An analysis of nonlinear ion transport model including diffusion and migration

The nonlocal identification problem related to nonlinear ion transport model including diffusion and migration is studied. Ion transport is assumed to be superposition of diffusion and migration under the influence of an electric field. Mathematical modeling of the experiment leads to an identification problem for a strongly nonlinear parabolic equation with nonlocal additional condition. Uniqueness of the nonlinear direct problem solution, and its continuity with respect to the total charge function is proved. An existence of a quasisolution of the identification problem is proved in the class of derived admissible coefficients. The nonlinear finite difference approximation of this problem, with an appropriate iteration algorithm, is derived. Numerical solutions of the identification problem are presented for various values of valences and diffusivities of oxidized and reduced oxidized species. The obtained results permits one to derive behaviour of the concentration and total charge depending on physical parameters.     

Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry

Ion transport problem related to controlled potential experiments in electrochemistry is studied. The problem is assumed to be superposition of diffusion and migration under the influence of an electric field. The comparative analysis are presented for three well-known models—pure diffusive (Cottrell’s), linear diffusion-migration, and nonlinear diffusion-migration (Cohn’s) models. The nonlinear model is derived by the identification problem for a nonlinear parabolic equation with nonlocal additional condition. This problem reduced to an initial-boundary value problem for nonlinear parabolic equation. The nonlinear finite difference approximation of this problem, with an appropriate iteration algorithm is derived. The comparative numerical analysis for all three models shows an influence of the nonlinear migration term, the valences of oxidized and reduced oxidized species, also diffusivity to the value of the total charge. The obtained results permits one to estimate bounds of linear and nonlinear ion transport models.     

Relationship between current response and time in ion transport problem including diffusion and convection. 2. Numerical approach

The mathematical model related to controlled potential experiments in electrochemistry is studied. Ion transport is regarded as the superposition of diffusion and migration under the influence of an electric field. Modeling of the experiment leads to the nonlocal identification problem for nonlinear parabolic equation. It is shown that in some cases the nonlocal identification problem can be transformed to an initial value problem for nonlinear parabolic equation. The finite diference approximation of this problem, with the appropriate iteration algorithm, is derived. Based on these algorithms the solution of the identification problem is presented. The obtained results permits one to derive the behaviour of the current response , depending on time, also the relationship between the current response and Gottrellian is obtained in explicit form. An influence of the valences oxidised and reduced species is also analyzed.      

On a unified treatment of kinetics and diffusion,and a connection to a nonlocal boltzmann equation

Fick's law of diffusion has been generalized to include kinetic processes, the transport term of the Boltzmann equation, and nonlocal interaction processes. It is shown that the collision interaction term can be obtained by the introduction of a quantum stochastic potential equation. Some approximations of a nonlocal Boltzmann equation can be solved exactly. The solutions can be applied to problems of molecular pattern in biology.     

Identification of unknown diffusion and convection coefficients in ion transport problems from flux data: an analytical approach

This article presents an analytical approach for identification problems related to ion transport problems. In the first part of the study, relationship between the flux j_L : = (D(x)u_x(0, t)_x=0{\varphi_L := (D(x)u_x(0, t)_{x=0}} and the current response I(t){{\mathcal I}(t)} is analyzed for various models. It is shown that in pure diffusive linear model case the flux is proportional to the classical Cottrelian I_C(t){{\mathcal I}_C(t)}. Similar relationship is derived in the case of nonlinear model including diffusion and migration. These results suggest acceptability of the flux data as a measured output data in ion transport problems, instead of nonlocal additional condition in the form an integral of concentration function. In pure diffusive and diffusive-convective linear models cases, explicit analytical formulas between inputs (diffusion or/and convection coefficients) and output (measured flux data) are derived. The proposed analytical approach permits one to determine the unknown diffusion coefficient from a single flux data given at a fixed time t ₁ > 0, and unknown convection coefficient from a single flux data given at a fixed time t ₂ > t ₁ > 0. Linearized model of the nonlinear ion transport problem with variable diffusion and convection coefficients is analyzed. It is shown that the measured output (flux) data can not be given arbitrarily.     

