Solution of the two-dimensional unsteady diffusion equation for vortex flow

A numerical method of solution based on the use of probability analogies is presented. An example of a calculation by the scheme developed is given.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 30, No. 6, pp. 1107–1111, June, 1976.     

A meshless method for two-dimensional diffusion equation with an integral condition

This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions. In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary condition is imposed directly. To treat the complicated integral boundary condition appearing in the problem, Simpson's composite numerical integration rule is applied. A time stepping scheme is employed to approximate the time derivative. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.     

Boundary integral equation formulations for free-surface flow problems in two and three dimensions

To compute the transient solution of free-surface flow problems in two and three dimensions boundary integral equation formulations are considered. Consistent lower and higher order approximations based on small curvature expansions are compared and applied to a time-dependent, linear free-surface wave problem.Enhanced version of a paper presented at the IUTAM-Symposium on advanced boundary element methods, applications in solid and fluid mechanics, April 13–16, 1987, San Antonio/Tx, USA     

The effects of flow rate and particle shape on the convective diffusion to a solid catalytic particle

Summary The flow of a fluid, containing a reactant, past a solid catalytic particle on which a reaction takes place is considered for large Péclet number. The concentration of the reactant is given by the diffusion boundary-layer equation, and this is solved in the case when the rate of reaction on the particle surface and the rate of diffusion of reactant onto the surface are of the same order of magnitude.For a spherical particle, a series solution for the concentration is found for the case of Stokes flow, and numerical solutions are found for Stokes flow and for flow at higher Reynolds numbers (up to Re=10). To examine the effect of particle shape, numerical solutions are found for prolate and oblate spheroids in Stokes flow.     

Boundary integral equation formulation for the generalized micro-polar thermoviscoelasticity

A formulation of the boundary integral equation method for generalized linear micro-polar thermoviscoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord-Shulman with one relaxation time, Green-Lindsay with two relaxation times, Green-Naghdi theories, and Chandrasekharaiah and Tzou with dual-phase lag, as well as to the dynamic coupled theory. The cases of generalized linear micro-polar thermoviscoelasticity of Kelvin-Voigt model, generalized linear thermoviscoelasticity and generalized thermoelasticity can be obtained from the given results.     

A solution of the diffusion equation

A nonstationary solution is obtained for the diffusion equation in the case of two gas containers of arbitrary volume, connected by a capillary. If the volume of the capillary is regarded as negligibly small in comparison with the flask volumes, the solution turns into the formula derived by Ney and Armsteady, and as the volume of one of the flasks approaches infinity, the solution turns into the one derived by Frank-Kamenetskii.     

Wave equation model for solving advection–diffusion equation

This paper presents a Wave Equation Model (WEM) to solve advection dominant Advection–Diffusion (A–D) equation. It is known that the operator-splitting approach is one of the effective methods to solve A–D equation. In the advection step the numerical solution of the advection equation is often troubled by numerical dispersion or numerical diffusion. Instead of directly solving the first-order advection equation, the present wave equation model solves a second-order equivalent wave equation whose solution is identical to that of the first-order advection equation. Numerical examples of 1-D and 2-D with constant flow velocities and varying flow velocities are presented. The truncation error and stability condition of 1-D wave equation model is given. The Fourier analysis of WEM is carried out. The numerical solutions are in good agreement with the exact solutions. The wave equation model introduces very little numerical oscillation. The numerical diffusion introduced by WEM is cancelled by inverse numerical diffusion introduced by WEM as well. It is found that the numerical solutions of WEM are not sensitive to Courant number under stability constraint. The computational cost is economical for practical applications.     

Solution of a boundary-value problem for the generalized diffusion equation

The boundary-value problem is solved for the complete diffusion equation and then for the same equation with a chemical reaction taken into account, for the case of a liquid flow for which the width and length are much larger than the thickness.     

Boundary value techniques for finite element solutions of the diffusion-convection equation

In this paper, the formulation of six-point and nine-point finite element equations for the solution of the diffusion-convection equation is presented. The six-point equation requires the solution of a tridiagonal system of equations and the nine-point centred equation is treated as a solution of a boundary value problem which leads to a large linear system of equations. Some numerical experiments are presented and the comparison with existing methods is included.     

