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Diffusion Equation (diffusion + equation)
Selected AbstractsA Simple Approach to the Solution of the Diffusion Equation at the Microcylinder Electrode,an Inspiration from the Film ProjectorCHEMPHYSCHEM, Issue 14 2009Yi-Min Fang 16 frames per second: A new and simple route for the solution of diffusion equation at three types of electrode (see picture) is based on a time-dependent diffusion layer approximation and time-dependent boundary conditions, without employing the traditional Laplace transform. [source] On the maximum principle and its application to diffusion equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2007T. Stys Abstract In this article, an analog of the maximum principle has been established for an ordinary differential operator associated with a semi-discrete approximation of parabolic equations. In applications, the maximum principle is used to prove O(h2) and O(h4) uniform convergence of the method of lines for the diffusion Equation (1). The system of ordinary differential equations obtained by the method of lines is solved by an implicit predictor corrector method. The method is tested by examples with the use of the enclosed Mathematica module solveDiffusion. The module solveDiffusion gives the solution by O(h2) uniformly convergent discrete scheme or by O(h4) uniformly convergent discrete scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] A cache-efficient implementation of the lattice Boltzmann method for the two-dimensional diffusion equationCONCURRENCY AND COMPUTATION: PRACTICE & EXPERIENCE, Issue 14 2004A. C. Velivelli Abstract The lattice Boltzmann method is an important technique for the numerical solution of partial differential equations because it has nearly ideal scalability on parallel computers for many applications. However, to achieve the scalability and speed potential of the lattice Boltzmann technique, the issues of data reusability in cache-based computer architectures must be addressed. Utilizing the two-dimensional diffusion equation, , this paper examines cache optimization for the lattice Boltzmann method in both serial and parallel implementations. In this study, speedups due to cache optimization were found to be 1.9,2.5 for the serial implementation and 3.6,3.8 for the parallel case in which the domain decomposition was optimized for stride-one access. In the parallel non-cached implementation, the method of domain decomposition (horizontal or vertical) used for parallelization did not significantly affect the compute time. In contrast, the cache-based implementation of the lattice Boltzmann method was significantly faster when the domain decomposition was optimized for stride-one access. Additionally, the cache-optimized lattice Boltzmann method in which the domain decomposition was optimized for stride-one access displayed superlinear scalability on all problem sizes as the number of processors was increased. Copyright © 2004 John Wiley & Sons, Ltd. [source] Mechanisms of transjunctional transport of NaCl and water in proximal tubules of mammalian kidneysACTA PHYSIOLOGICA, Issue 1 2002F. KIILArticle first published online: 30 APR 200 ABSTRACT Tight junctions and the intercellular space of proximal tubules are not accessible to direct measurements of fluid composition and transport rates, but morphological and functional data permit analysis of diffusion and osmosis causing transjunctional NaCl and water transport. In the S2 segment NaCl diffuses through tight junctions along a chloride gradient, but against a sodium gradient. Calculation in terms of modified Nernst,Fick diffusion equation after eliminating electrical terms shows that transport rates (300,500 pmol min,1 mm,1 tubule length) and transepithelial voltage of +2 mV are in agreement with observations. Diffusion coefficients are Dtj=1500 ,m2 s,1 in the S1 segment, and Dtj=90,100 ,m2 s,1 in the S2 segment where apical intercellular NaCl concentration is 132 mM, 1 mM below complete stop (Dtj=0 and Donnan equilibrium). Tight junctions with gap distance 6 Å are impermeable to mannitol (effective molecular radius 4 Å); reflection coefficients are ,=0.92 for NaHCO3 and ,=0.28 for NaCl, because of difference in anion size. The osmotic force is provided by a difference in effective transjunctional osmolality of 10 mOsm kg,1 in the S1 segment and 30 mOsm kg,1 in the S2 segment, where differences in transjunctional concentration contribute with 21 mOsm kg,1 for NaHCO3 and ,4 mOsm kg,1 for NaCl. Transjunctional difference of 30 mOsm kg,1 causes a volume flow of 2 nL min,1 mm,1 tubule length. Luminal mannitol concentration of 30 mM stops all volume flow and diffusive and convective transport of NaCl. In conclusion, transjunctional diffusion and osmosis along gradients generated by transcellular transport of other solutes account for all NaCl transport in proximal tubules. [source] Solution of the linear diffusion equation for modelling erosion processes with a time varying diffusion coefficientEARTH SURFACE PROCESSES AND LANDFORMS, Issue 10 2008Georgios Aim. Abstract In the present paper the differential equation of the temporal development of a landform (mountain) with a time dependent diffusion coefficient is solved. It is shown that the shape and dimensions of the landform at time t are independent of the specific variation of the diffusion coefficient with time; they only depend on the mean value of the diffusion coefficient in the time interval where the erosion process takes