Specific Features in Measuring Particle Size Distributions in Highly Disperse Aerosol Systems

The distribution of highly dispersed aerosols is studied. Particular attention is given to the diffusion dynamic approach, as it is the best way to determine particle size distribution. It shown that the problem can be divided into two steps: directly measuring particle penetration through diffusion batteries and solving the inverse problem (obtaining a size distribution from the measured penetrations). No reliable way of solving the so-called inverse problem is found, but it can be done by introducing a parametrized size distribution (i.e., a gamma distribution). The integral equation is therefore reduced to a system of nonlinear equations that can be solved by elementary mathematical means. Further development of the method requires an increase in sensitivity (i.e., measuring the dimensions of molecular clusters with radioactive sources, along with the activity of diffusion battery screens).     

Identification of unknown diffusion coefficient in pure diffusive linear model of chronoamperometry. II. Numerical implementation

This article presents numerical implementation of the approach proposed in the previous study (Identification of the unknown diffusion coefficient in ion transport problem. I. The theory, Math. Chem. (2009) (submitted)) for the coefficient inverse problems related to linear diffusion equation in chronoamperometry. The coarse-fine grid algorithm is used for determination of the unknown diffusion coefficient D(x) in the linear parabolic equation u _t  = (D(x)u _x ) _x from the measured output data (left flux). The main distinguished feature of the implemented algorithm is the use of a fine grid for the numerical solution of well-posed forward and backward parabolic problems, and a coarse grid for the interpolation of the unknown diffusion coefficient D(x). The nodal values of the unknown coefficient on the coarse grid are recovered sequentially, solving on each step of the coarse grid iteration algorithm the well-posed forward, and the sequence of backward pababolic problems. This guarantees a compromise between the accuracy and stability of the solution of the considered inverse problem. An efficiency and applicability of the proposed approach is demonstrated on various numerical examples with noisy free and noisy data.     

222Rn gas diffusion and determination of its adsorption coefficient on activated charcoal

The adsorption coefficient is the fundamental parameter characterizing activated charcoal"s ability to adsorb ²²²Rn. The adsorption coefficient is determined for ²²²Rn activated charcoal detectors. In addition, a diffusion and adsorption model is developed for the transport of ²²²Rn in a porous bed of activated charcoal. These processes can be described by parabolic second order differential equation. The equation is numerically solved using the finite differences method. With this model, the ²²²Rn activity adsorbed in the detector is calculated for diverse situations.     

A structure-preserving computational method in the simulation of the dynamics of cancer growth with radiotherapy

In this work, we consider a two-dimensional mathematical model that describes the growth dynamics of cancer when radiotherapy is considered. The mathematical model for the local density of the tumor is described by a parabolic partial differential equation with variable diffusion coefficient. The nonlinear reaction term considers both the logistic law of proliferation of tumor cells and the effect of a treatment against cancer. Suitable initial-boundary conditions are imposed on a bounded spatial domain, and a theorem on the existence and the uniqueness of solutions for the initial-boundary-value problem is proved. Motivated by this result, we design a finite-difference methodology to approximate the solutions of our mathematical model. The scheme is a linear method that is capable of preserving the positivity and the boundedness of the approximations. Some simulations are presented in order to illustrate the performance of the method. Among other conclusions, the numerical results show that the method is able to preserve the analytical features of the relevant solutions of the model.     

Contribution of convection,diffusion and migration to electrolyte transport through nanofiltration membranes

Transport mechanisms through nanofiltration membranes are investigated in terms of contribution of convection, diffusion and migration to electrolyte transport. A Donnan steric pore model, based on the application of the extended Nernst-Planck equation and the assumption of a Donnan equilibrium at both membrane-solution interfaces, is used. The study is focused on the transport of symmetrical electrolytes (with symmetric or asymmetric diffusion coefficients). The influence of effective membrane charge density, permeate volume flux, pore radius and effective membrane thickness to porosity ratio on the contribution of the different transport mechanisms is investigated. Convection appears to be the dominant mechanism involved in electrolyte transport at low membrane charge and/or high permeate volume flux and effective membrane thickness to porosity ratio. Transport is mainly governed by diffusion when the membrane is strongly charged, particularly at low permeate volume flux and effective membrane thickness to porosity ratio. Electromigration is likely to be the dominant mechanism involved in electrolyte transport only if the diffusion coefficient of coions is greater than that of counterions.     

