The backward group preserving scheme for 1D backward in time advection‐dispersion equation

In this article, we propose a backward group preserving scheme (BGPS) on the advection‐dispersion equation (ADE) for tackling of the contamination problems. The BGPS has been successfully applied on the backward heat conduction problems as well as the backward in time Burgers equation, but it has never been applied to solve the ADE. The BGPS is able to recover the spatial distribution of groundwater contaminant concentration in this work. Several numerical examples are worked out, and we show, based on those numerical examples, that the BGPS is applicable to the ADE and the method can also handle the ADE with piecewise constant dispersion coefficients. When a steep gradient is appeared in the solution, several steps of the BGPS can be used to retrieve the desired initial data and its result is better than the marching‐jury backward beam equation (MJBBE) method as far as our examples are concerned. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

浅水中污染物扩散的自适应有限元分析法

介绍浅水中污染物扩散分析中的有限元法.分析包括两个部分:1)流场速度、水面高度的计算;2)根据扩散模型计算污染物浓度场.联合使用了自适应网格技术以期提高解的精度,同时减少计算时间和计算机内存的消耗.通过几个有已知解的实例验证了有限元公式和计算机程序.最后,使用这种联合方法分析泰国Chao Phraya河附近海湾中的污染物扩散.     

Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer

This paper is concerned with the Rayleigh wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer under initial stress. Both the layer and half-space have subjected to the incompressible in nature. The particle motion of the Rayleigh type wave is elliptically polarized in the plane, which described by the normal to the surface and the focal point along with wave generation. The dispersion of waves refers typically to frequency dispersion, which means different wavelengths travel at a different velocity of phase. To deal with the analytical solution of displacement components of Rayleigh type waves in a layer over a half-space, we have taken the assistance of different methods like exponential, characteristic polynomial and undetermined coefficients. The dispersion relation has been derived based upon suitable boundary conditions. The finite difference scheme has been introduced to calculate the phase velocity and group velocity of the Rayleigh type waves. We also have derived the stability condition of the finite difference scheme (FDS) for the phase and group velocities. If a wave equation has to travel in the time domain, it is necessary to achieve both accuracy and stability requirements. In such cases, FDS is preferred because of its power, accuracy, reliability, rapidity, and flexibility. The effect of various parameters involved in the model like non-homogeneity, porosity, and internal pre-stress on the propagation of Rayleigh type waves have been studied in detail. Graphical representations for the effects of various parameters on the dispersion equation have been represented. Numerical results demonstrated the accuracy and versatility of the group and phase velocity depending on the stability ratio of the FDS.     

Effect of chemical reaction on the dispersion of a solute in a porous medium

A mathematical model is presented in this paper which describes the dispersion of a chemically active solute in the laminar flow in a sparsely packed porous medium. The validity of time-dependent dispersion coefficient is widened by using a generalized dispersion coefficient. The effect of porous parameter and chemical reaction on the dispersion coefficient is studied. The exact solution for the mean concentration distribution of a chemically active solute is obtained as a function of downwind distance and time. Results are also obtained for pure convection.     

Environmental dispersion in wetland flow

For the typical case of a pulsed contaminant emission into a shallow wetland channel, a theoretical analysis is presented in this paper for the decay of the width-averaged mean concentration under environmental dispersion. The velocity profile of a fully developed steady flow through the wetland channel is obtained with that for the well-known plane Poiseuille flow as a special case. An environmental dispersion model for the mean concentration is devised as an extension of Taylor’s classic analysis on dispersion, and corresponding environmental dispersivity is obtained by Aris’s method of moments and illustrated with an asymptotic time variation with stem dominated, transitional, and width-stem dominated stages. Analytical solution for the longitudinal decay of mean concentration due to environmental dispersion is rigorously derived and characterized with multiple time scales.     

Limit behavior of the global solutions to the Degasperis-Procesi-type equation

In this paper, we study the Degasperis-Procesi equation with a physically perturbation term—a linear dispersion. Based on the global existence result, we show that the solution of the Degasperis-Procesi equation with linear dispersion tends to the solution of the corresponding Degasperis-Procesi equation as the dispersive parameter goes to zero. Moreover, we prove that smooth solutions of the equation have finite propagation speed: they will have compact support if its initial data has this property.     

Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms

We report exact bright and dark solitary wave solution of the nonlinear Schrodinger equation (NLSE) in cubic–quintic non-Kerr medium adopting phase–amplitude ansatz method. We have found the solitary wave parameters along with the constraints under which bright or dark solitons may exist in such a media. Furthermore, we have also studied the modulation instability analysis both in anomalous and normal dispersion regime. The role of fourth order dispersion, cubic–quintic nonlinear parameter and self-steeping parameter on modulation instability gain has been investigated.     

A modified moments analysis of dispersion in steady and oscillatory flows

A modification of Aris's method of moments solution to the heat/mass dispersion equation has been developed. In addition to taking axial moments of concentration, the present approach also defines mean cross-sectionally weighted values of the concentration. This leads to a set of ordinary differential equations rather than the partial differential equations generated by Aris's method. Approximate solutions have been developed for the cases of steady and oscillatory flows in a flat channel and values of the dispersion coefficient have been in good agreement with exact analytical predictions calculated by Dewey and Sullivan and by Watson.

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Zusammenfassung Eine Modifaktion der Momenten-Methode von Aris für die Lösung der Dispersionsgleichung für Wärme/Masse ist entwickelt worden. Die neue Methode benützt nicht nur axiale Momente der Konzentration, sondern definiert zusätzlich im Querschnitt gewogene Durchschnittswerte der Konzentration. Dieses führt zu einem Satz gewöhnlicher Differentialgleichungen, an Stelle der partiellen Differentialgleichung nach der Methode von Aris. Angenäherte Lösungen sind entwickelt worden für die Fälle von stationären und oszillatorischen Strömungen in einem ebenen Kanal. Ergebnisse für Dispersions-Koeffizienten stimmen gut überein mit den exakten analytischen Werten, die von Dewey und Sullivan sowohl wie von Watson berechnet worden sind.

     

Dispersion models for annular climbing film flow

The mathematical theory of dispersion in annular climbing film flow is developed. Starting with dispersion in a uniform film, the theory is extended to incorporate successively the effects of a viscous sublayer, disturbance waves and interchange of material with entrained droplet. These effects are considered independently but their combined influence on the overall dispersion characteristics of the system is shown to be capable of analysis in terms of an interchange dispersion model (IDM). A solution method for this interchange model is given which may be used to obtain values for the dispersion parameter, P_f, and an ion fractionation coefficient, _f, by non-linear regression on experimental concentration distributions. Values for the dispersion parameter so obtained can be used to give an induction of the viscous layer thickness as well as other film characteristics.     

Axial dispersion in pressure perturbed flow through an annular pipe oscillating around its axis

A method of moment is employed to study the axial dispersion of passive tracer molecules released in an unsteady pressure-driven flow through an annular pipe which is oscillating around its longitudinal axis. The flow unsteadiness is caused by the oscillation of the tube around its axis as well as by a periodic pressure gradient. A finite difference implicit scheme is adopted to solve the Aris integral moment equations arising from the unsteady convective-diffusion equation for all time periods. The main objective is to study the nature of the dispersion coeffcient and mean concentration distribution under the sole as well as combined oscillation of the two driving forces. The behaviour of the dispersion coeffcient due to the variation of the aspect ratio, the absorption parameter for purely periodic flow has been examined and the sound response from dispersion coeffcient is found with the variation of these parameters in the sole presence of pressure pulsation. There is a remarkable difference in the behavior of the dispersion coeffcient depending on whether the ratio of two frequencies arising from the oscillations of the tube and the pressure gradient possesses a proper fraction or not. Oscillation of the tube produces much more dispersion than the pulsation of the pressure gradient and their combined effect leads to a further increase in dispersion. Tube oscillation shows a stronger effect on the dispersion coeffcient than the pressure pulsation though the effect of physical parameters are pronounced in the presence of pressure pulsation. The effect of the frequency parameter on the axial distribution of mean concentration is insensible when the oscillation of the annular tube is the only forcing. However this effect is much noticeable under the combined action of both forcing and much more effective under the sole influence of pressure pulsation.     