place. Studying the behaviour of the solution of the differential equation in the wave number domain, it is concluded that Fourier analysis may help in estimating, in quantitative terms, the initial dimensions, the age or, alternatively, the value of the diffusion coefficient of the landform. The theoretical predictions are tested on a hill of the southern part of the Ural mountainous region, in order to show how the results of the mathematical analysis can be used in describing, in quantitative terms, the morphological development of landforms due to erosion processes. Copyright © 2007 John Wiley & Sons, Ltd. [source] Required source distribution for interferometry of waves and diffusive fieldsGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2009Yuanzhong Fan SUMMARY The Green's function that describes wave propagation between two receivers can be reconstructed by cross-correlation provided that the receivers are surrounded by sources on a closed surface. This technique is referred to as ,interferometry' in exploration seismology. The same technique for Green's function extraction can be applied to the solution of the diffusion equation if there are sources throughout in the volume. In practice, we have only a finite number of active sources. The issues of the required source distribution is investigated, as is the feasibility of reconstructing the Green's function of the diffusion equation using a limited number of sources within a finite volume. We study these questions for homogeneous and heterogeneous media for wave propagation and homogeneous media for diffusion using numerical simulations. These simulations show that for the used model, the angular distribution of sources is critical in wave problems in homogeneous media. In heterogeneous media, the position and size of the heterogeneous area with respect to the sources determine the required source distribution. For diffusion, the sensitivity to the sources decays from the midpoint between the two receivers. The required width of the source distribution decreases with frequency, with the result that the required source distribution for early- and late-time reconstruction is different. The derived source distribution criterion for diffusion suggests that the cross-correlation-based interferometry is difficult to apply in field condition. [source] Diffusion through time-dependent mediaGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2000M. Holschneider Summary In this theoretical paper we show how to solve a time-dependent diffusion equation by means of a perturbation series. This technique is applied to the case of diffusion of a liquid through a time-dependent porous matrix. We compute to first order the phase and amplitude relations between the small deformation of the transporting matrix and the corresponding variation of the saturation at the surface. In particular we show that, for a large frequency range, there is a constant phase shift of ,/2 between the matrix and the surface saturation variations. Since the conductivity is to first approximation proportional to the saturation at the surface, this might explain the observed phase relations observed in an experiment in a cave near Abaratsubo (Japan). [source] Computational methods for optical molecular imagingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2009Duan Chen Abstract A new computational technique, the matched interface and boundary (MIB) method, is presented to model the photon propagation in biological tissue for the optical molecular imaging. Optical properties have significant differences in different organs of small animals, resulting in discontinuous coefficients in the diffusion equation model. Complex organ shape of small animal induces singularities of the geometric model as well. The MIB method is designed as a dimension splitting approach to decompose a multidimensional interface problem into one-dimensional ones. The methodology simplifies the topological relation near an interface and is able to handle discontinuous coefficients and complex interfaces with geometric singularities. In the present MIB method, both the interface jump condition and the photon flux jump conditions are rigorously enforced at the interface location by using only the lowest-order jump conditions. This solution near the interface is smoothly extended across the interface so that central finite difference schemes can be employed without the loss of accuracy. A wide range of numerical experiments are carried out to validate the proposed MIB method. The second-order convergence is maintained in all benchmark problems. The fourth-order convergence is also demonstrated for some three-dimensional problems. The robustness of the proposed method over the variable strength of the linear term of the diffusion equation is also examined. The performance of the present approach is compared with that of the standard finite element method. The numerical study indicates that the proposed method is a potentially efficient and robust approach for the optical molecular imaging. Copyright © 2008 John Wiley & Sons, Ltd. [source] Stability analysis of the Crank,Nicholson method for variable coefficient diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2007Charles Tadjeran Abstract The Crank,Nicholson method is a widely used method to obtain numerical approximations to the diffusion equation due to its accuracy and unconditional stability. When the diffusion coefficient is not a constant, the general approach is to obtain a discretization