Diffusion Equation For Transport Research Articles

Numerical model of Earth ionosphere F region based on three-dimensional transport and ambipolar diffusion equations

Abstract The paper provides a detailed description of the numerical implementation of the transport scheme in the Earth’s ionosphere three-dimensional dynamical model of the Institute of Numerical Mathematics (INMIM). The presented version of INM-IM model takes into account the global dynamical processes of ion transport and ambipolar diffusion in the altitude range between 100 and 500 km (corresponding to F region). Based upon the splitting method the model solves the equations of ambipolar diffusion on the first split step and incorporates specifically designed advective transport scheme on the second step. The accuracy of transport scheme implementation in the model has been investigated through analytical solutions. The stability of the numerical algorithm has been demonstrated even for cases when transport velocities approaching extreme values.

Features of ion exchange between the electrodes in metal-ion batteries during discharge

The importance and relevance of the storage of electrical energy is confirmed by events in the world and trends in the development and use of various electrical energy systems, household appliances, computer equipment, communication devices, etc. In addition to the growth of the metal-ion battery markets, there are trends towards a search for metals that in the future will be inexpensive and will have characteristics required for storage systems. This paper considers ion exchange between the electrodes of metal-ion batteries whose charge carriers are metal ions, which diffuse in the process of discharge from the negative electrode to the positive one. A mathematical model was developed and tested. The model is based on a system of diffusion transport equations with the Nernst–Planck–Poisson potential equation replaced by an equivalent conductivity potential equation. Quasi-equilibrium regimes are considered. The entire working area consists of a pore electrode space and a neutral separator. The mathematical model employed consists of potential distribution equations and an electrolyte concentration distribution equation supplemented by the dependence of the electrode surface current on the overvoltage and equations that determine the electrode pore structure depending on the masses transferred inside the electrode. The electric potential and diffuse component mass transfer equations are written within the framework of the modern theory of effective electrical conductivity in batteries with account for current exchange between the solid electrodes and the liquid electrolyte. The research results showed the following. A change in the resistance of the separator (a change in porosity) has little effect, if any, on the electrode current densities, but it causes some change in the potentials themselves. A change in the resistance of the electrolyte affects both the electrode potentials and the internal current distribution between the electrodes and the electrolyte.

Reactor physics of research reactors at BARC: Aspects and scopes

Research reactors have a wide range of applications in the fields like medicine, agriculture, industry, development and validation of reactor physics codes, condensed matter research, etc. Reactor physics plays very important role in the design and operation of nuclear reactors. The nuclear data, the methods of solution of neutron integral and integro-differential transport equation, diffusion equation, both steady and time dependent have undergoes tremendous improvement over last few decades due to availability of fast computing machines with large memory. The whole analysis of a reactor requires developments of lattice code for lattice level calculations, multigroup diffusion codes for core calculations and point kinetics & space time kinetics codes for reactivity-initiated transients, on-line reactivity measurements using inverse kinetics method, a point depletion code for estimating radioactivity or concentration of fission products/actinides, etc. Reactor physics also plays key role in developing the neutron gamma coupled shielding codes. The reactor safety, which requires the coupling of kinetics equations, heat conduction equations, coolant dynamics equations and hydrodynamics equations is also very important especially after Fukushima accidents. BARC has three functional research reactors currently. In this paper, we discuss important reactor physics aspects and the scope of development with respect to research reactors at Trombay.

