scholarly journals Analytical Solutions for a General Mixed Initial-Boundary Value Problem Corresponding to Hydromagnetic Flows of Second Grade Fluids through Porous Medium

2021 ◽  
Vol 5 (3) ◽  
pp. 225-236
Author(s):  
Constantin Constantin Fetecau ◽  
Dumitru Vieru
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Analytical Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Second Grade ◽  
Initial Boundary Value ◽  
Second Grade Fluids

scholarly journals Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium

2020 ◽  
pp. 72-76
Author(s):  
R.A. Virts ◽  
A.A. Papin ◽  
W.A. Weigant
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Second Order ◽  
Initial System ◽  
Effective Pressure ◽  
Initial Boundary Value

The paper considers a model for filtering a viscous incompressible fluid in a deformable porous medium. The filtration process can be described by a system consisting of mass conservation equations for liquid and solid phases, Darcy's law, rheological relation for a porous medium, and the law of conservation of balance of forces. This paper assumes that the poroelastic medium has both viscous and elastic properties. In the one-dimensional case, the transition to Lagrange variables allows us to reduce the initial system of governing equations to a system of two equations for effective pressure and porosity, respectively. The aim of the work is a numerical study of the emerging initial-boundary value problem. Paragraph 1 gives the statement of the problem and a brief review of the literature on works close to this topic. In paragraph 2, the initial system of equations is transformed, as a result of which a second-order equation for effective pressure and the first-order equation for porosity arise. Paragraph 3 proposes an algorithm to solve the initial-boundary value problem numerically. A difference scheme for the heat equation with the righthand side and a Runge–Kutta second-order approximation scheme are used for numerical implementation.

scholarly journals Numerical Solution of One Problem of Carbon Dioxide Injection into the Rock

2021 ◽  
pp. 81-85
Author(s):  
R.A . Virts
Keyword(s):  
Carbon Dioxide ◽  
Porous Medium ◽  
Boundary Value Problem ◽  
Numerical Study ◽  
Boundary Value ◽  
Original System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Carbon Dioxide Injection ◽  
Initial Boundary Value

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible liquid or gas in a porous medium. A unique feature of the model under consideration is the incorporation of poroelastic properties of the solid skeleton. From a mathematical point of view, the equations of mass conservation for liquid / gaseous and solid phases, Darcy's law, the rheological ratio for a porous medium, and the conservation law of the balance of forces are considered. The work is aimed at numerical study of the model initial-boundary value problem of carbon dioxide injection into the rock with minimum initial porosity. Also, it is necessary to find out the parameters at which the porosity will increase in the upper layers of the rock and, as a result, the gas will come to the surface. Section 1 contains a statement of the problem and a brief review of scientific papers related to this topic. In Section 2, the original system of constitutive equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and a first-order equation for porosity. Section 3 presents the results and conclusions of a numerical study of the initial-boundary value problem.

scholarly journals Numerical Solution of a Two-Dimensional Problem of Fluid Filtration in a Deformable Porous Medium

2021 ◽  
pp. 88-92
Author(s):  
R.A. Virts
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Deformable Porous Medium ◽  
Balance Of Forces ◽  
Initial Boundary Value

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible fluid in a deformable porous medium. The model is based on the equations of conservation of mass for liquid and solid phases, Darcy’s law, the rheological relationship for a porous medium, and the law of conservation of the balance of forces. In this article, the equation of the balance of forces is taken in full form, i.e. the viscous and elastic properties of the medium are taken into account. The aim of the work is a numerical study of a model initial-boundary value problem. Section 1 gives a statement of the problem and a brief review of the literature on works related to this topic. In item 2, the original system of equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and the first-order equation for porosity. Section 3 proposes an algorithm for the numerical solution of the resulting initial-boundary value problem. For the numerical implementation, a variable direction scheme for the heat equation with variable coefficients is used, as well as the Runge — Kutta scheme of the fourth order of approximation.

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.

scholarly journals An extension of Gronwall inequality in the theory of bodies with voids

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1161-1167
Author(s):  
Marin Marin ◽  
Praveen Ailawalia ◽  
Ioan Tuns
Keyword(s):  
Boundary Value Problem ◽  
Internal State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Internal Variables ◽  
State Variables ◽  
Gronwall’S Inequality ◽  
Initial Boundary Value

Abstract In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

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