Axial dispersion in pressure perturbed flow through an annular pipe oscillating around its axis

A method of moment is employed to study the axial dispersion of passive tracer molecules released in an unsteady pressure-driven flow through an annular pipe which is oscillating around its longitudinal axis. The flow unsteadiness is caused by the oscillation of the tube around its axis as well as by a periodic pressure gradient. A finite difference implicit scheme is adopted to solve the Aris integral moment equations arising from the unsteady convective-diffusion equation for all time periods. The main objective is to study the nature of the dispersion coeffcient and mean concentration distribution under the sole as well as combined oscillation of the two driving forces. The behaviour of the dispersion coeffcient due to the variation of the aspect ratio, the absorption parameter for purely periodic flow has been examined and the sound response from dispersion coeffcient is found with the variation of these parameters in the sole presence of pressure pulsation. There is a remarkable difference in the behavior of the dispersion coeffcient depending on whether the ratio of two frequencies arising from the oscillations of the tube and the pressure gradient possesses a proper fraction or not. Oscillation of the tube produces much more dispersion than the pulsation of the pressure gradient and their combined effect leads to a further increase in dispersion. Tube oscillation shows a stronger effect on the dispersion coeffcient than the pressure pulsation though the effect of physical parameters are pronounced in the presence of pressure pulsation. The effect of the frequency parameter on the axial distribution of mean concentration is insensible when the oscillation of the annular tube is the only forcing. However this effect is much noticeable under the combined action of both forcing and much more effective under the sole influence of pressure pulsation.     

Analytic continuation for the generalized Bohm criterion in a two-ion-species plasma

The generalized Bohm criterion in a two-ion-species plasma is analytically investigated. The universal iterative formulas of the approximative solutions have been affirmed for the ion acoustic wave (IAW) dispersion relation and the generalized Bohm criterion in the two-ion-species plasma for the first time. For a weakly collisional two-ion-species plasma, it is indicated that the choice of the initial solution and the repetitious iterative operation for the ion drift velocities is the linchpin of the optimized solution for the ion acoustic wave dispersion relation and the generalized Bohm criterion, considering the more general case and extensive physical characteristics.     

Diffusion of methylene blue in glass fibers – Application of the shrinking core model

Diffusion of mass in a solid cylinder with concentration dependent diffusivity (or temperature-dependent thermal conductivity in case of heat diffusion) does not admit of an analytical solution except in special cases. The ‘shrinking core model’ has been used to develop an approximate analytical solution in certain circumstances. The model, generally useful to describe heterogeneous solid–fluid reactions, is applied to theoretically analyze the adsorption–diffusion phenomena of methylene blue dye in a glass fiber in the present work. Theoretical equations have been derived for the case of diffusivity as an exponential function of concentration. The diffusivity parameters are evaluated by global minimization of the error between the experimental and the theoretical concentration history. Other forms of diffusivity, namely constant diffusivity and diffusivity varying linearly with concentration are found to involve larger errors. A parametric sensitivity analysis of the error has been done. The shrinking core model could satisfactorily interpret the experimental dye concentration profile in the substrate.     

Numerical method for advection diffusion equation using FEM and B-splines

In the present work, a comprehensive study of advection–diffusion equation is made using B-spline functions. Advection–diffusion equation has many physical applications such as dispersion of dissolved salts in groundwater, spread of pollutants in rivers and streams, water transfer, dispersion of tracers, and flow fast through porous media. Motivation behind the proposed scheme is to present a solution scheme which is easy to understand. Both linear and quadratic B-spline functions have been used in the present work to understand the basic aspect and advantages of the presented scheme. Along with this, some test examples are studied to observe the correctness of the numerical experiments. Finally, different comparisons are made to cross check the results obtained by the given scheme.     

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.     

