Kinds Of Boundary Conditions Research Articles

Atomic H over crystal surface: effective potential dependence on sample properties

Abstract We considered the behavior of the lowest electronic level of atomic H in a semi-infinite space bounded by a flat surface. 
We impose a third kind boundary condition on the electronic wave functions, where the boundary condition parameter models the adsorbent properties of the surface.
For the crystal surface, the double periodic function as the boundary parameter seems reasonable; therefore, this case is considered.
It is shown that there are two modes of atom adsorption on the sample surface depending on the parameters of the boundary condition. 
In the first case the effective atomic potential, considered as a function of the distance between H and the boundary plane, exhibits a well pronounced minimum at some finite distance and a relatively small effective range of interaction distances between the atoms and samples.
The second case occurs under the condition of a large positive affinity of the atomic electron to the sample boundary and low initial H-concentration inside the sample. 
In such a situation, the minimum of the effective potential is close to the sample surface, and a significant amount of energy can be emitted throughout the adsorption process.

General boundary conditions for a Boussinesq model with varying bathymetry

AbstractThis paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

RESEARCH OF THE DESTRUCTION CONDITIONS OF GLAZING OF ENCLOSING STRUCTURES UNDER THE CONDITIONS OF THERMAL INFLUENCE OF FIRE

The article presents the main results of mathematical modelling of the destruction of glazing of enclosing building structures with translucent elements under fire conditions. The authors analyse literature data on calculating the fire resistance of enclosing building structures with glazing. The results show that improving methods for predicting the fire resistance of glazing is relevant. Mathematical models for the numerical study of the behaviour of glazing under the thermal influence of the standard fire temperature regime are substantiated. Mathematical models of the heat transfer process involve using a non-stationary differential equation of heat conduction with boundary conditions of the third kind, accounting for the convective and radiant heat exchange between the air in the room with the fire and the glazing. As a criterion for the destruction of glazing under the influence of fire, the authors establish the temperature resistance parameter, which is determined based on the mechanical characteristics of the glass. The article highlights the relevant data on the study of temperature indicators and the mechanism of glazing failure, using mathematical modelling with well-founded mathematical models. It obtains the results of the thermal calculation by applying the finite difference method for solving the non-stationary differential equation of thermal conductivity. The methodology for determining the time of glazing failure under the influence of the standard temperature regime of a fire using the calculation parameters obtained based on well-founded mathematical models is substantiated. An analysis of the adequacy of the obtained calculation data was carried out by comparing them with the results of experimental studies using statistical criteria. As a result of the analysis, the authors confirmed the validity of the calculation data. A complete factorial experiment was conducted based on well-founded mathematical models, which revealed the regularities of the dependence of the parameters of thermomechanical processes in single-layer glass on its structural characteristics. The revealed regularities have the form of linear regression dependences of the destruction time on the thickness of the glazing and the maximum length of the glass panel without bars. Using the regularities of the dependence of the parameters of thermomechanical processes in single-layer glazing on its structural characteristics, the authors constructed reference tables for evaluating the fire resistance of enclosing structures with glazing. The research obtained new scientific data on the study of the behaviour of glazing of enclosing building structures under the thermal influence of fire through mathematical modelling, which is the basis for improving the methods of calculating the fire resistance of enclosing structures with translucent elements. Keywords: enclosing building structures, glazing, fire resistance limit, calculation method, glass heat resistance.

Comparison of the Position of the Maximum Humidification Zone when Using the Methods of Stationary and Non-Stationary Heat and Humidity Regime

The basic formulas for mathematical modeling of the heat and moisture regime of building envelopes in stationary and non-stationary settings are given. It is noted that the mathematical model uses third kind boundary conditions. Objective of the study: to check whether the moisture maximum plane position in the thickness of building envelopes is confirmed assessing non-stationary moisture conditions. Two methods for verification are used: the graphical method for determining the plane of maximum moisture and the method for assessing the non-stationary moisture regime, based on the Gagarin and Kozlov moisture potential. It was found that the maximum moisture is confirmed both for facade systems with mineral wool insulation and for facade systems with expanded polystyrene insulation. In the case of mineral wool, the maximum moisture is located outside the insulation layer. For expanded polystyrene insulation, the maximum moisture is determined inside the insulation layer. Thus, it was confirmed that the maximum moisture in the thickness of the enclosing structure, determined by the graphical method, is confirmed by mathematical modeling of a non-stationary heat and moisture regime.