for the PDE in the same manner as the case for constant coefficients. In this paper, we show that the manner of this discretization may impact the stability of the resulting method and could lead to instability of the numerical solution. It is shown that the classical Crank,Nicholson method will fail to be unconditionally stable if the diffusion coefficient is computed at the time gridpoints instead of at the midpoints of the temporal subinterval. A numerical example is presented and compared with the exact analytical solution to examine its divergence. Copyright © 2006 John Wiley & Sons, Ltd. [source] Numerical stability of unsteady stream-function vorticity calculationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 6 2003E. Sousa Abstract The stability of a numerical solution of the Navier,Stokes equations is usually approached by con- sidering the numerical stability of a discretized advection,diffusion equation for either a velocity component, or in the case of two-dimensional flow, the vorticity. Stability restrictions for discretized advection,diffusion equations are a very serious constraint, particularly when a mesh is refined in an explicit scheme, so an accurate understanding of the numerical stability of a discretization procedure is often of equal or greater practical importance than concerns with accuracy. The stream-function vorticity formulation provides two equations, one an advection,diffusion equation for vorticity and the other a Poisson equation between the vorticity and the stream-function. These two equations are usually not coupled when considering numerical stability. The relation between the stream-function and the vorticity is linear and so has, in principle, an exact inverse. This allows an algebraic method to link the interior and the boundary vorticity into a single iteration scheme. In this work, we derive a global time-iteration matrix for the combined system. When applied to a model problem, this matrix formulation shows differences between the numerical stability of the full system equations and that of the discretized advection,diffusion equation alone. It also gives an indication of how the wall vorticity discretization affects stability. Despite the added algebraic complexity, it is straightforward to use MATLAB to carry out all the matrix operations. Copyright © 2003 John Wiley & Sons, Ltd. [source] An accurate integral-based scheme for advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2001Tung-Lin Tsai Abstract This paper proposes an accurate integral-based scheme for solving the advection,diffusion equation. In the proposed scheme the advection,diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one-dimensional to multi-dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one-dimensional uniform flow; (ii) one-dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two-dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two-dimensional rigid-body rotating flow; and (v) three-dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time-centred implicit QUICK scheme and the fully time-centred implicit TCSD scheme for one-dimensional problem but also with the ADI-QUICK scheme, the ADI-TCSD scheme and the MOSQUITO scheme for two-dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd. [source] An a posteriori error estimator for the mimetic finite difference approximation of elliptic problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2008Lourenço Beirão da Veiga Abstract We present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form. We show that this estimator is reliable and efficient with respect to an energy-type error comprising both flux and pressure. Its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes, different permeability tensors, and different types of computational meshes. Copyright © 2008 John Wiley & Sons, Ltd. [source] A transmission line modelling (TLM) method for steady-state convection,diffusionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2007Alan Kennedy Abstract This paper describes how the lossy transmission line modelling (TLM) method for diffusion can be extended to solve the convection,diffusion equation. The method is based on the correspondence between the convection,diffusion equation and the equation for the voltage on a lossy transmission line with properties varying exponentially over space. It is unconditionally stable and converges rapidly to highly accurate steady-state solutions for a wide range of Peclet numbers from low to high. The method solves the non-conservative form of the convection,diffusion equation but it is shown how it can be modified to solve the conservative form. Under transient conditions the TLM scheme exhibits significant numerical diffusion and numerical convection leading to poor accuracy, but both these errors go to zero as a solution approaches steady state. Copyright © 2007 John Wiley & Sons, Ltd. [source] Insight into the flow-condition-based interpolation finite element approach: solution of steady-state advection,diffusion problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2005Haruhiko Kohno Abstract The flow-condition-based interpolation (FCBI) finite element approach is studied in the solution of advection,diffusion problems. Two FCBI procedures are developed and tested with the original FCBI method: in the first scheme, a general solution of the advection,diffusion equation is embedded into the interpolation, and in the second scheme, the link-cutting bubbles approach is used in the interpolation. In