Diffusion fronts in enzyme-catalysed reactions

In this paper the nature and validity of the mathematical formulation of Michaelis–Menten-type kinetics for enzyme-catalysed biochemical reactions is studied. Previous work has in the main concentrated on isolated, spatially uniform (well-mixed) reactions. The effects of substrate input and diffusion on this formulation, in particular, on the nature and validity of the quasi-steady-state-assumption for diffusion-driven fronts are investigated. It is shown that, provided the Michaelis–Menten constant K _M is sufficiently large, an appropriate quasi-steady-state assumption is valid at all points in space and for all times other than in a region that closely tracks the front itself. Moreover, it is shown that this region shrinks with time.     

Calculation of stress intensities at sharp notches in anisotropic media

We present a procedure to calculate mode I and II notch stress intensities in anisotropic media using the path-independent H-integral. The method is based on coupling the analysis of asymptotic stress and displacement fields near a sharp notch with a path independent integral that results from the application of Betti's reciprocal theorem to the notched solid. The approach is demonstrated for two loading/geometry combinations that arise naturally in etched single crystal silicon: mode I loading of a 70.53° notch and mixed mode I and II loading of a T-structure with a 90° notch. Results agree well with those obtained by correlating computed notch-flank displacements with the asymptotic solution.     

Heat and mass transfer in condensation of water vapor from moist air

Results of an analysis of heat and mass transfer in condensation of vapor from moist air are presented. The computational model is based on the solution of integral relations of boundary-layer energy and diffusion using the analogy of heat- and mass-transfer processes. The effect of the temperature and concentration boundary conditions on the relation of the components of the heat flux on the wall is analyzed. Results of calculations are compared to available experimental data. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 5, pp. 788–794, September–October, 1998.     

Numerical solution of stefan problems by the method of green functions

A method for reducing a multidimensional Stefan problem to a system of Hammerstein integral equations is proposed. Application of the proposed method to numerical solution of one-dimensional nonstationary Stefan problems formulated for the cases of an internal phase front, coincidence of the phase front with the external boundary, and a movable external boundary is considered. The efficiency of the method is tested on an exactly solvable Stefan problem. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 71, No. 3, pp. 564–570, May–June, 1998.     

Polynomial solutions in problems of controlling the heating of solids

We consider an approximate method of solving the problem of optimum control of the process that is described by a homogeneous heat-conduction equation. The governing parameter enters the boundary condition, while the integral quadratic expression is subjected to minimization. A solution of the problem is found for a class of controls described by a polynomial of finite degree. An example is considered. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 73, No. 5, pp. 907–910, September–October, 2000.     

A SIMPLE PATH-INDEPENDENT INTEGRAL FOR CALCULATING MIXED-MODE STRESS INTENSITY FACTORS

A path-independent integral is introduced for calculating stress intensity factors. The derivation of the integral is based on the application of the known Bueckner's fundamental field solution for a crack in an infinite body and on the reciprocal theorem. The method was applied to two-dimensional linear elastic mixed-mode crack problems. The key advantage of the present path-independent integral is that the stress intensity factor components for any irregular cracked geometry under any kind of loading can be easily obtained by a contour integral around the crack tip. The method is simple to implement and only the far field displacements and tractions along the contour must be known. The required stress analysis can be made by using any analytical or numerical method, e.g. the finite element method, without special consideration of the modelling of crack tip singularity. The application of this integral is also independent of the crack type, that is, there is no difference between an edge crack and an embedded crack, provided that the crack tip asymptotic behaviour exists.     

Analytical solution of heat-conduction problems with a variable initial condition based on determining a temperature perturbation front

With the aid of the integral heat-balance method an analytical solution of the problem of unsteady-state heat conduction has been obtained for an infinite plate with a variable initial condition. To increase the accuracy of solution by the integral method additional boundary conditions are introduced which are determined from the initial differential equation and basic boundary conditions, including those prescribed at the temperature perturbation front. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 3, pp. 27–35, May–June, 2007.     

