Variable Diffusion Coefficient Research Articles

Theoretical and numerical analysis of problems with a boundary turning point and a variable diffusion coefficient

Theoretical and numerical analysis of problems with a boundary turning point and a variable diffusion coefficient

Impact of spatially varying transport coefficients in EMC3-Eirene simulations of W7-X and assessment of drifts

Abstract Modelling the scrape-off layer of a stellarator is challenging due to the complex magnetic 3D geometry. The here presented study analyses simulations of the scrape-off layer (SOL) of the stellarator Wendelstein 7-X (W7-X) using spatially varying diffusion coefficients for the magnetic standard configuration, extending our previous study (Bold et al 2022 Nucl. Fusion 62 106011). Comparing the EMC3-Eirene simulations with experimental observations, an inconsistency between the strike-line width (SLW) and the upstream parameters was observed. While to match the experimental SLW a particle diffusion coefficient D ≈ 0.2 m2 s−1 is needed, D ≈ 1 m2 s−1 is needed to get experimental separatrix temperatures of 50 eV at the given experimental heating power. We asses the impact of physically motivated spatially varying transport coeffients. Agreement with experimental data can be improved, but various differences remain. We show that drifts are expected to help overcome the discrepancies and, thus, the development of SOL transport models including drifts is a necessary next step to study the SOL transport of the W7-X stellarator.

Treatment of 3D diffusion problems with discontinuous coefficients and Dirac curvilinear sources

Three-dimensional diffusion problems with discontinuous coefficients and unidimensional Dirac sources arise in a number of fields. The statement we pursue is a singular-regular expansion where the singularity, capturing the stiff behavior of the potential, is expressed by a convolution formula using the Green kernel of the Laplace operator. The correction term, aimed at restoring the boundary conditions, fulfills a variational Poisson equation set in the Sobolev space H1, which can be approximated using finite element methods. The mathematical justification of the proposed expansion is the main focus, particularly when the variable diffusion coefficients are continuous, or have jumps. A computational study concludes the paper with some numerical examples. The potential is approximated by a combined method: (singularity, by integral formulas, correction, by linear finite elements). The convergence is discussed to highlight the practical benefits brought by different expansions, for continuous and discontinuous coefficients.

Inverse resolution of spatially varying diffusion coefficient using physics-informed neural networks

Inverse resolution of spatially varying diffusion coefficient using physics-informed neural networks

Theoretical study of the expansion of Roper's theory on the laminar jet diffusion flame length in 2D and 3D in natural gas and hydrogen mixtures

In this work, three mathematical models were developed to study the laminar jet diffusion flame in mixtures of natural gas, hydrogen, and CO2 by extending and applying Roper's theory. Model M1 consists of derivation of the Roper's original equation described in the two-dimensional steady state without convection, thus showing the theoretical equations for the length of the diffusion flame for burners with circular, square, and rectangular cross sections and for regimes in which the flames are dominated by the momentum effects, the buoyancy effect, and in the transition regime. Models M2 and M3 extend the Roper's original theory by deriving analytical solutions of the two- and three-dimensional diffusion flame equation with convection in steady and transient states, taking into account the spatially variable velocity and diffusion coefficients. The flame lengths obtained with models M1 and M2 were compared with model M3 by Relative Percentage Difference (RPD) considering 5 mixtures of NG-H2-CO2. For t = 0.1, the maximum RPD in the circular cross-section for M1 in Gas20:60%NG-25%H2–15%CO2 was 14.75%, while for M2 in Gas14:70%NG-15%H2–15%CO2 was 6.47% and the minimum RPD in Gas1:100%NG was 9.78% and 4.83% for M1 and M2, respectively. The same analysis was developed for the square and rectangular (Fr ≪ 1) cross-sections and showed good agreement. The model M3 was tested under transient conditions, studying the effects of time and velocity on the distribution of the profile in the radial direction, in order to better understand the flame phenomenon, which cannot be observed with the Roper's original model M1. Finally, the models M2 and M3 were validated with experimental results and other analytical solutions showing a good performance considering the characteristics of each mixture.

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.