both procedures, as in the original FCBI method, no artificial parameters are included to reach stability for high Péclet number flows. The procedures have been implemented for two-dimensional analysis and the results of some test problems are presented. These results indicate good stability and accuracy characteristics and the potential of the FCBI solution approach. Copyright © 2005 John Wiley & Sons, Ltd. [source] An unconditionally stable three level finite difference scheme for solving parabolic two-step micro heat transport equations in a three-dimensional double-layered thin filmINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004Weizhong Dai Abstract Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equations are parabolic two-step equations, which are different from the traditional heat diffusion equation. In this study, we develop a three-level finite difference scheme for solving the micro heat transport equations in a three-dimensional double-layered thin film. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained. Copyright © 2003 John Wiley & Sons, Ltd. [source] A computational model for impact failure with shear-induced dilatancyINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2003Z. Chen Abstract It has been observed in plate impact experiments that some brittle solids may undergo elastic deformation at the shock wave front, and fail catastrophically at a later time when they are shocked near but below the apparent Hugoniot elastic limit. Because this phenomenon appears to have features different from those of usual inelastic waves, it has been interpreted as the failure wave. To design an effective numerical procedure for simulating impact failure responses, a three-dimensional computational damage model is developed in this paper. The propagation of the failure wave behind the elastic shock wave is described by a non-linear diffusion equation. Macroscopic shear-induced dilatancy is assumed and treated as a one-to-one measure of the mean intensity of microcracking. The damage evolution in time is determined based on the assumption that the deviatoric strain energy in the elastically compressed material (undamaged) is converted, through the damaging process, into the volumetric potential energy in the comminuted and dilated material. For the ease in large-scale simulations, the coupled damage diffusion equation and the stress wave equation are solved via a staggered manner in a single computational domain. Numerical solutions by using both the finite element method and the material point method, i.e. with and without a rigid mesh connectivity, are presented and compared with the experimental data available. It is shown that the model simulations capture the essential features of the failure wave phenomenon observed in shock glasses, and that the numerical solutions for localized failure are not mesh-dependent. Copyright © 2003 John Wiley & Sons, Ltd. [source] Time accurate consistently stabilized mesh-free methods for convection dominated problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2003Antonio Huerta Abstract The behaviour of high-order time stepping methods combined with mesh-free methods is studied for the transient convection,diffusion equation. Particle methods, such as the element-free Galerkin (EFG) method, allow to easily increase the order of consistency and, thus, to formulate high-order schemes in space and time. Moreover, second derivatives of the EFG shape functions can be constructed with a low extra cost and are well defined, even for linear interpolation. Thus, consistent stabilization schemes can be considered without loss in the convergence rates. Copyright © 2003 John Wiley & Sons, Ltd. [source] Performance of very-high-order upwind schemes for DNS of compressible wall-turbulenceINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2010G. A. Gerolymos Abstract The purpose of the present paper is to evaluate very-high-order upwind schemes for the direct numerical simulation (DNS) of compressible wall-turbulence. We study upwind-biased (UW) and weighted essentially nonoscillatory (WENO) schemes of increasingly higher order-of-accuracy (J. Comp. Phys. 2000; 160:405,452), extended up to WENO17 (AIAA Paper 2009-1612, 2009). Analysis of the advection,diffusion equation, both as ,x,0 (consistency), and for fixed finite cell-Reynolds-number Re,x (grid-resolution), indicates that the very-high-order upwind schemes have satisfactory resolution in terms of points-per-wavelength (PPW). Computational results for compressible channel flow (Re,[180, 230]; M,CL,[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid-resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(,t2) time-integration and O(,x2) discretization of the viscous terms, comparative studies of various orders-of-accuracy for the convective terms demonstrate that very-high-order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd. [source] Insights on a sign-preserving numerical method for the advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2009E. SousaArticle first published online: 17 DEC 200 Abstract In this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one-dimensional unsteady advection,diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first-order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second-order terms of the truncation error and the other considers additionally the third-order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd. [source] A lattice Boltzmann-BGK algorithm for a