Haar wavelet method for solving some nonlinear Parabolic equations

Wavelet transform or wavelet analysis is a recently developed mathematical tool in applied mathematics. In this paper, we develop an accurate and efficient Haar transform or Haar wavelet method for some of the well-known nonlinear parabolic partial differential equations. The equations include the Nowell-whitehead equation, Cahn-Allen equation, FitzHugh-Nagumo equation, Fisher’s equation, Burger’s equation and the Burgers-Fisher equation. The proposed scheme can be used to a wide class of nonlinear equations. The power of this manageable method is confirmed. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive.     

Analytical formulaes and comparative analysis for linear models in chronoamperometry under conditions of diffusion and migration

Mathematical models of ion transport in a potential field are analyzed. Ion transport is regarded as the superposition of diffusion and migration. The explicit analytical formulaes are obtained for the concentration of the reduced species and the current response in the case of pure diffusive as well as diffusion–migration model, for various initial conditions. The comparitive analysis of these formulaes for current responses and deviation from the classical Cottrellian are derived. The proposed approach can predict an influence of ionic diffusivities, valences, initial and boundary concentrations to the behaviour of current response. In addition to these, the analytical formulaes obtained can also be used for numerical and digital simulation methods for Nernst-Planck equations.     

Estimation of concentration-dependent diffusion coefficient in pressure-decay experiment of heavy oils and bitumen

Molecular diffusion has been considered to be an underlying mechanism for many of oil recovery processes like miscible and immiscible gas injection projects. Reliable estimation of the molecular diffusion coefficient as a transport property is therefore important in studying the performance of such systems. Interpretation of pressure-decay data has been traditionally used to estimate the molecular diffusion coefficient and usually to simplify the interpretation, its concentration dependency has been neglected. A pressure-decay model with concentration-dependent diffusion coefficient leads to a non-linear problem in which an analytical solution is difficult if not impossible to obtain. In this study, we used the Heat Integral Method (HIM) to solve the non-linear diffusion problem as a forward model. Using that forward model, we have developed a simple methodology for estimating the diffusion coefficient regardless of the form of function used for the concentration dependency of the molecular diffusion. Three different forms of functions for diffusion coefficient were considered. In its simplest form, the diffusion coefficient is set to be a constant value. In the two other forms, the diffusion coefficient was evaluated as a concentration-dependent two parameter equation using exponential and power-law functions, respectively. The proposed methodology is verified and tested using direct numerical solutions of the non-linear diffusion problem. Many numerical examples with a wide range of input parameters demonstrate the effectiveness of the proposed approach.     

Migration study of iodine in Beishan granite by a column method

The fate and migration behavior of radionuclides in environment are influenced by a series of physical and chemical processes such as advection, hydrodynamic dispersion (including mechanical dispersion and molecular diffusion), retention, chemical reaction and so on. In this study, the migration of ¹²⁵I^? in Beishan granite and the potential retention of iodine by silver halide additives were investigated by a pulse injection column method. All breakthrough curves were analyzed according to the analytical solution of transport equation and the dispersion coefficient (D), and first-order sorption constant (k) were obtained. For conservative nuclide, the dispersion behavior is only related to the dispersion medium. Silver halides were proved having sorption ability for ¹²⁵I^? in the order of AgCl > AgBr > AgI. The transport of iodine in the crushed granite column can be adequately described by the advection–dispersion equation with a first-order, irreversible sorption term. The pulse injection column method can be used as a fast method to evaluate the sorption or retention ability of solid phase.     

Diffusion and Brownian motion in Lagrangian coordinates

In this paper we consider the convection-diffusion problem of a passive scalar in Lagrangian coordinates, i.e., in a coordinate system fixed on fluid particles. Both the convection-diffusion partial differential equation and the Langevin equation are expressed in Lagrangian coordinates and are shown to be equivalent for uniform, isotropic diffusion. The Lagrangian diffusivity is proportional to the square of the relative change of surface area and is related to the Eulerian diffusivity through the deformation gradient tensor. Associated with the initial value problem, we relate the Eulerian to the Lagrangian effective diffusivities (net spreading), validate the relation for the case of linear flow fields, and infer a relation for general flow fields. Associated with the boundary value problem, if the scalar transport problem possesses a time-independent solution in Lagrangian coordinates and the boundary conditions are prescribed on a material surface/interface, then the net mass transport is proportional to the diffusion coefficient. This can be also shown to be true for large Peclet number and time-periodic flow fields, i.e., closed pathlines. This agrees with results for heat transfer at high Peclet numbers across closed streamlines.     