Two stable methods with numerical experiments for solving the backward heat equation

This paper presents results of some numerical experiments on the backward heat equation. Two quasi-reversibility techniques, explicit filtering and structural perturbation, to regularize the ill-posed backward heat equation have been used. In each of these techniques, two numerical methods, namely Euler and Crank-Nicolson (CN), have been used to advance the solution in time.Crank-Nicolson method is very counter-intuitive for solving the backward heat equation because the dispersion relation of the scheme for the backward heat equation has a singularity (unbounded growth) for a particular wave whose finite wave number depends on the numerical parameters. In comparison, the Euler method shows only catastrophic growth of relatively much shorter waves. Strikingly we find that use of smart filtering techniques with the CN method can give as good a result, if not better, as with the Euler method which is discussed in the main text. Performance of these regularization methods using these numerical schemes have been exemplified.     

Taylor dispersion in a packed tube

Presented in this paper is a theoretical analysis for longitudinal scalar spread of mean concentration under a fully developed flow in a tube packed with porous media. A general form of momentum equation for superficial flow in porous media is introduced as a combination of the Navier–Stokes equation and Darcy’s law plus a superficial dispersion term due to phase discontinuity between the fluid flow and solid frame. The analytical solution presented for the fully developed superficial flow includes that for the Poiseulle flow in an evacuated tube as a limiting case. As an extension of Taylor’s classical work on dispersion of soluble matter in solvent flowing slowly through an evacuated tube, a one-dimensional dispersion equation valid for overall environmental assessment of contaminant is rigorously derived by cross-sectionally averaging the superficial mass equation and introducing a closure relation for a new unknown out of the averaging procedure, and corresponding Taylor dispersivity determined is shown to be a generalization of Taylor’s well-known result for the Poiseulle flow.     

Anisotropic characters in two-dimensional hexagonal crystal

The anisotropic characters of two-dimensional hexagonal crystal are investigated in this paper. The dispersion relation has been studied numerically. It is shown that the dispersion relation strongly dependents on the directions of wave propagation. Generally, the direction of waves has the inclination angle with respect to particle displacement. There are compressional waves  = 0 or transverse waves  = π/2 for long wavelength. However, for short wavelength compressional and transverse waves exists only for some special directions. For most cases, neither compressional nor transverse waves exists. The nonlinear waves in this crystal have also been studied. The characters of solitons, such as amplitude and width, have been investigated.     

An explicit finite difference solution to the convection-dispersion equation

An explicit finite difference equation has been development for the solution of the convection-dispersion equation. This equation has been over the entire range of 2D/vΔx between zero and one, region where no completely satisfactory method has been previously available. No oscillations or numerical dispersion were observed in any of the solutions.     

Advection-Diffusion in Reversing and Oscillating Flows: 1. The Effect of a Single Reversal

Motivated by the need to understand effluent dispersion in shallowtidal waters, a two-dimensional analysis of advection and diffusionin a reversing flow has been carried out. The flow speed varieslinearly with time, passing through zero at time t=0. A pointsource discharges contaminant into this flow at a steady rate,so that water which is close to the source around the time offlow reversal will become highly contaminated. Thus a peak inthe contaminant concentration field will appear, moving downstreamafter the reversal at a speed close to that of the flow. Thisconcentration peak has certain characteristics similar to acloud of contaminant from an instantaneous discharge at t=0.The solution of the advection-diffusion equation is in the formof an integral of concentration fields due to instantaneousreleases of contaminant at all previous times. At large timesafter the flow reversal, asymptotic analysis yields good approximationsto this integral. The use of Laplace's method is equivalentto ignoring longitudinal diffusion (the boundary-layer approximation);however, the expansion obtained in this way is not uniformlyvalid near the concentration peak, indicating that longitudinaldiffusion plays an important role in the development of thispeak. Uniformly valid expansions are obtained for the concentrationaround the peak, and also around the source where the boundary-layerapproximation always breaks down. Numerical integration hasalso been carried out, the results being used to produce contourplots of concentration for various times either side of theflow reversal.