A Lattice-Boltzmann-based modelling chain for traffic-related atmospheric pollutant dispersion at the local urban scale

Urban traffic-related air pollution is a major source of environmental and health damage and is difficult to quantify due to its inherent physical complexity. We construct a CFD-based simulation framework coupling an efficient numerical method for turbulent fluid flows with a microscopic traffic model and an emissions model to simulate road transport pollutant dispersion at the urban microscale. We improve the open-source Lattice-Boltzmann based CFD software OpenLB to overcome its original stability deficiencies for high Reynolds number flows. A stable recursive regularization procedure with a double distribution function approach is proposed to solve an advection diffusion equation for passive scalar transport at high Reynolds number. The code is successfully validated on three reference cases of increasing complexity and the traffic model SUMO along with a physical engine emissions model are coupled with OpenLB to simulate traffic-induced pollution from a road network in a realistic complex geometry. Transient flow features are analysed and the time-averaged concentration levels in different neighbourhoods of the considered geometry are evaluated: high concentration levels are observed close to the streets but also inside specific building infrastructures due to complex wind dynamics. Analyses of altitudinal concentration variations show that flow recirculations located close to traffic lights can drive pollutant over the buildings and increase concentration levels inside inner courtyards. Time-averaged concentration maps are constructed using both spatially uniform and non-uniform line sources and it is shown that using uniform sources leads to up to 20% local concentration overestimations inside the urban canopy.

Convergence of the heat kernel approximate solutions of the transport-diffusion equation in the half-space

A family of approximate solutions to transport-diffusion equation in the half-space R+d is constructed by using the heat kernel and the translation representing transport on each step of time discretization. The uniform convergence of these approximate solutions and that of their first derivatives with respect to the space variables as well as the point-wise convergence of their second derivatives with respect to the space variables are proved. We also show that the limit function satisfies the transport-diffusion equation in the half-space with homogenous boundary conditions. From the technical point of view it is essential to obtain for the third derivatives of approximate solutions with respect to the space variables a necessary estimate for the passage to the limit, taking into account the influence of the boundary conditions.

Convection effect on solidification microstructure: dendritic arm spacing and microsegregation

Convection affect primary crystalline structure, particularly, the dendrite crystal velocity and dendrite tip radius. The present work aims on the influence of the convective flow on the primary dendrite spacing between neighbouring crystals within the dendrite ensemble. Solidification of a binary alloy is considered within the model of stagnant boundary layer, under imposed thermal gradient which influence on the crystal microstructure and chemical microsegregation. Chemical composition in the solidifying liquid and crystalline solid is derived from the solution of the solute diffusion transport equation taking into account the convective flow. The model results are consistent with the Scheil-Brody-Flemings model, experimental data and computational results.

Velocity averaging for diffusive transport equations with discontinuous flux

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to ‘recognise’ changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for non-linear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at t = 0 $t=0$ for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.

A Proposal for a Novel Formulation Based on the Hyperbolic Cattaneo’s Equation to Describe the Mechano-Transduction Process Occurring in Bone Remodeling

In this paper, we propose a model for the mechanical stimulus involved in the process of bone remodeling together with its evolution over time. Accumulated evidence suggests that bone remodeling could be interpreted as a feedback control process in which the mechanical state of the bone tissue is monitored, then appropriate signals are derived from the daily mechanical usage of the bone, these signals are transmitted into the surrounding region, and then they are detected by other agents whose purpose is to adapt the bone mass to the mechanical requirements of the environment. Therefore, we employ the diffusion equation for mass transport which is improved with Cattaneo’s correction to model the stimulus. This last improvement considers the effects of relaxation and non-locality, which we believe play essential roles in signaling messengers transport phenomena and are essential to match the evidence that suggests time-dependent excitations provide a more significant response at specific frequencies. To illustrate this particular behavior, numerical simulations have been performed in a 2D framework. The results fit the central aspect addressed, related to the dependency of the time of the adaptive process of bone, suggesting that our model is promising and deserves further investigation, both theoretical and experimental.

Rigidity aspects of singular patches in stratified flows

We explore the local well-posedness theory for the 2d inviscid Boussinesq system when the vorticity is given by a singular patch. We give a significant improvement of \cite{Hassainia-Hmidi} by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We establish a uniform LWP theory with respect to the vanishing conductivity. This issue is much more delicate than the inviscid case and one should carefully deal with various difficulties related to the diffusion effects which tend to alter some local structures. The weak a priori estimates are not trivial and refined analysis on transport-diffusion equation subject to a logarithmic singular potential is required. Another difficulty stems from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing effects of transport-diffusion equation advected by a velocity field which scales slightly below the Lipschitz class.

