Nonstationary Equations Research Articles

Axisymmetric Resonant Vibrations and Vibration Heating of an Inelastic Cylindrical Shell Compliant to Shear with Piezoelectric Actuators and Rigidly Fixed End Faces

We consider the problem of axisymmetric resonance vibrations of an inelastic cylindrical shell compliant to shear with piezoelectric actuators under the action of electromechanical monoharmonic loading. We take into account transverse shear strains, rotation inertia of a normal element, and temperature dependence of complex modules of piezopassive and piezoactive materials. The nonlinear problem is solved by using the iterative approach with respect to time within the framework of which the system of ordinary differential equations for vibrations is reduced to the integration of nonstationary heat-conduction equation by the finite-difference method. By using piezoelectric actuators, we analyze the influence of shear strains and the temperature of vibration heating on the amplitude and temperature-frequency characteristics and active damping of the resonance vibrations of the shell.

No-Slip Boundary Condition for Vorticity Equation in 2D Exterior Domain

In this article we derive the no-slip boundary condition for a non-stationary vorticity equation. This condition generates the affine invariant manifold and no-slip integral relations on vorticity can be transferred to a Robin-type boundary condition.

An Engineering Method of Analyzing the Dynamics of Mass Transfer during Concrete Corrosion Processes in Offshore Structures.

The environment of an underground structure is much more complex than the above-ground environment. Erosion processes are underway in soil and groundwater; groundwater seepage and soil pressure are also typical for underground environments. Alternating layers of dry and wet soil have a strong effect on concrete, and they reduce its durability. Corrosion of cement concretes is caused by the diffusion of free calcium hydroxide, located in the pores of concrete, from the volume of the cement stone to its surface, bordering on an aggressive environment, and the further transition of the substance through the phase boundary solid (concrete)-soil-aggressive environment (liquid). Due to the fact that all minerals in cement stone exist only in saturated or close-to-saturated solutions of calcium hydroxide, a decrease in the content of which in the pores of concrete as a result of mass transfer processes causes a change in the phase and thermodynamic equilibrium in the body of concrete and leads to the decomposition of highly basic compounds of cement stone and, consequently, to the deterioration of the mechanical properties of concrete (reduction in strength, modulus of elasticity, etc.). A mathematical model of mass transfer in a two-layer plate imitating the "reinforced concrete structure-layer of the soil-coastal marine area" system is proposed as a system of nonstationary partial derivative differential equations of the parabolic type with Newmann's boundary conditions inside the building and at the interface between the soil and the marine environment and with conjugating boundary conditions at the interface between the concrete and the soil. When the boundary problem of mass conductivity in the "concrete-soil" system is solved, expressions are obtained to determine the dynamics of the concentration profiles of the target component (calcium ions) in the volumes of the concrete and soil. As a result, one can select the optimum composition of concrete, having high anticorrosion properties, to extend the durability of the concrete constructions of offshore marine structures.

Game-theoretical problems for fractional-order nonstationary systems

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system’s trajectory to a terminal set, whereas the other strives to prevent it. Using the technique of set-valued maps and their selections, sufficient conditions for reaching the terminal set in a finite time are derived. Theoretical results are supported by a model example.

Non-hydrostatic layer-averaged approximation of Euler system with enhanced dispersion properties

A new family of non-hydrostatic layer-averaged models for the non-stationary Euler equations is presented in this work, with improved dispersion relations. They are a generalisation of the layer-averaged models introduced in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018), named LDNH models, where the vertical profile of the horizontal velocity is layerwise constant. This assumption implies that solutions of LDNH can be seen as a first order Galerkin approximation of Euler system. Nevertheless, it is not a fully (x, z) Galerkin discretisation of Euler system, but just in the vertical direction (z). Thus, the resulting model only depends on the horizontal space variable (x), and therefore specific and efficient numerical methods can be applied (see Escalante-Sanchez et al. in J Sci Comput 89(55):1–35, 2021). This work focuses on particular weak solutions where the horizontal velocity is layerwise linear on z and possibly discontinuous across layer interfaces. This approach allows the system to be a second-order approximation in the vertical direction of Euler system. Several closure relations of the layer-averaged system with non-hydrostatic pressure are presented. The resulting models are named LIN-NH_k models, with k=0,1,2. Parameter k indicates the degree of the vertical velocity profile considered in the approximation of the vertical momentum equation. All the introduced models satisfy a dissipative energy balance. Finally, an analysis and a comparison of the dispersive properties of each model are carried out. We show that Models LIN-NH_1 and LIN-NH_2 provide a better dispersion relation, group velocity and shoaling than LDNH models.

