Моделирование течения воздуха в упруго-деформируемой пористой среде, аппроксимирующей легкие человека: структура модели, ее основные уравнения и разрешающие соотношения

Abstract

This paper is devoted to refining the mathematical model of the human respiratory system designed to predict the occurrence of respiratory pathologies caused by the negative effects of atmospheric air pollution. It is difficult to identify individual air channels in the human lungs, consisting of small respiratory tracts, called alveoli, through which oxygen from the air passes into blood. Therefore, the human lungs are modeled as a two-phase elastically deformable saturated porous medium, in which one phase is the deformable skeleton of the medium (lung tissue) described by the model of deformable solid body, and the other phase is gas filling the porous space. We construct the model of air flow in the human lungs, which consists of the sub-model of elastic deformation of the porous lung medium, experiencing large displacement gradients, and the sub-model of air filtration through the deformable porous medium. The model takes into account the relationships between sub-models of the respiratory system, i.e. we consider the coupled problems of deformation and filtration in differential form. Since at present the analytical solution of the stated essentially nonlinear problem is unfeasible, the problem is formulated in a generalized weak form to allow subsequent application of numerical methods using the Galerkin method. For the numerical solution of the nonlinear problem of deformation of the two-phase lung medium, it is suggested to use the finite element method, for the filtration problem - the finite volume method. The resolving relations needed to solve the formulated sub-problems are derived. To solve the nonlinear problem, we developed algorithms and a set of programs. The description of these computational tools, as well as analysis of the results obtained will be presented in our the next paper.