diffusion equation with Robin boundary condition,application to NMR relaxationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2009A. Hiorth Abstract We present a lattice Boltzmann-BGK (LBGK) algorithm for a diffusion equation together with a Robin boundary condition, which we apply in the case of nuclear magnetic resonance relaxation. The boundary condition we employ is independent of the direction of the wall. This makes the algorithm very suitable for complicated geometries, such as porous media. We discuss the effect of lattice topology by using, respectively, an eight-speed and a four-speed lattice. The numerical algorithm is compared with analytical results for a square and an equilateral triangle. The eight-speed lattice performs well in both cases. The four-speed lattice performs well for the square, but fails in the case of an equilateral triangle. Comparison with a random walk algorithm is also included. The LBGK algorithm presented here can also be used for a convective diffusion problem if the speed of the fluid can be neglected close to the boundary. Copyright © 2008 John Wiley & Sons, Ltd. [source] A higher-order predictor,corrector scheme for two-dimensional advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Chuanjian Man Abstract A higher-order accurate numerical scheme is developed to solve the two-dimensional advection,diffusion equation in a staggered-grid system. The first-order spatial derivatives are approximated by the fourth-order accurate finite-difference scheme, thus all truncation errors are kept to a smaller order of magnitude than those of the diffusion terms. Therefore, there is no need to add an artificial diffusion term to balance the unwanted numerical diffusion. For the time derivative, the fourth-order accurate Adams,Bashforth predictor,corrector method is applied. The stability analysis of the proposed scheme is carried out using the Von Neumann method. It is shown that the proposed algorithm has good stability. This method also shows much less spurious oscillations than current lower-order accurate numerical schemes. As a result, the proposed numerical scheme can provide more accurate results for long-time simulations. The proposed numerical scheme is validated against available analytical and numerical solutions for one- and two-dimensional transport problems. One- and two-dimensional numerical examples are presented in this paper to demonstrate the accuracy and conservative properties of the proposed algorithm by comparing with other numerical schemes. The proposed method is demonstrated to be a useful and accurate modelling tool for a wide range of transport problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Positivity-preserving, flux-limited finite-difference and finite-element methods for reactive transportINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2003Robert J. MacKinnon Abstract A new class of positivity-preserving, flux-limited finite-difference and Petrov,Galerkin (PG) finite-element methods are devised for reactive transport problems. The methods are similar to classical TVD flux-limited schemes with the main difference being that the flux-limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite-element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity-preserving property. Analysis of the latter scheme shows that positivity-preserving solutions of the resulting difference equations can only be guaranteed if the flux-limited scheme is both implicit and satisfies an additional lower-bound condition on time-step size. We show that this condition also applies to standard Galerkin linear finite-element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time-step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd. [source] A class of higher order compact schemes for the unsteady two-dimensional convection,diffusion equation with variable convection coefficientsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002Jiten C. Kalita Abstract A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection,diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter , and fourth order accurate in space. For 0.5,,,1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection,diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection,diffusion problem and three flows of varying complexities governed by the two-dimensional incompressible Navier,Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 John Wiley & Sons, Ltd. [source] Moisture adsorption by milk whey protein filmsINTERNATIONAL JOURNAL OF FOOD SCIENCE & TECHNOLOGY, Issue 3 2002C. M. P. Yoshida Edible films, using whey protein as the structural matrix, were tested for water vapour diffusion properties. Whey protein films were prepared by dispersing 6.5% whey protein concentrate (WPC) in distilled water with pH kept at 7.0. Glycerol was the plasticizer agent. Film slabs (13.5 × 3.5 cm) were put in a chamber at 25 °C and 75% relative humidity, being held in vertical planes for different periods of time. The mass gain was determined throughout the experiment. We show that moisture adsorption by milk whey protein films is well described by a linear diffusion equation model. After an adsorption experiment was performed the solution of the diffusion equation was fitted to the data to determine the diffusion coefficient of the material. [source] Fat Migration in Chocolate: Diffusion or Capillary Flow in a Particulate Solid?,A Hypothesis PaperJOURNAL OF FOOD SCIENCE, Issue 7 2004J. M. Aguilera ABSTRACT: The exact mechanism