The Dissipation of the Kinetic Energy for 2D Bounded Flow by Using Moment-Based Boundary Conditions with Burnett Order Stress for LBM

In this article, the lattice Boltzmann method with two relaxation time (TRT) for the D2Q9 model is used to investigate numerical results for 2D flow. The problem is performed to show the dissipation of the kinetic energy rate and its relationship with the enstrophy growth for 2D dipole wall collision. The investigation is carried out for normal collision and oblique incidents at an angle of . We prove the accuracy of moment -based boundary conditions with slip and Navier-Maxwell slip conditions to simulate this flow. These conditions are under the effect of Burnett-order stress conditions that are consistent with the discrete Boltzmann equation. Stable results are found by using this kind of boundary condition where dissipation of the kinetic energy is found to be proportional to in the first regime and it is in the second part of the regime as expected. An excellent agreement with the benchmark data is observed.

Devising a calculation method for modern structures of current-conducting elements in large electric machines in a three-dimensional statement

The jumpers of rotor pole-to-pole connections are highly stressed elements in a hydraulic generator structure. These assemblies often fail due to deformation that exceeds the size of an air gap. Existing methods do not take into account the thermal component and attempts to improve the design are not based on mathematical models that make it possible to perform calculations with an accuracy of more than 50 %. The method devised in this work makes it possible to obtain boundary conditions of the third kind on the basis of three-dimensional mathematical modeling of the ventilation system of the hydraulic unit without simplifications. The method accuracy is explained by taking into account the spatial thermal component. The heat transfer coefficient determined by this method in the pole-to-pole connections area was ~250 W/(m2·K). Using FEM, mathematical modeling of the thermal stress state of pole-to-pole connections was carried out, taking into account mechanical and thermal factors. This made it possible to design the improved connection structure with additional fastening elements, which make it possible to reduce the displacement to 0.03 mm, and the stress to 53 MPa at the rotor rotation frequency of 880 rpm. This design makes it possible to enable reliable operation of the hydraulic unit under the condition of increasing the rotor rotation frequency to overspeed with disconnected combinatorial dependence, provided that the actual stresses are 0.95 % of the material yield strength. The convergence of the values obtained by the proposed method and by the HSS method exceeded 99 %. The practical result is the proposals for the hydraulic generator design modernization

ANALYTICAL SOLUTION OF THE DIFFERENTIAL EQUATION OF HEAT CONDUCTIVITY FOR DAMAGED THERMAL INSULATION OF PIPELINES

For heat supply systems of Ukraine, the determination and forecasting of thermal energy losses during the transportation of heat carrier is an urgent problem. The thermal lines of the heating networks are long and its insulation is damaged. Installation of heat energy metering devices at all sources and all consumers (i.e. buildings) without exception allows determining and forecasting real heat losses in the heat network, but this is a difficult task, not all consumers and sources are actually covered by heat consumption accounting. The task of determining heat losses is also relevant for energy management systems of heat supply systems, energy companies and industry. The solution to the problem is proposed by a separate mathematical modeling of the temperature state of the section of the damaged insulating layer with the determination of the heat flow through it. It is proposed to solve the problem by the method of analytical solution of the differential equation of heat conduction with boundary conditions of the third kind. With the uniform distribution of characteristic damages along the total length of the pipeline, knowing the limits of damage influence, the share of insulation damage and the number of damages on the pipeline, it is possible to determine the real heat flow from the outer surface of the pipelines, including coefficient of increase of heat losses on the section of the heat pipe in relation to those determined by regulatory documents. Traditionally, one-dimensional models or determinations of the two-dimensional temperature field and actual heat fluxes in a cross-section to the axis of a characteristic damaged section of insulation are considered. But this does not take into account the two-dimensionality of the temperature field of the damaged layer of insulation along the length (that is, along the axis) of the pipeline. Therefore, the purpose of this work is to develop a methodology for determining heat losses through pipelines, taking into account the damage to their insulation and the distribution of characteristic damages along the length. The following simulation was carried out for one of the areas. Also, experimental and numerical studies using the finite difference method in combination with the method of running variable directions for similar models confirmed the coincidence of the analytical solution of the proposed model and the finite difference model within the permissible error.

