Conversely, in the small $\gamma$ regime, the density field behaves like pure diffusion with distorted isopycnals. Lastly, we show that the classical thin film equation aligns with the results obtained using the proposed expansion in the small $\gamma$ regime but fails to accurately describe the dynamics of the density field for large $\gamma$ .

Shear-driven diffusion with stochastic resetting

External flows, such as shear flow, add directional biases to particle motion, introducing anisotropic behavior into the system. Here, we explore the non-equilibrium dynamics that emerge from the interplay between linear shear flow and stochastic resetting. The particle diffuses with a constant diffusion coefficient while simultaneously experiencing linear shear and being stochastically returned to its initial position at a constant rate. We perturbatively derive the steady-state probability distribution that captures the effects of shear-induced anisotropy on the spatial structure of the distribution. We show that the dynamics, which initially spread diffusively, will at late times reach a steady state due to resetting. At intermediate timescales, the system approaches this steady state either by passing through a superdiffusive regime (in the shear-dominated case) or by exhibiting purely sub-diffusive behavior (in the resetting-dominated case). The steady state also gains cross correlations, a feature absent in simpler resetting systems. We also show that the skewness has a non-monotonic behavior when one passes from the shear-dominated to the resetting-dominated regime. We demonstrate that at small resetting rates, the energetic cost of maintaining the steady state becomes significantly higher due to the displacement caused by shear, a unique scaling not seen without shear. Surprisingly, if only the x-position is reset, the system can maintain a Brownian yet non-Gaussian diffusion pattern with non-trivial tails in the distribution.

Reaction-diffusion for fish populations with realistic mobility

Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations.

Determining the constant diffusion coefficient by Fourier series solution of the diffusion equation

The purpose of this study was to obtain the diffusion coefficient by calculating directly from the Fourier series solution of the diffusion equation, which has not been studied so far. The Fourier series solution is the sum of a steady-state solution and a transient solution. The steady-state solution is a linear function of distance. The transient solution is the product of a distance function and a time function. The Fourier sine series solution, a function of distance only, is negative for the steady-state solution. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate (PET) film using the film-roll method, the constant surface concentration and the diffusion distance were determined from the quadratic regression curve for the concentration distribution at a long time. The concentration distribution of the transient solution is a downwardly convex curve with negative values. The concentration distribution of the Fourier series solution is also a downwardly convex curve. The diffusion coefficient in the exponential term, a function of time only, was determined from the value of the exponential term obtained by the surface concentration and the diffusion distance. The constant diffusion coefficients determined from the Fourier series solution are reliable because they are very similar to those obtained from the error function solution and have excellent linearity of the linear regression lines for their Arrhenius plots.

Cosmic-Ray Feedback on Bistable Interstellar Medium Turbulence

While cosmic rays (E ≳ 1 GeV) are well coupled to a galaxy’s interstellar medium (ISM) at scales of L > 100 pc, adjusting stratification and driving outflows, their impact on small scales is less clear. Based on calculations of the cosmic-ray diffusion coefficient from observations of the grammage in the Milky Way, cosmic rays have little time to dynamically impact the ISM on those small scales. Using numerical simulations, we explore how more complex cosmic-ray transport could allow cosmic rays to couple to the ISM on small scales. We create a two-zone model of cosmic-ray transport, with the cosmic-ray diffusion coefficient set at the estimated Milky Way value in cold gas but smaller in warm gas. We compare this model to simulations with a constant diffusion coefficient. Quicker diffusion through cold gas allows more cold gas to form compared to a simulation with a constant, small diffusion coefficient. However, slower diffusion in warm gas allows cosmic rays to take energy from the turbulent cascade anisotropically. This cosmic-ray energization comes at the expense of turbulent energy which would otherwise be lost during radiative cooling. Finally, we show our two-zone model is capable of matching observational estimates of the grammage for some transport paths through the simulation.

Real-space diffusion theory from quantum mechanics using analytic continuation

We show that a physical correspondence between Brownian motion and quantum mechanics can be established by formal analytic continuation if Wick rotation (t→it) is replaced by the mirror symmetric, complex conjugate time transformation t→±it. Invariance of the square modulus of the wave function under this transformation reveals a tight connection between Born's interpretation of the probability density and the proper time t2=(−it)(+it), as being both the result of a time symmetry breaking in the informational content of the microscopic (quantum) world, which takes place as we move to a macroscopic level. We find that under the transformation t→±it, the Schrödinger equation and its complex conjugate conforms with a stochastic-mechanical model of the Brownian motion of a particle. From the present analysis, Nelson's osmotic velocity arises naturally and the quantum phase function admits a classical interpretation in terms of a continuous (scalar) potential field that influences the motion of the particle. The maximum phase variation defines the direction of the osmotic and current velocity. Whereas the paper focuses on the correlation between quantum mechanics and Ficks's second law for a constant diffusion coefficient, a discussion is also provided on the possibility of a quantum mechanical formulation for anomalous diffusion, where the diffusion coefficient is a function of space and/or time.