Adaptive approach for nonlinear sensitivity analysis of reaction kinetics

We present a unified approach for linear and nonlinear sensitivity analysis for models of reaction kinetics that are stated in terms of systems of ordinary differential equations (ODEs). The approach is based on the reformulation of the ODE problem as a density transport problem described by a Fokker-Planck equation. The resulting multidimensional partial differential equation is herein solved by extending the TRAIL algorithm originally introduced by Horenko and Weiser in the context of molecular dynamics (J. Comp. Chem. 2003, 24, 1921) and discussed it in comparison with Monte Carlo techniques. The extended TRAIL approach is fully adaptive and easily allows to study the influence of nonlinear dynamical effects. We illustrate the scheme in application to an enzyme-substrate model problem for sensitivity analysis w.r.t. to initial concentrations and parameter values.     

Anomalous nonlinear dielectric and Kerr effect relaxation steady state responses in superimposed ac and dc electric fields

It is shown how the rotational diffusion model of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended to anomalous nonlinear dielectric relaxation and the dynamic Kerr effect by using a fractional kinetic equation. This fractional kinetic equation (obtained via a generalization of the noninertial kinetic equation of conventional rotational diffusion to fractional kinetics to include anomalous relaxation) is solved using matrix continued fractions yielding the complex nonlinear dielectric susceptibility and the Kerr function of an assembly of rigid dipolar particles acted on by external superimposed dc E0 and ac E1(t)=E1 cos omegat electric fields of arbitrary strengths. In the weak field limit, analytic equations for nonlinear response functions are also derived.     

Hopping transport and energy transfer in substitutionally disordered media

We present an alternative derivation of configuration-averaged equations governing transport and trapping in a three-component disordered system. It is assumed that migration of the initial excitation is described by the usual master equation. Using the Zwanzig projection operator technique, we were able, after averaging, to separate the contribution of the transport and transfer processes. We found finally that the averaged master equation including sinks is equivalent to a hierarchy of coupled equations.     

Darken's equation and other diffusion relations in the light of atomistic kinetic concepts

The atomistic kinetic approach of the DOCC sites concept (meaning, very simplified, that the dopant migration in solids progresses via sites which are suitable for occupation by dopant corpuscles summarized as dopant-occupiable sites, i.e. DOCC sites) is here used as a basis on which several diffusion models are thoroughly analysed. Since it is able to cover all effects determining dopant migrations, so that it may be valid in general, it proves other statements on dopant transport to be incorrectly formulated. Following this conception, Darken's equation leads to a link between the Fickian diffusion coefficient of an ideal solution and the activity coefficient of the non-ideal solution, which has up to now been ignored. Contrasting with Darken's hypothesis, Einstein's relation between the Fickian diffusion coefficient and the mobility of dopant particles proves true even in cases of non-ideal solutions. The supposed vacancy wind effect and the diffusion of dopant-defect pairs as molecule-like joined complexes are shown to be physically unrealistic. Orlowski's dopant flux formula proves false. Roth's and Plummer's model on oxidation-enhanced diffusion in silicon is shown to involve incorrectnesses. Received: 27 November 1998 / Accepted: 25 March 1999     

Improved 1.5-sided model for the weakly nonlinear study of Bénard-Marangoni instabilities in an evaporating liquid layer

Bénard-Marangoni instability, with coupled gravity and surface tension effects, in an evaporating liquid layer surmounted by its vapor and an inert gas is investigated theoretically. We show that this system can be described by a model that consists in the liquid layer equations plus the diffusion equation for the vapor in the gas (the so-called 1.5-sided model) and that this model is equivalent to a one-sided model when the vapor mass fraction field can be considered as quasi-stationary, provided that the equivalent Biot number is a nonlocal function of the interface temperature. A comparison of weakly nonlinear results for the 1.5-sided model with a previous one-sided model [M. Dondlinger, J. Margerit, P.C. Dauby, J. Colloid Interface Sci. 283 (2) (2005) 522-532] that considered a Biot number depending on the wavenumber evaluated at the threshold is performed. Very good agreement is found between both models. For this reason, the present analysis can also be considered as a detailed theoretical justification for the use of a one-sided model in the study of evaporative thermoconvection.