On one method for solving a mixed boundary value problem for a parabolic type equation using operators AT <sub>ƛ, J</sub>

A new method for obtaining a generalized solution of a mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition is proposed. Generalized functions are understood in the sense of a sequential approach. The representative of the class of sequences, which is a generalized function, is obtained using the function interpolation operator, constructed using solutions of the Cauchy problem. The solution is obtained in the form of a series that converges uniformly inside the domain of the solution.

Imperfectly conducting eigenflows in a vertical fluid layer

The stability of buoyant flow in an infinite vertical fluid layer bounded by imperfectly conducting rigid walls, called imperfectly conducting eigenflows, is discussed. The third kind boundary conditions describing heat transfer to the external environment are applied to perturbations in temperature. The linear stability analysis is carried out numerically by employing the Chebyshev collocation method. Instability arises when the Grashof number G exceeds its critical value, which depends on the Prandtl number Pr and the Biot number Bi. It is found that the onset of instability changes dramatically depending on the magnitude of Prandtl and Biot numbers particularly when the instability is through the traveling-wave mode. The numerical results show that the Biot number plays a pivotal role in determining the transition Prandtl number PrT at which the instability switches over from one mode to another mode. The novel outcomes suggest the presence of a single PrT for Bi<2.1739 while three distinct values of PrT for Bi≥2.1739. The departure from the conventional single value typically observed at isothermal boundaries signifies the complexity of the instability mechanism.

Heat Transfer through Double-Chamber Glass Unit with Low-Emission Coating

The numerical modeling of radiation and convective heat transfer through a double-chamber glass unit was carried out to substantiate the increase in the heat transfer resistance of this unit via the application of low-emission coatings to glass surfaces. In the space between the panes of a window without low-emission coatings, the amount of heat transferred via radiation exceeds the amount of heat transferred via thermal conductivity and convection. The question of the effect of low-emissivity coatings on reducing heat loss through a window has not yet been sufficiently studied. This problem is also not sufficiently reflected in the literature. In this regard, this paper presents the results of numerical simulation aimed at studying the effect of low-emissivity coatings on heat transfer through a double-chamber glass unit. Simulation is carried out by numerically solving a system of equations of fluid dynamics and energy for the air gap and glass. Boundary conditions of the fourth kind are set on the internal surfaces of the chambers, taking into account the radiation and conduction components of the total heat flux emanating from the glass. The results of modeling heat transfer through a glass unit with ordinary glass show that about 60% of the heat is transferred by radiation. Therefore, an effective measure to reduce heat loss through windows is to reduce the radiation component of the total heat flux by applying a low-emissivity coating to the internal surfaces of the glass unit. This allows for the reduction of the overall heat flux (and, accordingly, heat loss to the environment) by 20–34%, depending on the number of glass surfaces with such a coating.

Инженерная методика расчета установки извлечения бетулина из бересты березы

The analysis of the current state of the process of betulin extraction from birch bark has shown the relevance of the implementation of periodic extraction technology using toluene as a solvent at small enterprises of the timber industry complex. This article presents the scheme of an extraction plant consisting of an extractor, an evaporator, a condenser, a florentine flask, an extract collector, as well as the principle of the plant operation. The extraction process is carried out in two stages: when the raw material is at rest in relation to toluene and when the raw material is extracted by continuous feeding of fresh extractant. The engineering procedure for calculating the extractor has been developed, allowing to determine its overall dimensions (diameter and height) and the duration of individual stages of the extraction process. The duration of the first stage has been calculated by solving Fick’s differential equation of mass conductivity under boundary conditions of the first kind. The duration of the second stage has been calculated considering the fact that betulin extract is continuously withdrawn from the extractor and fresh extractant is injected. In the latter case, the betulin content in the extractant varies not only in time, but also with the height of the extractor. The developed mathematical description and subsequent modeling made it possible to obtain the calculated curves of betulin distribution in birch bark at different points in time, as well as to compare the experimental and calculated data on the kinetics of the average betulin content in birch bark and toluene for each stage. Meanwhile, the kinetic curves of the average betulin content in birch bark at the second stage are presented for different heights of the extractor. It has been established that the optimal duration of the first stage is 20 min, and with the continuous feeding of fresh extractant with a flow rate of 22.5 kg/h at the second stage, the process speed increases by 6 times. This significantly reduces the total extraction time under the specified process parameters. The necessity for the introduction of the third stage (the stabilization period of the betulin content over the layer height and the cross– section of particles) has been shown.