Heat-dependent properties of methane diffusion in coal: an experimental study and mechanistic analysis.

Coalbed methane thermodynamic extraction, as an emerging ECBM recovery method, can effectively improve gas recovery rates. And clarifying methane diffusion and migration law in coal under thermal stimulation is crucial for the selection of its process parameters. Based on laboratory methane adsorption-release experiments, the evolution law of methane diffusion characteristics with temperature and pressure was studied, and the control mechanism of heat-dependent methane diffusion behavior was explored. The results show that both thermal stimulation and high adsorption pressure accelerated the methane diffusion rate in coal. Adsorption pressure had little effect on methane diffusion percentage, but thermal stimulation promoted a significant increase in diffusion percentage and improved the net methane yield. The influence mechanisms of adsorption pressure and thermal stimulation on methane diffusion characteristics are elucidated in relation to the amount and proportion of methane-activated molecules in the diffusion process. The constant diffusion coefficient of methane is heat-dependent, based on which a diffusion model is derived to accurately predict the methane release process in coal. Additionally, temperature has a more important effect on transient diffusion coefficient than pressure. Thermal stimulation leads to a net increase rather than a decrease in diffusion coefficient in the early diffusion stages and can also accelerate the attenuation of diffusion coefficient, with this intensifying effect becoming more pronounced at higher temperatures. The research results can provide some reference for the determination of coal seam gas content and the selection of heat injection process parameters.

A nonlocal convection–diffusion model with Gaussian‐type kernels and meshfree discretization

AbstractNonlocal models have demonstrated their indispensability in numerical simulations across a spectrum of critical domains, ranging from analyzing crack and fracture behavior in structural engineering to modeling anomalous diffusion phenomena in materials science and simulating convection processes in heterogeneous environments. In this study, we present a novel framework for constructing nonlocal convection–diffusion models using Gaussian‐type kernels. Our framework uniquely formulates the diffusion term by correlating the constant diffusion coefficient with the variance of the Gaussian kernel. Simultaneously, the convection term is defined by integrating the variable velocity field into the kernel as the expectation of a multivariate Gaussian distribution, facilitating a comprehensive representation of convective transport phenomena. We rigorously establish the well‐posedness of the proposed nonlocal model and derive a maximum principle to ensure its stability and reliability in numerical simulations. Furthermore, we develop a meshfree discretization scheme tailored for numerically simulating our model, designed to uphold both the discrete maximum principle and asymptotic compatibility. Through extensive numerical experiments, we validate the efficacy and versatility of our framework, demonstrating its superior performance compared to existing approaches.

Mechanical Failure of Core-Shell Cathode Particles: The Effects of Concentration-Dependent Material Properties and Phase Field Fracture Modelling

The use of core-shell cathode particles in lithium-ion batteries is an attractive approach to enhancing energy density whilst retaining lifetime, through reducing undesired reactions at the electrode-electrolyte interface and limiting the volume change of the electrode particles. However, mechanical failure through the fracture and debonding of the core-shell interface is a major challenge. In this work, we employ a coupled finite-element model to predict and mitigate the mechanical failure of core-shell cathode structures, taking as example a particle of NMC811 (core) coated with NMC111 (shell). In particular, we focus on two aspects:The first one involves the assumptions of material properties as inputs of the model. The material properties are often considered constant by battery modelling researchers, yet these parameters can vary significantly during charge/discharge. For example, experiments have unveiled a three-orders-of-magnitude drop in the diffusion coefficient of NMC materials during discharge [1]. Here, we incorporate material properties obtained from experimental data, including concentration-dependent diffusion coefficient obtained from GITT measurement and partial molar volume derived from in situ X-ray diffraction data. Our results indicate that when assuming a concentration-dependent partial molar volume, the maximum values of tensile hoop stress in the shell are nearly three times lower than those predicted with constant average properties, diminishing the likelihood of fracture.When accounting for concentration-dependent diffusion coefficient, large concentration gradient is observed near the outer surface of the core due to reduced lithium mobility at high states of lithiation, hindering full electrode capacity utilisation. The significant concentration gradient and capacity underutilisation align with direct observations from experiments [2]. Notably, these phenomena cannot be captured in modelling if assuming constant diffusion coefficient. These findings offer new perspectives on the performance and design of core-shell electrode particles. Safe design maps in terms of core size and shell thickness to mitigate fracture and debonding are obtained.The second focus of this work is the utilisation of phase field fracture model in predicting cracking behaviours within the core-shell particle. Phase field modelling provides a continuous representation of cracks, enabling the tracking of complex cracking patterns [3]. We predict cracking behaviors caused by volume changes during charge/discharge, as well as inaccessible capacity and newly created active surface area caused by cracking. Our multiphyics model can hopefully act as a catalyst, accelerating the development process of the core-shell structured cathode particles and shortening the time to market of advanced batteries with extended lifetime.

