ELSIM—a problem-solving environment for electrochemical kinetic simulations. Version 3.0-solution of governing equations associated with interfacial species,independent of spatial coordinates or in one-dimensional space geometry

A third version of the program ELSIM (for IBM compatible PCs) for the simulation of electrochemical transient methods has been elaborated upon. The program has been equipped with numerical algorithms that allow solution of kinetic problems characterized by the presence of interfacial species. Problems of this kind play an important role in electrochemical kinetic studies, among others in connection with electrocatalysis, modified electrodes or oscillatory systems. The user-written governing equations may take the form of differential—algebraic equations (for the concentrations of interfacial species) independent of spatial coordinates, or partial differential equations (for the concentrations of bulk species) in one-dimensional space geometry, coupled with differential—algebraic equations. Problem formulation has also been facilitated by the inclusion of a reaction compiler, which automatically generates the text of the above governing equations for user-written reaction mechanisms. Other new features include an option for the least-squares fitting of simulated transients to experimental curves, a hypertext help facility, and enhanced performance owing to the revised formula translation based on the three-address internal code generation, and on automatic differentiation.     

Emulating cellular automata in chemical reaction–diffusion networks

Chemical reactions and diffusion can produce a wide variety of static or transient spatial patterns in the concentrations of chemical species. Little is known, however, about what dynamical patterns of concentrations can be reliably programmed into such reaction–diffusion systems. Here we show that given simple, periodic inputs, chemical reactions and diffusion can reliably emulate the dynamics of a deterministic cellular automaton, and can therefore be programmed to produce a wide range of complex, discrete dynamics. We describe a modular reaction–diffusion program that orchestrates each of the fundamental operations of a cellular automaton: storage of cell state, communication between neighboring cells, and calculation of cells’ subsequent states. Starting from a pattern that encodes an automaton’s initial state, the concentration of a “state” species evolves in space and time according to the automaton’s specified rules. To show that the reaction–diffusion program we describe produces the target dynamics, we simulate the reaction–diffusion network for two simple one-dimensional cellular automata using coupled partial differential equations. Reaction–diffusion based cellular automata could potentially be built in vitro using networks of DNA molecules that interact via branch migration processes and could in principle perform universal computation, storing their state as a pattern of molecular concentrations, or deliver spatiotemporal instructions encoded in concentrations to direct the behavior of intelligent materials.     

Fractional differential equations in electrochemistry

Electrochemistry was one of the first sciences to benefit from the fractional calculus. Electrodes may be thought of as “transducers” of chemical fluxes into electricity. In a typical electrochemical cell, chemical species, such as ions or dissolved molecules, move towards the electrodes by diffusion. Likewise, other species are liberated into solution by the electrode reaction and diffuse away from the electrode into the bulk solution. It is demonstrated in this paper that the electric current is linearly related to the temporal semiderivative of the concentrations, at the electrode, of the species involved in the electrochemical reaction. More usefully, the semiintegral of the current provides immediate access information about concentrations.     

Heat and mass transfer phenomena in the flow of Casson fluid over an infinite oscillating plate in the presence of first-order chemical reaction and slip effect

The objective of this article is to study the physics of slip effect at the boundary of a vertical plate in starting the flow of Casson fluid with the combined effect of radiative heat and mass transfer in the presence of first-order chemical reaction. The problem has been modeled in terms of partial differential equations along with appropriate initial and boundary conditions. The dimensionless governing equations have been solved by means of the Laplace transform technique. Exact solutions have been obtained for velocity, temperature and concentration profiles. The obtained velocity has been computed in tabular form for steady and transient velocities. The physics of velocity profile has been studied for various physical parameters through numerical computation and displayed in graphs. From obtained solutions, the well-known published results in the open literature have been recovered and displayed in graphs and tables.