On the nonstandard maximum principle and its application for construction of monotone finite-difference schemes for multidimensional quasilinear parabolic equations

UDC 517.9 We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary value problem for multidimensional quasilinear parabolic equation with an unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L 2 -norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind

Difference methods for solving some classes of multidimensional loaded parabolic equations with boundary conditions of the first kind

Об интегральном подходе при использовании метода коллокации

The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

Разностная схема повышенного порядка аппроксимации для уравнения Аллера с переменными коэффициентами

The initial boundary value problem for the one-dimensional Aller’s equation with variable coefficients and boundary conditions of the first kind is studied. The problem under study describes the processes of heat transfer in a heterogeneous environment, moisture transfer in soils, and fluid filtration in fractured porous media. To numerically solve the problem posed, a difference scheme of high order of accuracy was constructed - fourth order of accuracy in h and second order of accuracy in τ. Using the method of energy inequalities, an a priori estimate of the solution in a difference treatment is obtained. From this estimate it follows that the solution is unique and stable with respect to the right-hand side and initial data. Under the assumption of the existence of an exact solution to the original differential problem in the class of sufficiently smooth functions, and also due to the linearity of the problem under consideration, the obtained a priori estimate implies that the solution of the constructed difference problem converges to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme. The goal and scientific novelty of the work is to obtain a new numerical scheme of a higher order of approximation when solving the Dirichlet problem for the Aller’s equation with variable coefficients.

Heat treatment of ground surface using an electric arc plasma generator

The article presents a generalization of the results of experimental and theoretical studies on the intensive thermal effect on the surface of foundations from cohesive soils (as opposed to deep heat treatment) using an electric arc generator of low-temperature plasma (plasmatron). A plasmatron with massive consumable electrodes made of borosilicated graphite and a wide "blurred" plasma torch was used for heat treatment. Physical and mathematical modeling of the thermal effect on the surface of clay soil by a moving high-temperature source has been performed. For the first time, a special installation was manufactured and experimental studies were carried out on the heat treatment of ground surfaces on the site with an electric arc plasma torch. It is established that the effective heat flux density reaches 2.5-.3.5 kW/cm2. The main part of the thermal energy for the used plasmatron enters the material due to radiant heat transfer. An analytical mathematical model of the temperature field of a three-dimensional semi-bounded body in the technological process under consideration is given by an essentially nonlinear boundary value problem for a nonlinear heat equation with nonlinear boundary conditions of the second kind. When solving thermophysical problems, the ground semi-bounded space was considered as a quasi-homogeneous medium having a constant initial temperature and initial thermophysical parameters that change in the process of temperature increase. The effective values of the thermophysical characteristics were determined on the basis of experimental data. Calculations and experiments have shown that the temperature boundary leading to significant changes in the structural properties of soils drops to a depth of 4-6 cm from the surface, despite the boiling of the ground melt with a temperature of 2500 ...2800 K on the ground surface. Therefore, a new technological principle of heat treatment is proposed, which consists in building up the ground melt in layers of 4-5 cm up from the initial surface. The impossibility of obtaining a positive effect even with the use of a powerful high-temperature source of exposure in the case of traditional surface heat treatment technology has been confirmed. The new plasma technology of surface heat treatment of ground surfaces to the stage of silicate melt allows to obtain a common layer of the required thickness. At the same time, the efficiency of surface heat treatment is significantly increased. The new technology of layer-by-layer surface deposition reduces cracking in the layer when the melt cools, but does not eliminate this negative process.