Dynamic simulation of unsteady Casson–Carreau fluid flow considering temperature-dependent variable-properties and thermo-solutal mixed convection over flat and slendering sheets

The present investigation delves into the intricate interplay between surface effects and fluid dynamics, focusing on the flow characteristics of Casson and Carreau fluids endowed with temperature-dependent thermophysical properties. Specifically, the study explores the ramifications of radiative, mixed convection, and magnetized processes on heat and mass transfer phenomena. A comprehensive mathematical model is developed to analyze the behavior of Casson–Carreau fluids in conjunction with microscopic particles and gyrotactic microorganisms over both flat and slender surfaces. Leveraging similarity transformations, the governing equations are transformed into dimensionless form, facilitating numerical solutions using the bvp5c solver in MATLAB. The ensuing numerical and graphical analyses elucidate key physical quantities, including local skin friction coefficients, gyrotactic microorganism density, Nusselt number, and Sherwood number, across a spectrum of physical parameters. Notably, the investigation underscores the profound influence of flat surfaces on flow patterns and highlights the heightened consumption of fluid particles with increasing variable diffusivity and thermal conductivity. Moreover, enhanced values of parameters such as variable motile microorganism diffusion coefficient, Brownian motion parameter, and temperature-dependent thermal conductivity parameter are found to augment the Nusselt number, indicative of intensified heat transfer rates. This research underscores the imperative of comprehending the intricate dynamics of fluid flow and heat transfer, offering actionable insights for enhancing engineering designs and addressing multifaceted challenges across domains encompassing chemical engineering, environmental science, and biomedical applications.

Utilization of variable thermal conductivity and diffusion coefficient on non-Newtonian Prandtl model with modified heat and mass fluxes

PurposeThis study aims to analyze the Prandtl fluid flow in the presence of better mass diffusion and heat conduction models. By taking into account a linearly bidirectional stretchable sheet, flow is produced. Heat generation effect, thermal radiation, variable thermal conductivity, variable diffusion coefficient and Cattaneo–Christov double diffusion models are used to evaluate thermal and concentration diffusions.Design/methodology/approachThe governing partial differential equations (PDEs) have been made simpler using a boundary layer method. Strong nonlinear ordinary differential equations (ODEs) relate to appropriate non-dimensional similarity variables. The optimal homotopy analysis technique is used to develop solution.FindingsGraphs analyze the impact of many relevant factors on temperature and concentration. The physical parameters, such as mass and heat transfer rates at the wall and surface drag coefficients, are also displayed and explained.Originality/valueThe reported work discusses the contribution of generalized flux models to note their impact on heat and mass transport.

Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusion

We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/04/abstr.html

The diffusion scattering parameters identification for a modified model of viral infection in the conditions of logistic dynamics of immunological cells

Based on the modification of the infectious disease model, taking into account diffusion disturbances and logistic dynamics of immunological cells, separate approaches to the diffusion scattering parameters identification for different types of functional dependence of diffusion coefficients and given redefinition conditions are proposed. A special step-by-step procedure for numerically asymptotic approximation of the solution to the corresponding singularly perturbed model problem with a delay has been improved. The results of computer experiments on identifying the unknown diffusion scattering parameters are presented. It is noted that the identification and application of variable diffusion coefficients will provide a more accurate prediction of the dynamics of an infectious disease, which is significant in decision-making regarding the use of various medical procedures.

A numerical approximation for the Caputo-Hadamard derivative and its application in time-fractional variable-coefficient diffusion equation

A numerical approximation for the Caputo-Hadamard derivative and its application in time-fractional variable-coefficient diffusion equation

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.

Projection-Grid Schemes on Irregular Grids for a Parabolic Equation

A family of projection-grid schemes has been constructed for approximating parabolic equations with a variable diffusion coefficient in tensor form. The schemes are conservative and retain the self-adjointness of the original differential operator and are destined for calculations on 3D irregular difference grids, including tetrahedral, mixed (grids of arbitrary polyhedra), and locally adaptive (octal-tree type).

Fractional diffusion model generalised by the distributed-order operator involving variable diffusion coefficients

The diffusion process plays a crucial role in various fields, such as fluid dynamics, microorganisms, heat conduction and food processing. Since molecular diffusion usually takes place in complex materials and disordered media, there still exist many challenges in describing the diffusion process in the real world. Fractional calculus is a powerful tool for modelling complex physical processes due to its non-local property. This research generalises a fractional diffusion model by using the distributed-order operator in time and the Riesz fractional derivative in space. Moreover, variable diffusion coefficients are introduced to better capture the diffusion complexity. The fractional diffusion model is discretised by the finite element method in space. The approximation of the distributed-order operator is implemented by Simpson’s rule and the L2-1σ formula. A numerical example is provided to verify the effectiveness of the proposed numerical methods. This generalised fractional diffusion model may offer more insights into characterising diffusion behaviours in complex and disordered media.

