scholarly journals 3D MATHEMATICAL MODEL CHARACTERIZING THE DYNAMICS OF THE TEMPERATURE FIELD OF A WALL STRUCTURE WITH A DOUBLE-SIDED FACING FROM A SAPROPEL-HEMP COMPOSITE MATERIAL

2021 ◽  
Vol 3 ◽  
pp. 95-103
Author(s):  
Sharif E. Guseynov ◽  
Janis Rimshans ◽  
Jekaterina V. Aleksejeva ◽  
Aleksandrs Bereznojs ◽  
Stanislavs Pleiksnis
Keyword(s):  
Mathematical Model ◽  
Temperature Field ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Heat Propagation ◽  
Wall Structure ◽  
Temperature Function ◽  
Layer Composite ◽  
Initial Boundary Value

In this paper, a 3D mathematical model is proposed to determine the dynamics of the temperature field in a three-layer composite sapropel-hemp slab. The proposed model consists of a system of three initial-boundary value problems with respect to the temperature function for each layer, respectively, and one initial-boundary value problem with respect to the unknown velocity of heat propagation along the thickness dimension of the composite sapropel-hemp slab. 

The Initial Boundary-Value Problem for a Mathematical Model for Granular Medium

2015 ◽  
Vol 725-726 ◽  
pp. 863-868
Author(s):  
Vladimir Lalin ◽  
Elizaveta Zdanchuk
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Stress Tensor ◽  
Granular Medium ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cosserat Continuum ◽  
Suitable Model ◽  
Initial Boundary Value

In this work we consider a mathematical model for granular medium. Here we claim that Reduced Cosserat continuum is a suitable model to describe granular materials. Reduced Cosserat Continuum is an elastic medium, where all translations and rotations are independent. Moreover a force stress tensor is asymmetric and a couple stress tensor is equal to zero. Here we establish the variational (weak) form of an initial boundary-value problem for the reduced Cosserat continuum. We calculate the variation of corresponding Hamiltonian to obtain motion differential equation.

scholarly journals MODELING PROCESS OF DESTRUCTION OF PARTICLES OF CEMENT AT INITIAL STAGE OF HYDRATION

2019 ◽  
Vol 62 (11) ◽  
pp. 138-142
Author(s):  
Olga N. Filimonova ◽  
Marina V. Enyutina ◽  
Anatoly A. Khvostov ◽  
Viktor I. Ryashskih
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Coordinate System ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Spherical Coordinate ◽  
Initial Stage ◽  
Average Size ◽  
Initial Boundary Value

The substantiation the existence of the destruction period of cement particles during the hydration of concrete into fragments, in size significantly larger than the average size of molecules is presented. An idealized structure of the cell representation for the cement grain and the water surrounding it in the form of nested spheres is adopted. The outer sphere does not change its coordinates, and the inner sphere imitating the cement grain decreases in the course of the process due to the transfer of the formed fragments into the inter-spherical space filled with water. The assumption is made that the concentration field is uniform and varies along the radius. The crushing process is described using a physical model of "pseudo-dissolution" and is based on the formulation of Fick's law. By introducing the rate of destruction associated with the movement of the interfacial boundary, a mathematical model is formulated in the form of an initial boundary value problem for the diffusion-type equation in a spherical coordinate system. By introducing a special coordinate system, the formulated initial-boundary value problem is transformed into a problem with fixed boundaries. Integration of the obtained system is performed numerically using an explicit finite-difference scheme. The computational experiment confirmed the efficiency of the proposed algorithm, which allowed to conduct a qualitative analysis of the model, which showed the correctness of the assumptions made in the formulation of the mathematical model. It is found that in a spherical cell containing an average cement grain, the fragments are localized near the interfacial surface due to the weak influence of the diffusion transfer mechanism. The estimation of the average size of the fragments, which are significantly larger than the average molecular size, further confirmed the hypothesis of the existence of the period of destruction of cement grains in the initial stage of concrete hydration.

scholarly journals MATHEMATICAL MODELING OF WATER WAVE MOTION IN A RESERVOIR CAUSED BY THE INTRUSION OF A LANDSLIDE ARRAY

2020 ◽  
Vol 12 (4) ◽  
pp. 555-564
Author(s):  
Illarion MUZAEV ◽  
◽  
Valery SOZANOV ◽  
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Boundary Value Problem ◽  
Flow Rate ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Special Cases ◽  
Initial Boundary Value ◽  
Reservoir Basin