Generalized diffusion equation for impurity atoms in semiconductor crystals

A diffusion equation is obtained for impurity atoms migrating by means of the formation of equilibrium complexes with intrinsic point defects, the distribution of which is nonequilibrium and nonuniform.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 57, No. 5, pp. 805–810, November, 1989.     

The motion of a particle in a nonuniform moving viscous medium

The motion of particles is examined in a nonuniform moving viscous medium in the regions Re < 1 and 1 < Re 300. Solutions are presented for the differential equations which make it possible to calculate the trajectories of particle motion in a nonuniform moving viscous medium.     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

Axial particle diffusion in rotating cylinders

We study the interface dynamics of a binary particle mixture in a rotating cylinder numerically. By considering only the particle motion in axial direction, it is shown that the initial dynamics can be well described by a one-dimensional diffusion process. This allows us to calculate a macroscopic diffusion constant and we study its dependence on the inter-particle friction coefficient, the rotation speed of the cylinder and the density ratio of the two components. It is found that radial segregation reduces the drift velocity of the interface. We then perform a microscopic calculation of the diffusion coefficient and investigate its dependence on the position along the cylinder axis and the density ratio of the two particle components. The latter dependence can be explained by looking at the different hydrostatic pressures of the two particle components at the interface. We find that the microscopically calculated diffusion coefficient agrees well with the value from the macroscopic definition when it is measured in the middle of the cylinder.     

Boundary element solution of the transonic integro-differential equation

The transonic integro-differential equation for two-dimensional flows is solved by boundary element methods. In addition to constant and quadrilateral elements we develop hybrid elements based on constant elements in the streamwise direction and variable elements in the transverse direction. Computation is carried out for parabolic-arc and NACA0012 airfoils and the results, which converge fast, compare favourably with finite-difference solutions. The hybrid elements are to be preferred because they yield results which are more accurate than constant elements without the computational complexity associated with quadrilateral elements. Moreover, they can be applied with a small number of nodes by using only one strip of rectangular elements.     

Some asymptotic solutions of the diffusion equation

The equations describing the variation of the radius of a water droplet with time are investigated for quasi-stationary and nonstationary evaporation processes. Solutions of these equations are found in the form of asymptotic series in powers of a small dimensionless parameter. Some properties of the solutions are determined.     

Boundary layer refinements in convective diffusion problems

The paper outlines a numerical procedure for the finite element solution of convective diffusion problems with significant convective terms using conventional (not upwinded) Galerkin methods in connection with ‘boundary-layer type’ elements. The underlying argument in the sequel is that the poor stability properties of conventional Galerkin methods are caused by the insufficient approximation of eigensolutions. These are located at some sections of the boundary and are only present within a generally very thin layer. Consequently, the identification of these layers and the satisfactory approximation of the eigensolutions are necessary and totally sufficient for a satisfactory solution. In the following we intend to present this procedure, its theoretical background and selected numerical results.     

Modeling of silicon diffusion in gallium arsenide. 3. Numerical method for solving the diffusion equation

A finite-difference approximation is obtained and investigated and an algorithm for numerical solution of the diffusion equation for silicon atoms in gallium arsenide is constructed.Belarussian State University. Belarussian State University of Informatics and Radioelectronics, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 4, pp. 484–489, April, 1994.     

A flow equation for anisotropic ice

Twenty-three samples of anisotropic glacier ice were tested in simple shear at a constant stress and constant temperature to investigate the effects of crystal fabric and crystal size on ice rheology. Results indicate that minimum strain rate is sensitive to variations in both crystal fabric and crystal size; as fabric intensity increases from isotropic to strong single maximum, minimum strain rate increases by a factor of four, and if crystal size is doubled, minimum strain rate increases about nine times. Statistical treatment of the data yield an empirical flow law of the form:

$\text{γ}\text{?}\text{min}\text{= d}{}^{\mathrm{l?m}}\text{(}\text{τ}\text{β}{}^1\text{)}{}^{\mathrm{n}}\text{exp}\text{(}\text{?Q}\text{RT}\text{)}$

where $\text{γ}\text{?}{}_{\mathrm{<mtext>min</mtext>}}$ is the minimum octahedral shear-strain rate, d is the average crystal size, f is the fabric intensity, τ is the octahedral-shear stress, Q is the activation energy for creep, R is the gas constant, T is the absolute temperature, and l, m, n, and β′ are constants.