of fat and oil migration in chocolate and chocolate coatings is still unknown. Nevertheless, the so-called "diffusion equation" derived from Fick's 2nd law has been extensively used to model the phenomenon, giving the impression that molecular diffusion is the single transport mechanism. We propose that chocolate may be microstructurally regarded as a particulate medium formed by an assembly of fat-coated particles (for example, cocoa solids, sugars crystals, and milk powder). Within this matrix the liquid fraction of cocoa fat (which increases with temperature) is likely to move under capillary forces through interparticle passages and connected pores. Based on available evidence (microstructure, kinetic data, temperature dependence of liquid fat fraction, and so on) we demonstrate that capillary forces may have an important role to play in bulk flow of liquid fat and oils. The Lucas-Washburn equation for capillary rise fits available data under most reported experimental conditions. Detailed microstructural analysis in actual products as well as data on key parameters (surface tension, contact angle, viscosity) is necessary to confirm this hypothesis. Bulk flow due to capillary effects, highly disregarded in structured foods, should be considered as a mass transfer mechanism in liquid-filled porous or particulate foods. [source] Diffusion through ordered force fields in nanopores represented by Smoluchowski equationAICHE JOURNAL, Issue 6 2009Fu Yang Wang Abstract The classical Einstein or Fick diffusion equation was developed in random force fields. When the equation is applied to gas transport through coal, significant discrepancies are observed between experimental and simulation results. The explanation may be that the random force field assumption is violated. In this article, we analyze molecular transport driven by both random and ordered (directional) forces in nanopores. When applied to CO2 transport through cone-shaped carbon nano-tubes (CNTs) and Li+ doped graphite pores, computational results show that directional force fields may significantly affect porous media flow. Directional forces may be generated by potential gradients arising from a range of non-uniform characteristics, such as variations in the pore-sizes and in local surface compositions. On the basis of the simulation and experimental results, the Smoluchowski and Fokker-Planck equations, which account for the directional force fields, are recommended for diffusion through ordered force fields in nanopores. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source] Transdermal Delivery of the Potent Analgesic Dihydroetorphine: Kinetic Analysis of Skin Permeation and Analgesic Effect in the Hairless RatJOURNAL OF PHARMACY AND PHARMACOLOGY: AN INTERNATI ONAL JOURNAL OF PHARMACEUTICAL SCIENCE, Issue 12 2000SATOSHI OHMORI Dihydroetorphine is an extraordinarily strong opioid analgesic. To assess its effectiveness after topical application in hairless rats we have examined the kinetic analysis of skin permeation through excised skin and the in-vitro reservoir effect of skin, and have investigated the predictability of plasma concentration and analgesic effect following in-vivo transdermal application. Dihydroetorphine was moderately permeable from an aqueous suspension through excised hairless rat skin. Dihydroetorphine flux from drug-dispersed pressure-sensitive adhesive tape was threefold that from the applied aqueous suspension. The fluxes through the abdominal and the dorsal skin during tape application fitted the Fickian diffusion equation well after the tape was removed peeling off the outer layer of the stratum corneum. The relationship between the plasma concentration and the analgesic effect was examined for four different rates of infusion of dihydroetorphine. A non-linear pharmacokinetic disposition was observed. Following abdominal (0.28 cm2, 20,g) and dorsal (0.50 cm2, 35,g) applications of the dihydroetorphine tape, plasma concentration (0.2-0.8 ng mL,1) and analgesic effect were maintained at a suitable level, for more than 8h, until removal of the tape. These profiles were predictable using the combined equation for percutaneous absorption, disposition and the analgesic effect, but the analgesic effect was slightly lower than the predicted value. The results show that it was possible to control the plasma concentration and the analgesic effect of dihydroetorphine by topical application of the analgesic using pressure-sensitive adhesive tape in the hairless rat. It was possible to predict the result using mathematical modelling. [source] Mean-Variance Hedging for Stochastic Volatility ModelsMATHEMATICAL FINANCE, Issue 2 2000Francesca Biagini In this paper we discuss the tractability of stochastic volatility models for pricing and hedging options with the mean-variance hedging approach. We characterize the variance-optimal measure as the solution of an equation between Doléans exponentials; explicit examples include both models where volatility solves a diffusion equation and models where it follows a jump process. We further discuss the closedness of the space of strategies. [source] Positivity and time behavior of a linear reaction,diffusion system, non-local in space and timeMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2008Andrii Khrabustovskyi Abstract We consider a general linear reaction,diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction,diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd. [source] | |||||||||||||||||||||||||