Parameterizing Vertical Mixing Coefficients in the Ocean Surface Boundary Layer Using Neural Networks

AbstractVertical mixing parameterizations in ocean models are formulated on the basis of the physical principles that govern turbulent mixing. However, many parameterizations include ad hoc components that are not well constrained by theory or data. One such component is the eddy diffusivity model, where vertical turbulent fluxes of a quantity are parameterized from a variable eddy diffusion coefficient and the mean vertical gradient of the quantity. In this work, we improve a parameterization of vertical mixing in the ocean surface boundary layer by enhancing its eddy diffusivity model using data‐driven methods, specifically neural networks. The neural networks are designed to take extrinsic and intrinsic forcing parameters as input to predict the eddy diffusivity profile and are trained using output data from a second moment closure turbulent mixing scheme. The modified vertical mixing scheme predicts the eddy diffusivity profile through online inference of neural networks and maintains the conservation principles of the standard ocean model equations, which is particularly important for its targeted use in climate simulations. We describe the development and stable implementation of neural networks in an ocean general circulation model and demonstrate that the enhanced scheme outperforms its predecessor by reducing biases in the mixed‐layer depth and upper ocean stratification. Our results demonstrate the potential for data‐driven physics‐aware parameterizations to improve global climate models.

Fractal relaxation model with a nonlinear diffusion coefficient for fitting anomalous diffusion data in magnetic resonance imaging

In this paper a relaxation model of signal attenuation in diffusion-weighted magnetic resonance imaging (dMRI) with varying diffusion coefficient in terms of fractal derivative is proposed, in which the diffusion coefficient is a power law of the effective diffusion time. The relaxation model provides measures of diffusion constant, fractal dimension of diffusive trajectory of water molecule and the time power-law behavior of the diffusion coefficient. The proposed model was used to describe the magnetic resonance attenuation signal of the bullfrog sciatic nerve, and the corresponding spectral entropy was calculated to detect the environmental complexity in bullfrog sciatic nerve for water molecular diffusion. The results showed that the fractal derivative relaxation model (the VDC model) can accurately depict the diffusion pattern of water molecules in complex heterogeneous biological media at large b values. The VDC model provides an alternative theoretical reference for biological tissue detection based on time-dependent diffusion of water molecules.

Fractal derivative model with time dependent diffusion coefficient for chloride diffusion in concrete

More research and experiments have indicated that, under different water cement ratios, the binding effect of concrete on chloride ions and the blocking effect of hydration on chloride ions are clearly dominant. During this time period, the diffusion rate of chloride ions will increase or decrease with time, resulting in the classical Fickian diffusion law no longer being applicable. This study establishes an improved fractal model with a time-dependent diffusion coefficient for chloride diffusion in concrete. An explicit solution is derived in terms of the complementary error function. The proposed model is verified by fitting the chloride ion concentration data from different concrete immersion tests. The results show that the fractal model can fit the data in different spatial scales, and the corresponding prediction accuracy is rather high when compared to the three integer order models with constant, variable diffusion coefficient, and curing age. Therefore, the fractal derivative model is feasible in describing the anomalous diffusion of chloride ions in concrete.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

Numerical Solution of a Subdiffusion Equation with Variable Order Time Fractional Derivative and Nonlinear Diffusion Coefficient

Numerical Solution of a Subdiffusion Equation with Variable Order Time Fractional Derivative and Nonlinear Diffusion Coefficient

A compact ADI finite difference method for 2D reaction–diffusion equations with variable diffusion coefficients

Reaction–diffusion systems on a spatially heterogeneous domain have been widely used to model various biological applications. However, solving such partial differential equations (PDEs) analytically is rarely possible. Therefore, efficient and accurate numerical methods for solving such PDEs are desired. In this paper, we apply the well-known Padé approximation-based operator splitting techniques and develop a fourth-order compact alternative directional implicit (ADI) scheme. The new scheme is compact and fourth-order accurate in space. Combined with the Richardson extrapolation, the method can be improved to fourth-order accuracy in time. Stability analysis shows that the method is unconditionally stable; thus, a large time step can be used to improve the overall computational efficiency. Numerical examples have also demonstrated the new scheme’s high efficiency and high order accuracy.