Stress intensity near a crack in the composition of a half space and a layer under harmonic loading

We consider a three-dimensional problem for an elastic bimaterial body formed by a half space and a layer and containing a circular crack. The surface of the crack is subjected to the action of stationary torsional forces. The problem is reduced to the solution of a system of boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. The dynamic stress concentration in the vicinity of the crack contour is investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 27–32, March–April, 2008.     

Elastic Wave Propagation in a Class of Cracked,Functionally Graded Materials by BIEM

Elastic wave propagation in cracked, functionally graded materials (FGM) with elastic parameters that are exponential functions of a single spatial co-ordinate is studied in this work. Conditions of plane strain are assumed to hold as the material is swept by time-harmonic, incident waves. The FGM has a fixed Poisson’s ratio of 0.25, while both shear modulus and density profiles vary proportionally to each other. More specifically, the shear modulus of the FGM is given as μ (x)=μ ₀ exp (2ax ₂), where μ ₀ is a reference value for what is considered to be the isotropic, homogeneous material background. The method of solution is the boundary integral equation method (BIEM), an essential component of which is the Green’s function for the infinite inhomogeneous plane. This solution is derived here in closed-form, along with its spatial derivatives and the asymptotic form for small argument, using functional transformation methods. Finally, a non-hypersingular, traction-type BIEM is developed employing quadratic boundary elements, supplemented with special edge-type elements for handling crack tips. The proposed methodology is first validated against benchmark problems and then used to study wave scattering around a crack in an infinitely extending FGM under incident, time-harmonic pressure (P) and vertically polarized shear (SV) waves. The parametric study demonstrates that both far field displacements and near field stress intensity factors at the crack-tips are sensitive to this type of inhomogeneity, as gauged against results obtained for the reference homogeneous material case     

Effect of solute concentration on Portevin-Le Chatelier effect in Al-Cu alloys

The solute concentration and precipitated phase content in alloys were altered by solution treatment (ST) at different temperatures. A series of serrated load curves was obtained in tensile tests at a constant applied strain rate. The influences of solute concentration and precipitated phase content on dislocation movement were analyzed with tensile test results, and the micro-mechanism of the Portevin-Le Chatelier (PLC) effect is discussed in this paper. In the tests, when ST temperature was reduced from 500°C, the amplitude of serrated flow decreased. It reached the minimum when ST temperature was reduced to 300°C. On the other hand, the amplitude of serrated flow increased with ST temperature decreasing from 300 to 100°C. Experimental results show that when ST temperature is higher than 300°C, solute concentration is a governing factor to PLC effect; whereas when ST temperature is lower than 300°C, the precipitated phase content significantly affects the PLC effect. Translated from Acta Metallurgica Sinica, 2006, 42(12): 1248–1252 (in Chinese)     

Crack extension force in a dielectric medium

The authors' previously derived path-independent integral [1], which is related to the energy-release rate, is evaluated for a crack placed in an infinite dielectric medium subjected to a far field uniaxial tension and an external electric field. The governing equations and boundary conditions for a fully nonlinear elastic dielectric are linearized to a quasi-linear form. Electric field quantities are then decomposed into a rigid body state and a deformed state. This leads to a set of linear differential equations, boundary conditions and constitutive relations. An integral transform technique is used to obtain the solution in closed form. From this solution the crack extension force is evaluated.     

A plane periodic boundary-value problem of elasticity theory for a half-plane with curvilinear edge

We deduce a singular integral equation of a plane periodic problem of the theory of elasticity for a half plane with loaded curvilinear edge. The numerical solution of the integral equation is obtained by the method of quadratures for various configurations of the boundary of the half plane. The stress concentration factors are computed for a half plane with sinusoidal edge and for a periodic system of rounded V-shaped notches. On this basis, as a result of the limit transition, we get the stress intensity factors at the tips of sharp V-shaped notches and analyze their dependence on the opening angle of the notch. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 4, pp. 5–12, July–August, 2008.     