Classic Evans’s Drop Corrosion Experiment Investigated in Terms of a Tertiary Current and Potential Distribution

Background: Evans’s drop is a classic corrosion experiment that is nearly 100 years old, and it is analogous to other corrosion systems promoted by O2 gradients. The availability of more robust finite element software packages opens the possibility to reach a deeper understanding of these kind of corrosion systems. Methodology: In order to solve the problem, the model includes the governing mass transport diffusion and migration equation and the material balance in a nonsteady state by the finite element method. This is performed using COMSOL Multiphysics to predict the tertiary current and potential distribution considering the geometry, reaction kinetics, and mass transport for each ionic species. Significant Findings: A simulation of the tertiary current and potential distribution of the Evans’s drop corrosion experiment on an iron surface is presented. An oxygen concentration difference of 0.18 mol m−3 between the center and the drop periphery sets up a potential difference of 60 mV which acts as a corrosion driving force. Reaction kinetics are described by Tafel equations. Results include the evolution of concentration profiles for OH−, Fe2+, Fe3+, Fe(OH)2, and Fe(OH)3.

Modeling ultrafast demagnetization and spin transport: The interplay of spin-polarized electrons and thermal magnons

We theoretically investigate laser-induced spin transport in metallic magnetic heterostructures using an effective spin transport description that treats itinerant electrons and thermal magnons on an equal footing. Electron-magnon scattering is included and taken as the driving force for ultrafast demagnetization. We assume that in the low-fluence limit the magnon system remains in a quasi-equilibrium, allowing a transient nonzero magnon chemical potential. In combination with the diffusive transport equations for the itinerant electrons, the description is used to chart the full spin dynamics within the heterostructure. In agreement with recent experiments, we find that in case the spin-current-receiving material includes an efficient spin dissipation channel, the interfacial spin current becomes directly proportional to the temporal derivative of the magnetization. Based on an analytical calculation, we discuss that other relations between the spin current and magnetization may arise in case the spin-current-receiving material displays inefficient spin-flip scattering. Finally, we discuss the role of (interfacial) magnon transport and show that, a priori, it cannot be neglected. However, its significance strongly depends on the system parameters.

Numerical modeling of two dimensional non-capacity model for sediment transport by an unstructured finite volume method with a new discretization of the source term

The main goal of this work is the resolution of the two-dimensional shallow water equations of water–sediment mixture coupled to the transport diffusion equation for the total sediment load, and bed change rate equation by Roe scheme with a new discretization of the source term. The proposed discretization is well-balanced with the flux gradient and uses data right and left on the interfaces between two control volumes and satisfies the C-property. The numerical method uses unstructured meshes and incorporates minmod limiter and Runge–Kutta method to reach second order spatial and temporal accuracy. We also use an adaptive mesh based on gradient concentration of sediments to refine the study domain with a lower computational cost. We present some numerical results in order to verify and validate the performance of the numerical scheme, particular attention is given to the treatment of the dam-break problem over mobile beds. The numerical scheme demonstrates the intended accuracy and robustness to modelize dam-break flows over erodible bed.

Convergence of approximate solutions by heat kernel for transport-diffusion equation in a half-plane

In this paper, by using the heat kernel and the transport operator on each step of time discretization, approximate solutions for the transport-diffusion equation on the half-plane+2are constructed, and their convergence to a function which satisfies the transport-diffusion equation and the initial and boundary conditions is proved. These approximate solutions can be considered as a deterministic version of (the approximation of) the stochastic representation of the solution to parabolic equation, realized by the relationship between the heat kernel and the Brownian motion. But as they are defined only by an integral operator and transport, their properties and their convergence are proved without using probabilistic notions. The result of this paper generalizes that of recent papers about the convergence of analogous approximate solutions on the whole spacen. In case of the half-plane, it is necessary to elaborate (not trivial) estimates of the smoothness of the approximate solutions influenced by boundary condition.

