An initial-boundary value problem on the convection of a viscous weakly compressible fluid with axial symmetry II. Stability of generalized solutions

1991 ◽  
Vol 43 (1) ◽  
pp. 81-86 ◽  
Author(s):  
V. B. Moseenkov
Keyword(s):  
Boundary Value Problem ◽  
Compressible Fluid ◽  
Axial Symmetry ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solutions ◽  
Weakly Compressible ◽  
Initial Boundary Value

scholarly journals Constructing the Approximate Solution of Axisymmetric Problem on the Dynamics of Anisothermal Moisture Transfer

2020 ◽  
pp. 41-52
Author(s):  
Olga Marchenko ◽  
Tetiana Samoilenko
Keyword(s):  
Heat Transfer ◽  
Finite Elements ◽  
Boundary Value Problem ◽  
Moisture Transfer ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Elements Method ◽  
Generalized Solutions ◽  
Initial Boundary Value

Introduction. Calculation of dynamics of the anisothermal moisture transfer processes in axisymmetric formulation is essential in the study of wet soils condition around, for example, vertical drains, wells, piles, etc. In this paper, we formulate the initial boundary value problem for the system of moisture and heat transfer nonstationary equations. The problem is considered for isotropic medium in cylindrical coordinate system under the inhomogeneous mixed boundary conditions. The obtained results are important for future research in cylindrical coordinates of problems that model the migration of moisture during the seasonal freezing of the soil, taking into account phase transitions from unfrozen water to ice in the entire volume of the soil mass without highlighting the crystallization front. In this case moisture exchange and heat transfer characteristics appear as functions of the total humidity. Consequently, the equation of moisture transfer is written relative to the "fictitious" moisture content. Because of the main direction of moisture migration relative to the freezing/melting front, the convective heat transfer along the vertical coordinate axis is considered to be essential that leads to sufficient coincidence with the experimental data. The purpose of the paper is to formulate the appropriate generalized problem in the Galorkin form for the axisymmetric initial-boundary value problem. The important goal is to investigate the accuracy of the continuous in time and completely discrete approximate generalized solutions based on the finite elements method. Results. The algorithm for constructing of approximate generalized solution of the axisymmetric initial-boundary value problem for the system of filtration and heat transfer equations is proposed. The estimates of the convergence rate for the continuous in time and discrete approximate solutions based on the finite elements method are obtained.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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