The construction of dams and reservoirs in mountainous and foothill areas poses a number of urgent tasks for designers and researchers related to the wave movement of water in a reservoir with a complex geometric configuration. The collapse of significant masses of rock into the reservoir basin as a result of a landslide phenomenon provokes high surface waves, leading to catastrophic natural disasters in the form of victims and destruction. Designers, construction organizations and maintenance services are required to assess the expected wave increase in water level through the dam crest, as well as the zone and degree of flooding of the area in the reservoir bays and along the river gorge, depending on the geometric, kinematic and dynamic characteristics of potentially possible landslide massifs, mudslides and avalanche flows. In this way, you can predict and then prevent or mitigate the consequences and damage that can cause the formation of destructive waves. In addition to the above problem, this article develops a mathematical model of the wave movement of water in a reservoir, when the movement is caused by the intrusion of a landslide rock mass or a high-speed mudflow into its bowl. The model is the initial boundary value problem of hydrodynamics and hydraulics of gravitational waves. In contrast to previous works, the developed mathematical model takes into account the planned non-prismatic configuration of the reservoir and changes in the water depth in the longitudinal direction of the reservoir. Taking these factors into account significantly clarifies the numerical values of the amplitude of the wave formed, as well as the values of the flow rate, speed and volume of water poured over the dam crest. In the General initial-boundary value problem, the coefficients of the main differential equation are variables that depend on the spatial coordinate. In the General formulation, this creates great difficulties in the analytical solution of the problem and has not been solved by anyone in the whole world to date. In this article was found and applied original and effective substitution, i.e. replacing a variable that in two special cases based on the coefficients from the spatial coordinates that led basic differential equation initial-boundary value problem to the equation with constant coefficients, and thus substantially easier way of solving the initial-boundary value problem. In the above-mentioned special cases, the initial-boundary value problem is solved by strict analytical methods of mathematical physics. For each case, a set of calculation formulas is obtained for calculating the amplitude of the waves formed, as well as for the flow rate and volume of water poured over the dam crest.

scholarly journals Research of a mathematical model of low-concentrated aqueous polymer solutions

2006 ◽  
Vol 2006 ◽  
pp. 1-27 ◽  
Author(s):  
Mikhail V. Turbin
Keyword(s):  
Mathematical Model ◽  
Weak Solution ◽  
Boundary Value Problem ◽  
Polymer Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Aqueous Polymer ◽  
Initial Boundary Value ◽  
The Mathematical Model

The initial-boundary value problem for the mathematical model of low-concentrated aqueous polymer solutions is considered. For this initial-boundary value problem a concept of a weak solution is introduced and the existence theorem for such solutions is proved.

scholarly journals Numerical simulation of the thermodynamics of a fast neutron reactor

2018 ◽  
Keyword(s):  
Mathematical Model ◽  
Boundary Value Problem ◽  
Fuel Element ◽  
Fast Neutron ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
International Forum ◽  
Initial Boundary Value ◽  
Fuel Elements

The article deals with one reactors design, which, under the International Forum, are attributed to the 4th generation of the GIF-IV (Generation IV International Forum) of fast neutron reactors with a helium coolant and a closed fuel cycle (GFR). Although the use of helium as a coolant in reactors of this type and has great advantages in comparison with other coolants, for example, CO2 gas, however, due to the great difficulties encountered in the implementation of such a project, only prototypes of similar reactors are currently implemented. Due to the complexity of gas flow in the collectors and backfill, the averaged flow of the coolant is considered throughout the proposed mathematical model. It is assumed that the averaged flow is symmetric everywhere relative to the common axis of the cylinders forming the annular domain, and, consequently, is axisymmetric, that is, two-dimensional. One such annular cylindrical cavity will be called a fuel element. The mathematical model of a cassette of several such fuel elements connected by common distributed and gathering collectors is considered in the article. The algorithm for solving the arising non-stationary initial-boundary value problem is proposed in the article, as well as the results of some computational experiments that are obtained using the PC program, compiled and debugged by the author of the article. The experiments were carried out both for one fuel element, and for cassettes of 2, 3 and 4 fuel elements. The algorithm for solving the arising non-stationary initial-boundary value problem is proposed in the article, as well as the results of some computational experiments that are obtained using the PC program, compiled and debugged by the author of the article.

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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