Параболічні крайові задачі математичної фізики в кусково-однорідному клиновидному циліндрично-круговому півпросторі з порожниною

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable r wedge-shaped by the angular variable φ cylindrical-circular half-space with a cavity were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a finite integral Fourier transform on the Cartesian semiaxis (0; +∞) by an applicative variable z and a Weber hybrid integral transform type on the polar axis (R0; +∞) with n points of conjugation by a radial variable were used to construct solutions of investigated boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The consistent application of inverse integral transforms to the obtained solution in the space of images restores the solutions of the considered parabolic boundary value problems through their integral image in an explicit form in the space of the originals. At the same time, the main solutions to the problems were obtained in an explicit form.

Natural Vibrations of a Gas in a Helmholtz Resonator with a Periodically Varying Cross-section

Within the framework of the long-wave approximation, the frequencies and shapes of gas natural oscillations in a Helmholtz resonator having the shape of a pipe of periodic cross-section have been studied. The problem is reduced to the Sturm–Liouville problem with boundary conditions of the first kind, the solution of which is carried out by the method of accelerated convergence. A detailed analysis of the dependences of eigenvalues and eigenfunctions on pipe parameters was carried out. A “self-similar” type of dependence of the natural frequency for various modes has been revealed. The values of the resonator periodicity parameters at which a sharp change in the natural frequency occurs are determined.

On a stable calculation of the normal to a surface given approximately

The paper proposes a stable method for constructing a normal to a surface given approximately. The normal is calculated as the gradient of the function in the surface equation. As is known, the problem of calculating the derivative is ill-posed. In the paper, an approach is adopted to solving this problem as to the problem of calculating the values of an unbounded operator. To construct its stable solution, the principle of minimum of the smoothing functional in Morozov’s formulation is used. The normal is obtained in the form of a Fourier series in the expansion in terms of eigenfunctions of the Laplace operator in a rectangle with boundary conditions of the second kind. The functional stabilizer uses the Laplacian, which makes it possible to obtain a normal in the form of a Fourier series that converges uniformly to the exact normal vector as the error in the surface definition tends to zero. The resulting approximate normal vector can be used to solve various problems of mathematical physics using surface integrals, normal derivatives, simple and double layer potentials.

MODELS OF FIRE EXTINGUISHING WHEN FLAMMABLE LIQUID COMBUSTION

The process of extinguishing a class B fire with sprayed water is described by the differential equation of heat conduction with boundary conditions of the second kind. The solution of this differential equation in a dimensionless form allows to make a transition to the operative form of representation of the mathematical model of the quenching process - to the transfer function. The peculiarity of such a fire transfer function when extinguishing it with sprayed water is its irrationality, which causes difficulties in its use. One of the ways out of this situation is to switch to an equivalent representation of an irrational function of a complex variable in the form of a small-rational function of such a variable. Such a transition is carried out with the help of the minimax approximation using the Remez algorithm. Determination of the maximum order of the characteristic polynomial of the fire transfer function is carried out using the Hurwitz stability criterion. It is shown that for the transfer function of class B fire, the order of the Hurwitz polynomials does not exceed four, the approximation error does not exceed 3.6% using the method of undetermined coefficients, the transfer functions of the fire when they are extinguished with sprayed water are constructed in the form of a superposition of the transfer functions of aperiodic links. Numerical values of the parameters of such transfer functions were obtained. For the case when the intensity of the sprayed water supply is described by the Heaviside function, expressions describing the temperature of the surface of the burning liquid were constructed. It is noted that these expressions correspond to the expressions that describe the dynamic properties of class B fire in the time domain. It is shown that the maximum value of the relative error when using such transfer functions does not exceed 4.0%, if the general transfer function represents the superposition of the transfer functions of two aperiodic links. It is noted that when using the Padé approximation, the transfer function in the form of the transfer functions of eight aperiodic links corresponds to this value of the divergence error. Keywords: flammable liquid, fire extinguishing, sprayed water, transfer function.