     

A BASIC program for the numerical solution of the transient kinetics of complex biochemical models

A highly optimized software for the kinetic analysis of complex chemical models is presented. The program is applied to the analysis of a vectorial biochemical reaction, where many species are linked by multiple equilibria of any order. The reaction stimulates the Ca2(+)-transport-linked ATPase reaction taking place in a suspension of vesicular fragments of isolated sarcoplasmic reticulum membranes, as described in many experimental reports. The model includes 12 reactants and intermediate chemical species, 14 kinetic constants, compartmentalization, and thermodynamic adjustment. The concentrations of all the model components, at any time, starting from a known initial condition, are calculated. The transient concentrations of the species are obtained by numerical integration of the appropriate differential equations, using an optimized version of the Runge-Kutta-Gill algorithm, with the aid of a Digital PDP11/23 computer and a standard BASIC-11 software, which could be fast and easily fitted to work with any microcomputer and/or alternative language or faster working compiled BASIC version. The errors of the calculations are evaluated.     

Diffusion of a chemically reactive species of a power-law fluid past a stretching surface

A numerical solution for the steady magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a continuously moving surface with species concentration and chemical reaction has been obtained. The viscous flow is driven solely by the linearly stretching sheet, and the reactive species emitted from this sheet undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. Using a similarity transformation, the governing non-linear partial differential equations are transformed into coupled nonlinear ordinary differential equations. The governing equations of the mathematical model show that the flow and mass transfer characteristics depend on six parameters, namely, the power-law index, the magnetic parameter, the local Grashof number with respect to species diffusion, the modified Schmidt number, the reaction rate parameter, and the wall concentration parameter. Numerical solutions for these coupled equations are obtained by the Keller-Box method, and the solutions obtained are presented through graphs and tables. The numerical results obtained reveal that the magnetic field significantly increases the magnitude of the skin friction, but slightly reduces the mass transfer rate. However, the surface mass transfer strongly depends on the modified Schmidt number and the reaction rate parameter; it increases with increasing values of these parameters. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Thin film evaporation in microchannels with interfacial slip

We investigate the role of interfacial slip on evaporation of a thin liquid film in a microfluidic channel. The effective slip mechanism is attributed to the formation of a depleted layer adhering to the substrate–fluid interface, either in a continuum or in a rarefied gas regime, as a consequence of intricate hydrophobic interactions in the narrow confinement. We appeal to the fundamental principles of conservation in relating the evaporation mechanisms with fluid flow and heat transfer over interfacial scales. We obtain semi-analytical solutions of the pertinent governing equations, with coupled heat and mass transfer boundary conditions at the liquid–vapor interface. We observe that a general consequence of interfacial slip is to elongate the liquid film, thereby leading to a film thickening effect. Thicker liquid films, in turn, result in lower heat transfer rates from the wall to liquid film, and consequently lower mass transfer rates from the liquid film to the vapor phase. Nevertheless, the total mass of evaporation (or equivalently, the net heat transfer) turns out to be higher in case of interfacial slip due to the longer film length. We also develop significant physical insights on the implications of the relative thickness of the depleted layer with reference to characteristic length scales of the microfluidic channel on the evaporation process, under combined influences of the capillary pressure, disjoining pressure, and the driving temperature differential for the interfacial transport.     

Field modeling with sampled distances

Traditional mesh-based approaches to the modeling and analysis of physical fields within geometric models require some form of topological reconstruction and conversion in the mesh generation process. Such manipulations tend to be tedious and error-prone manual processes that are not easily automated. We show that most field problems may be solved directly by using approximate distance fields computed from designed or sampled geometric data, thus avoiding many of the difficult reconstruction and meshing problems. With distances we can model fields that satisfy boundary conditions while approximating the governing differential equations to arbitrary precision. Because the method is based on sampling, it provides natural control for multi-resolution both in geometric detail of the domain and in accuracy of the computed physical field. We demonstrate the field modeling capability with several heat transfer applications, including a typical transient problem and a ‘scan and solve’ approach to the simulation of a physical field in a real-world artifact.     

