DIFFERENCE METHODS TO ONE AND MULTIDIMENSIONAL INTERDIFFUSION MODELS WITH VEGARD RULE

Lucjan Sapa,Marek Danielewski,Bogusław Bożek

doi:10.3846/mma.2019.018

Abstract

In this work we consider the one and multidimensional diffusional transport in an s-component solid solution. The new model is expressed by the nonlinear parabolic-elliptic system of strongly coupled differential equations with the initial and the nonlinear coupled boundary conditions. It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. We construct the implicit finite difference methods (FDM) generated by some linearization idea, in the one and two-dimensional cases. The theorems on existence and uniqueness of solutions of the implicit difference schemes, and the theorems concerned convergence and stability are proved. We present the approximate concentrations, drift and its potential for a ternary mixture of nickel, copper and iron. Such difference methods can be also generalized on the three-dimensional case. The agreement between the theoretical results, numerical simulations and experimental data is shown.

Highlights

Quantitative description of the diffuse mass transport is essential for materials processing and hydrodynamics
In the case of the onedimensional binary closed mixture with constant concentration, c1 +c2 = const, the Darken method allows to transform the system of two partial differential equations modeling the process
In the paper we study the model of interdiffusion introduced in [22] and in some special case in [24], in the one-dimensional and the multidimensional cases

Summary

Introduction

Quantitative description of the diffuse mass transport is essential for materials processing and hydrodynamics. S, where Ωk means the partial molar volume of the kth component (see Section 2), with the initial and the nonlinear coupled boundary conditions It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. A detailed analysis of a concept of the drift velocity, a choice of the reference frame, as well as the other physical, mathematical and numerical consequences of the proposed formalism can be found in [4,7,8,11,16,25,26] and in the references therein In those papers concentration of a mixture must be constant, while the Vegard rule assumed by us admits the overall concentration depending on time and a space. The examples of physical problems and numerical experiments are given in these sections

Strong formulation of an interdiffusion model

Implicit difference method

Example and numerical experiments