Mathematical Model of Fluids Motion in Poroelastic Snow-ice Cover

The dynamics of a snow-ice cover is considered within the theory of poroelasticity. The snow-ice cover is modeled by a three-phase medium consisting of water, air and ice. The governing equations are the equations of mass conservation for each phase with phase transitions, the equations of conservation of phase momentum in the form of Darcy’s law, the equation of conservation of momentum of the whole system, the rheological equation for porosity and the equation of heat balance of snow. In the full formulation the liquid and air pressures are functions of the temperature and the corresponding densities, and the viscosity and compressibility coefficients of ice are functions of the temperature. The problem of two-dimensional nonstationary filtration of water in a thin poroelastic ice plate is considered in the model case. The solution is obtained in quadratures

Study of some mathematical models for nonstationary filtration processes

Рассматриваются математические модели, связанные с изучением нестационарных процессов фильтрации в подземной гидродинамике. Они представляют собой нелинейные задачи для параболических уравнений с неизвестной функцией источника в правой части. Одна из постановок является системой, которая состоит из краевой задачи с граничными условиями первого рода и из уравнения, задающего закон изменения по времени искомой функции источника. В другой постановке соответствующая система включает в себя краевую задачу с граничными условиями второго рода. Указанные постановки существенно отличаются от обычных краевых задач для параболических уравнений. Цель исследования - установить для этих нелинейных параболических задач условия однозначной разрешимости в классе гладких функций на основе априорных оценок метода Ротэ. We consider some mathematical models connected with the study of nonstationary filtration processes in underground hydrodynamics. These models involve nonlinear problems for parabolic equations with unknown source functions. One of the problems is a system consisting of a boundary value problem of the first kind and an equation describing a time dependence of the sought source function. In the other problem, the corresponding system is distinguished from the first one by boundary conditions of the second kind. These problems essentially differ from usual boundary value problems for parabolic equations. The aim of our study is to establish conditions of unique solvability in a class of smooth functions for the considered nonlinear parabolic problems. The proposed approach involves the proof of a priori estimates for the Rothe method.

Analysis of pressure difference changes in respirator filters while dusting

Objective is to determine the relation between pressure difference of a filter and parameters of a filtering layer to lengthen the protection period. The study involved elements of system analysis and mathematical modeling. Basic statements of the theory of nonstationary filtration and aerohydrodynamics were applied to develop a dust-loaded filter model. Dependence of pressure difference in a filter upon certain changes in packaging density of dust-loaded fibers has been determined; the dependence makes it possible to define minimum fiber packaging density to provide maximum dust capacity. To provide maximum dust capacity and high protective efficiency, the number of filtering layers in multilayered filters depends upon dust concentration within the air of the working zone, specified protection level of a respirator, air loss, and maximum dust volume which may be accumulated in the finishing filtering layer in terms of fiber packaging density being determined according to the minimum pressure difference in a filter at final time of respirator operation. A model of changes in pressure difference in filtering respirators in the process of aerosol particles depositing on filters has been improved; contrary to other available models, that one takes into consideration changes in fiber packaging density while dusting, filtration coefficient, and the amount of aerosol particles.