The Analytical Theory of Water Activity in the Paddy Rice Drying Process

Drying involves the evaporation of moisture, accompanied by simultaneous heat transfer, mass transfer, and momentum transfer. While the diffusion law is considered an applicable model for explaining the drying phenomenon, the actual drying process cannot be accurately predicted using an analytical solution with a constant diffusion coefficient. Energy efficiency in the drying process is low due to an insufficient understanding of the mechanisms governing moisture migration from solids to air. The development of drying theory has stalled due to an unsolvable discrepancy between experimental results and analytical results. This study analyzes the effect of the binding energy of moisture in paddy rice on the diffusion coefficient. The theoretical relationship between water activity and drying rate in paddy rice was investigated, and the drying process was successfully explained by analyzing free energy transfer and transition theory. The mechanism of rice grain drying was described using a new theoretical solution for the drying process. These results provide new insights into the development of a scientific evaluation standard for assessing the efficiency of actual drying processes.

Simulation and optimization of fatty acid extraction parameters from Nannochloropsis sp. using supercritical carbon dioxide

Microalgae, which are rich in fatty acids, have potential applications in various sectors such as bioenergy, health, food, and biomaterials. The Supercritical Fluid Extraction (SFE) method is commonly used to extract microalgae. This research estimated the process parameters of desorption rate constant (kd) and binary diffusion coefficient (DAB) for SFE fatty acid from Nannochloropsis sp. using a mathematical model called as hot sphere diffusion. Desorption models were used to model the release of fatty acids into the solvent (supercritical carbon dioxide). The parameter estimation process was conducted at temperatures of 313 and 333 K and pressures of 12.5, 20, and 30 MPa. The value of kd increased with increasing pressure and temperature and DAB values were obtained at varying pressures and temperatures.

Macroscopic finite-difference scheme and modified equationsof the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Invariant analysis of the two-cell tumor growth model in the brain

In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model’s behavior. Notably, our findings demonstrate that the tumor’s growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model’s exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.

Mechanistic Investigation of Enzyme Triggered Release from a Xyloglucan Matrix Tablet for Controlled Colonic Drug Delivery

The objective of this study was to investigate the mechanisms underlying drug release from a controlled colonic release (CCR) tablet formulation based on a xyloglucan polysaccharide matrix and identify the factors that control the rate of release for the purpose of fundamentally substantiating the concept and demonstrating its robustness for colonic drug delivery. Previous work demonstrated in vitro limited release of 5-aminosalicylic acid (5-ASA) and caffeine from these tablets in small intestinal environment and significant acceleration of release by xyloglucanase, an enzyme of the colonic microbiome. Targeted colonic drug delivery was verified in an animal study in vivo. In the present work, interaction of the xyloglucan matrix tablets with aqueous dissolution media containing xyloglucanase was found to lead to the spontaneous formation of a hydrated highly viscous gummy layer at the surface of the matrix which had a reduced drug content compared to the underlying regions and persisted with a nearly constant thickness that was inversely correlated to the enzyme concentration throughout the duration of the release process. Enzymatic hydrolysis of xyloglucan was determined to take place at the surface of the matrix leading to matrix erosion and a relation for the rate of enzymatic reaction as a function of bulk enzyme concentration and the concentration of dissolved xyloglucan in the gummy layer was derived. A mathematical model was developed encompassing aqueous medium ingress, matrix metamorphosis due to xyloglucan dissolution and matrix swelling, enzymatic hydrolysis of the polysaccharide and concomitant drug release due to matrix erosion and simultaneous drug diffusion. The model was fitted to data of reducing sugar equivalents in the medium reflecting matrix erosion and released drug amount. Enzymatic reaction parameters and reasonable values of medium ingress velocity, xyloglucan dissolution rate constant and drug diffusion coefficient were deduced that provided an adequate approximation of the data. Erosion was shown to be the overwhelmingly dominant drug release mechanism while the role of diffusion marginally increased at low enzyme concentration and high drug solubility. Changing enzyme concentration had a rather weak effect on matrix erosion and drug release rate as demonstrated by model simulations supported by experimental data, while xyloglucan dissolution was slow and had a stronger effect on the rate of the process. Therefore, reproducible colonic drug delivery not critically influenced by inter- and intra-individual variation of microbial enzyme activity may be projected.