Pattern formation in enzyme inhibition and cooperativity with parallel cellular automata

Enzyme reactions with inhibition and cooperativity are modelled in terms of a pair of coupled nonlinear reaction–diffusion equations. The governing equations are solved using stochastic cellular automata with local rules derived from the corresponding nonlinear partial differential equations. The parallel cellular automaton is implemented using domain decomposition according to the nature of the locality of its update rules. Numerical simulations show stable 2-D and 3-D pattern formation, and complex patterns have the interesting feature of self-organized criticality. The numerical results of cellular automata are also compared with results obtained from finite difference and finite element methods.     

含微结构二维固体中非对称孤立波的存在条件

根据Mindlin微结构理论重新推导了含微结构的二维固体中孤立波传播的控制方程.利用行波变换,把复杂的非线性偏微分方程组简化为一非线性常微分方程.最后用动力系统定性分析理论,分析了含微结构的二维固体中孤立波的存在条件及其几何特性,证明了当介质中的某些参数满足适当条件时,在含微结构的二维固体中可以存在一种非对称孤立波.     

Nonlinear control of a coupled PDE/ODE system modeling a switched power converter with a transmission line

We consider an infinite dimensional system modeling a boost converter connected to a load via a transmission line. The governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations modeling the converter. The coupling is given by the boundary conditions and the nonlinear controller we introduce. We design a nonlinear saturating control law using a Lyapunov function for the averaged model of the system. The main results give the well-posedness and stability properties of the obtained closed loop system.     

A mathematical model for foreign body reactions in 2D

The foreign body reactions are commonly referred to the network of immune and inflammatory reactions of human or animals to foreign objects placed in tissues. They are basic biological processes, and are also highly relevant to bioengineering applications in implants, as fibrotic tissue formations surrounding medical implants have been found to substantially reduce the effectiveness of devices. Despite of intensive research on determining the mechanisms governing such complex responses, few mechanistic mathematical models have been developed to study such foreign body reactions. This study focuses on a kinetics-based predictive tool in order to analyze outcomes of multiple interactive complex reactions of various cells/proteins and biochemical processes and to understand transient behavior during the entire period (up to several months). A computational model in two spatial dimensions is constructed to investigate the time dynamics as well as spatial variation of foreign body reaction kinetics. The simulation results have been consistent with experimental data and the model can facilitate quantitative insights for study of foreign body reaction process in general.     

Sensitivity of the numerical analysis of the three-fluid plasma mixed initial-boundary value problem

The sensitivity of the numerical solution of the nonlinear three-fluid equations governing the effect of forcing a time dependent disturbance at a point in a plasma is investigated. With the equations transformed into a diagonal form it is shown that only certain variables may be prescribed as functions of time at $\text{x=0}$, where these specified functions must satisfy certain compatibity conditions. With forward differences used to replace the time derivative and either forward or backward differences used to replace the spatial derivatives, the difference equations formulated are consistent and their solution converges to the solution of the differential equations. This convergence is true as long as the domain of dependence concept is adhered to. A lineary analysis provides a guide to the actual stability of the system of equations. From this analysis it is seen that the magnitude of the collission frequences, as well as the speed of light, restricts the size of the steps which may be used. Furthermore, it is shown that the solution is extremely sensitive to the boundary and initial conditions specified.     

Orthotropic cylindrical shells under line load along a generator

A technique for elastic analysis of an orthotropic cylindrical shell subjected to a uniform line load along a generator is developed. An accurate form of governing differential equations is derived and a mathematically discrete element method is used for its solution. The shell is divided into a finite number of longitudinal strips and the derivatives with respect to the circumferential coordinate in the governing equation are replaced by their finite difference relationships. The solution of the resulting equations is written in closed form. A computer program to implement this technique is developed and the computed results are compared with published experimental and analytical results. An excellent agreement is obtained. Some new results for a shell with fixed end boundary conditions are also presented.     