Parameter estimation techniques for a chemotaxis model inspired by Cancer-on-Chip (COC) experiments

The present work is inspired by laboratory experiments, investigating the cross-talk between immune and cancer cells in a confined environment given by a microfluidic chip, the so called Organ-on-Chip (OOC). Based on a mathematical model in form of coupled reaction–diffusion-transport equations with chemotactic functions, our effort is devoted to the development of a parameter estimation methodology that is able to use real data obtained from the laboratory experiments to estimate the model parameters and infer the most plausible chemotactic function present in the experiment. In particular, we need to estimate the model parameters representing the convective and diffusive regimes included in the PDE model, in order to evaluate the diffusivity of the chemoattractant produced by tumor cells and its biasing effect on immune cells.The main issues faced in this work are the efficient calibration of the model against noisy synthetic data, available as macroscopic density of immune cells. A calibration algorithm is derived based on minimization methods which applies several techniques such as regularization terms and multigrids application to improve the results, which show the robustness and accuracy of the proposed algorithm.

Influence of Joule effect on thermal response of nano FinFET transistors

Integrated circuits based on FinFET nanodevices suffer from thermal processes related to heat dissipation caused by phonon transport bottlenecks. In this work, we report a numerical study of quasi-ballistic transport behavior in 20 nm FinFET transistors. Using a mathematical formulation, numerically implemented by the finite element method , we successfully validate the transfer characteristics and temporal temperature evolution compared with experimental and numerical works. We found that the proposed model, given by a calibrated charge drift-diffusion model coupled with the enhanced ballistic diffusive heat transport equation, predicts the experimental I–V characteristics and the transient temperature evolution well. The error given by this comparison is less than 10% in the I DS -V GS and T-t characteristics, respectively. We show that the proposed equations provide an improved approximation compared to the dual-phase lag and the Cattaneo-Vernotte models. From an engineering viewpoint, we found that the self-heating is concentrated at the end of the channel region at the drain side; this phenomenon was confirmed from literature by experimental results. • We report a numerical study of the quasi-ballistic transport behavior in 20 nm FinFET devices. • The model is able to predict the I(V) characteristics and heating phenomenon compared to experimental and numerical data. • The self-heating is concentrated at the end of the channel region at the drain side of the devices.

Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

Solving one-dimensional advection diffusion transport equation by using CDV wavelet basis

On a Mathematical model for traveling sand dune

Our aim in this paper is to introduce and study a mathematical model for the description of traveling sand dunes. We use surface flow process of sand under the effect of wind and gravity. We model this phenomenon by a nonlinear diffusion–transport equation coupling the effect of transportation of sand due to the wind and the avalanches due to the gravity and the repose angle. The avalanche flow is governed by the evolution surface model and we use a nonlocal term to handle the transport of sand face to the wind.

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.

Macrotransport theory for diffusiophoretic colloids and chemotactic microorganisms

Abstract

Anomalous diffusion and Noether's second theorem.

Despite the fact that conserved currents have dimensions that are determined solely by dimensional analysis (and hence no anomalous dimensions), Nature abounds in examples of anomalous diffusion in which L∝t^{γ}, with γ≠1/2, and heat transport in which the thermal conductivity diverges as L^{α}. Aside from breaking of Lorentz invariance, the true common link in such problems is an anomalous dimension for the underlying conserved current, thereby violating the basic tenet of field theory. We show here that the phenomenological nonlocal equations of motion that are used to describe such anomalies all follow from Lorentz-violating gauge transformations arising from Noether's second theorem. The generalizations lead to a family of diffusion and heat transport equations that systematize how nonlocal gauge transformations generate more general forms of Fick's and Fourier's laws for diffusive and heat transport, respectively. In particular, the associated Goldstone modes of the form ω∝k^{α},α∈R are direct consequences of fractional equations of motion.