A new dynamic Monte Carlo method satisfying n-particle diffusion equation with position-dependent diffusion coefficient, free energy, and intermolecular interactions.

A dynamic Monte Carlo (MC) method recently proposed by us [Nagai et al., J. Chem. Phys. 156, 154506 (2022)] to describe single-particle diffusion of a molecule in a heterogeneous space with position-dependent diffusion coefficient and free energy is generalized here to n-particle dynamics, where n molecules diffuse in heterogeneous media interacting via their intermolecular potential. Starting from the master equation, we give an algebraic proof that the dynamic MC transition probabilities proposed here produce particle trajectories that satisfy the n-particle diffusion equation with position-dependent diffusion coefficient D0i(ri), free energy F1i(ri), and intermolecular interactions Vij(ri, rj). The MC calculations based on this method are compared to molecular dynamics (MD) calculations for two-dimensional heterogeneous Lennard-Jones test systems, showing excellent agreement of the long-distance global diffusion coefficient between the two cases. Thus, the particle trajectories produced by the present MC transition probabilities satisfy the n-particle diffusion equation, and the diffusion equation well describes the long-distance trajectories produced by the MD calculations. The method is also an extension of the conventional equilibrium Metropolis MC calculation for homogeneous systems with a constant diffusion coefficient to the dynamics in heterogeneous systems with a position-dependent diffusion coefficient and potential. In the present method, interactions and dynamics of the real systems are coarse-grained such that the calculation cost is drastically reduced. This provides an approach for the investigation of particle dynamics in very complex and large systems, where the diffusing length is of sub-micrometer order and the diffusion time is of the order of milliseconds or more.

A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and its macroscopic finite-difference scheme for two-dimensional diffusion equations

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the corresponding macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain a macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis on the finite-difference scheme and the mesoscopic MRT-LB model, and find that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis.

An inverse problem for nonlocal reaction-diffusion equations with time-delay

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature.

Computational study on the influence of variable diffusion coefficients on non-fourier transport in shear-rate-dependent fluid with solute particles

Heat transfer in fluids occurs in numerous industrial processes. It is a fact that diffusion coefficients during heat transfer in fluids do not remain constant. Therefore, consideration of the constant diffusion coefficient is not a good approximation. In view of this, generalized modelling of heat and mass transfer in fluids of variable properties is done, and these generalized problems are solved by finite element method (FEM). After pre- and post-calculations, numerical runs are executed to observe the influence of variation in conductance on transport mechanisms in the fluid. The impact of other parameters is also studied. The outcomes are recorded in the form of graphs and numerical data. Ohmic dissipation results in an increase in the temperature of the fluid for variable and constant viscosity. The thermal relaxation time has shown significant effects on the flow and thickness of the thermal boundary layer region. The temperature of the fluid decreases with thermal relaxation time. The thermal conductance of the fluid varies with an increase in temperature. Once the temperature rises, the ability of the fluid to conduct heat is increased, due to which more heat from the heated surface diffuses into the fluid. Hence, heat flux decreases.The behavior of the variable viscosity of the fluid on the diffusion of momentum into the fluid is studied, and it is now known that the diffusion of wall momentum into the fluid is compromised because of a decrease in the viscosity, which is inversely proportional to the temperature. Hence, the diffusion of wall momentum into the fluid in the isothermal case is greater than that in the temperature-varying case.

MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS

The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements).

Application of image-based diffusion coefficients in multi-scale simulation of coalbed methane reservoirs

Gas diffusion in coalbed methane (CBM) reservoirs is a multi-scale process. It can be a dominant factor in long-term gas production performance. Producing coalbed methane requires a detailed understanding of diffusion behaviours and their contribution to production. This paper develops a core-scale model based on micro-CT images. Our model considers free and adsorbed gas diffusion, sorption, and viscous flow. It also includes the effect of swelling, gas moisture, and effective stress. Following that, it is used to simulate gas flow on micro-CT images. The simulation results quantify the influence of effective stress on gas diffusion in coal. The core-scale model is then coupled to the large-scale simulation, where we compare the gas production curves against the result based on the assumption of a constant diffusion coefficient. This model can be used to calculate the effective diffusion coefficient from a micro-CT image and to predict the production performance at the reservoir scale reliably using a coupled multi-scale approach.