甲醇合成平行反应体系催化剂效率因子的求解(I)—维Green函数法

由CO,CO_2加氢合成甲醇的反应是一个复合反应体系,其催化剂颗粒内的温度、浓度分布的表征方程为二阶常微分方程组,本文针对所涉及方程的特殊性,运用Green函数法,采用双曲型的动力学方程和以实验为基础得到的催化剂有效导热系数解此方程组,得到效率因子的数值解,并与实验数据进行对此,吻合良好。     

Computational study of band-crossing reactions

A numerical study of band-crossing reactions is conducted using a quasi-one-dimensional (1-D) computational model that accounts for species bulk advection, electromigration velocities, diffusion, and chemical reaction. The model is used to simulate chemical reactions between two initially distinct sample zones, referred to as "bands," that cross each other due to differences in electromigration velocities. The reaction is described in terms of a single step, reversible mechanism involving two reactants and one product. A parametric study is first conducted of the behavior of the species profiles, and results are interpreted in terms of the Damko/spl uml/hler number and of the ratios of the electromigration velocities of the reactant and product. Computed results are then used to explore the possibility of extracting forward and backward reaction rates based on time resolved observation of integral moments of species concentrations. In particular, it is shown that in the case of fast reactions, robust estimates can be obtained for high forward rates, but that small reverse rates may not be accurately observed.     

A mathematical model for foreign body reactions in 2D

The foreign body reactions are commonly referred to a network of immune and inflammatory reactions of human or animals in response to foreign objects being placed in tissues. They are basic biological processes, and are also highly relevant to bioengineering applications in implants, as fibrotic tissue formations surrounding medical implants have been found to substantially reduce the effectiveness of devices. Despite the intensive research on determining the mechanisms governing such complex responses, few mechanistic mathematical models have been developed to study such foreign body reactions. This study focuses on a kinetics-based predictive tool to analyse the outcomes of multiple interactive complex reactions of various cells/proteins and biochemical processes and to understand transient behaviour during the entire period (up to several months). A computational model in two spatial dimensions is constructed to investigate the time dynamics as well as the spatial variation of foreign body reaction kinetics. The simulation results have been consistent with the experimental data and the model can facilitate quantitative insights into the study of foreign body reaction process, in general.     

柔性悬臂梁非线性非平面运动的多脉冲轨道分析

首先建立了柔性悬臂梁非线性非平面运动的偏微分方程;然后运用Galerkin和多尺度方法得到平均方程．并利用规范形理论进一步将方程化简;最后用能量相位法求出多脉冲跳跃的能量函数序列．Dynamics软件数值计算表明：在系统中确实存在着由多脉冲跳跃而导致的Smale马蹄型混沌．     

Modified algorithms for nonconforming taylor discretization

Algorithms for solving partial differential equations which extend previous applications of the nonconforming Taylor discretization method (NTDM) are presented. In one modification the number of interrelated grid points is variable, thus enabling additional geometric flexibility. Another modification is the approximation of the governing differential equation using the method of weighted residuals. A simple one-dimensional test case with a known analytic solution is solved using this code. The results demonstrate that precision is enhanced when using the method of weighted residuals with an increased number of interrelated points. The algorithm is applied as a general purpose two-dimensional code for nonlinear steady state heat-conduction. Two-dimensional examples with complex geometry and boundary conditions are then solved both by the NTDM and by the finite elements method (FEM). The results obtained by the two methods are compared.     

Matrix stability of the backward differentiation formula in electrochemical digital simulation

The matrix stability of the backward differentiation formula (BDF) algorithm for the numerical solution of the reaction–diffusion partial differential equations arising in electrochemistry, as a function of the number of time levels k and under several boundary conditions, was studied. The study included also the two-species catalytic mechanism, and unequal intervals (nonuniform spatial grid). The method is unconditionally stable in all cases for k⩽7 (that is, order ⩽6) irrespective of the rate of homogeneous chemical reactions or boundary conditions. Homogeneous chemical reactions, except in the case of the two-species catalytic reaction, were found to have a stabilising effect.