Boundary integral methods in low frequency acoustics

Abstract

This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

An advanced meshless LBIE/RBF method for solving two-dimensional incompressible fluid flows

The present work presents a meshless local boundary integral equation (LBIE) method for the solution of two-dimensional incompressible fluid flow problems governed by the Navier–Stokes equations. The method uses, for its meshless implementation, nodal points spread over the analyzed domain and employs in an efficient way the radial basis functions (RBF) for the interpolation of the interior and boundary variables. The inverse matrix of the RBF is computed only once for every nodal point and the interpolation functions are evaluated by the inner product of the inverse matrix with the weight vector associated to the integration point. This technique leads to a fast and efficient meshless approach, the locality of the method is maintained and the system matrices are banded with small bandwidth. The velocity–vorticity approach of the Navier–Stokes equations is adopted and the LBIEs are derived for the velocity and the vorticity field, resulting in a very stable and accurate implementation. The evaluation of the volume integrals is accomplished via a very efficient and accurate technique by triangularizing the local area of the nodal point to the minimum number of well formed triangles. Numerical examples illustrate the proposed methodology and demonstrate its accuracy.     

Unsteady heat transfer in a system of three coaxial finite cylinders

An integral transformation with the aid of which a solution of the problem of unsteady-state heat transfer in a system of three coaxial finite cylinders with different boundary conditions on their surfaces depending on space and time is presented. Each of the cylinders evolves heat of a certain intensity, depending on time and coordinates. A numerical solution of one variant of the boundary conditions is given and illustrated by figures. The method of transforming the solution of the problem with other boundary conditions is shown. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 1, pp. 140–148, January–February, 2007.     

Cell‐based volume integration for boundary integral analysis

The evaluation of volume integrals that arise in boundary integral formulations for non‐homogeneous problems was considered. Using the “Galerkin vector” to represent the Green's function, the volume integral was decomposed into a boundary integral, together with a volume integral wherein the source function was everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near‐singular integrals were handled by partial analytic integration. The method employed a Galerkin approximation and was presented in terms of the three‐dimensional Poisson problem. An axisymmetric formulation was also presented, and in this setting, the solution of a nonlinear problem was considered. Copyright © 2012 John Wiley & Sons, Ltd.     

Diffusive-convective transport into a porous membrane. A comparison of theory and experiment using scanning electrochemical microscopy operated in reverse imaging mode

Molecule transfer at the interface between a single ion-selective micropore and aqueous solutions is quantitatively investigated using scanning electrochemical microscopy operated in reverse imaging mode (SECM-RIM). Accumulation of two electroactive solute molecules, acetaminophen and ferrocenylmethyltrimethylammonium, at the pore/solution interface is observed when an electrical current is passed through the pore. Slow interfacial transfer of solute relative to the solvent as the solution is driven across the membrane by electroosmosis is responsible for solute accumulation. A theoretical expression for the concentration distribution of solute molecules above an individual pore opening is obtained by analytical solution of the convective-diffusive flux equation. The fluid velocity through the pore at constant electroosmotic force is determined by fitting the theoretical expression to SECM-RIM concentration profiles and is found, as anticipated, to be independent of the solute species and the bulk solute concentration. The results provide a theoretical basis for the SECM-RIM imaging of biological membranes as well as a general method for characterizing interfacial molecule/ion transfer kinetics.     

Boundary element analysis of ductile crack growth using the traction-free fundamental solution

A new boundary element formulation in two-dimensional rate-independent plasticity is given. This new formulation uses a so-called traction-free fundamental solution so that the resulting boundary integral equation converges in the normal sense, and more important, a formal differentiation of the boundary integral equation leads to a valid integral representation for the in-plane stress component on the boundary. No finite difference approximation is needed to construct the stress recovery routine. The new boundary element method is then used to solve the problem of quasi-static ductile crack growth. Numerical simulations based on a set of experimental data have been carried out to evaluate a new path-independent integral,T* _M . TheT* _M ,-integral is a modified version of Atluri'sT*-integral. This modified version has an advantage of having a less singular domain integral near the crack flank so that it is numericaly preferable toT*.