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Quasichemical Models of Multicomponent NonlinearDiffusion

Published online by Cambridge University Press:  10 August 2011

A.N. Gorban ,
H.P. Sargsyan  and
H.A. Wahab
A.N. Gorban*
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
H.P. Sargsyan
Affiliation:
UNESCO Chair – Life Sciences International Postgraduate Educational Center (LSIPEC), Yerevan, Republic of Armenia
H.A. Wahab
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
*
⋆ Corresponding author. E-mail: ag153@le.ac.uk
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Abstract

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusionshould be nonlinear if there exist non-diagonal terms. The vast variety of nonlinearmulticomponent diffusion equations should be ordered and special tools are needed toprovide the systematic construction of the nonlinear diffusion equations formulticomponent mixtures with significant interaction between components. We develop anapproach to nonlinear multicomponent diffusion based on the idea of the reaction mechanismborrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This isthe stoichiometric algebra supplemented by the simple kinetic law. The results of thisinvention are now applied in many areas of science, from particle physics to sociology. Inour work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

Keywords

diffusionreaction mechanismentropy productiondetailed balancecomplex balancetransport equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 184 - 262
Copyright
© EDP Sciences, 2011

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