Математическое моделирование натекания водорода в гермзоне реакторного здания АЭС с ВВЭР-1200

Introduction. The patterns of hydrogen-air mixtures formation during the flow of hydrogen into the upper part of a large volume room in the initial moments of time are insufficiently studied. Therefore, the determination of regularities of formation of local explosive and fire-hazardous hydrogen-air mixtures when hydrogen flows into the lower part of the dome space of the reactor building is important. Goals and objectives. The purpose of the article is a theoretical study of occurrence of local fire and explosive zones of the hydrogen-air mixture generated during hydrogen leakage in the containment area of the reactor building in order to substantiate parameters of concentration sensors of the hydrogen concentration control system. To achieve it, a zone mathematical model of hydrogen concentration calculation in a pressurized room has been developed. Numerical experiments on regularities of hydrogen-air mixtures formation have been carried out. Theoretical basis. The generalized three-dimensional non-stationary differential equation of the laws of conservation of mass, momentum and energy is used to calculate local hydrogen concentration fields. The developed zone model makes it possible to determine hydrogen concentrations in the convective column and in the ceiling layer. Results and discussion. The characteristic fields of hydrogen mass concentrations in the volume of the dome space are obtained. It has been shown that at the initial stage of hydrogen leakage, a zone of the ceiling layer is formed under the ceiling of the dome space, which confirms the validity of the usage of the zone model. The hydrogen concentrations obtained by the field and zone models are compared. The distributions of hydrogen mass concentration along the convective column height at different Reynolds numbers in the hydrogen leakage hole have been obtained. It has been shown that hydrogen concentration sensors can detect a hydrogen leakage mode in the dome space only in a narrow area of Reynolds Re = 900–5,000. There is a leakage mode (Re = 3,358), in which the maximum hydrogen volume concentration is generated at the location of the concentration sensors at the maximum size of the fire and explosive hydrogen-air mixture zones in the room. Conclusions. The hydrogen concentration sensors used in the dome space of the NPP reactor building with water-­water reactors may not detect hydrogen at the top point of the dome at a sensitivity threshold of 2 % vol. In this case, in terms of the height of the convective column, hydrogen-air mixtures are formed within the fire and explosive concentration limits.

The study of heat exchange processes during induction heating of the casing string in relation to the determination of behind-the-casing flows

One of the promising methods of geophysical survey of existing wells is active thermometry. The technology for conducting research using this method includes induction heating of a section of a metal casing string, registration and analysis of temperature changes in the range of induction exposure. As a result of heat exchange with the heated section of the column, a thermal disturbance is created in the fluid flow moving inside the column or in the behind-the-casing flow channel. The analysis of non-stationary temperature in the process of induction action allows solving actual practical problems, for example, determining the presence of fluid overflows in the space behind the casing string. In this work, on the basis of numerical simulation, the features of the formation of a temperature field in the process of inductive heating of the casing string are studied in relation to the determination of behind-the-casing fluid flows. Numerical simulation was performed using the Ansys Fluent software package. Fluid movement in the well is described by the Navier–Stokes equation in the Boussinesq–Oberbeck approximation, and its temperature is calculated taking into account forced and free convection. To calculate the temperature in the inductor, casing string, rocks and cement sheath, a non-stationary heat equation is used. The application of induction heating in diagnosing behind-the-casing flows in the sump, localized below the working perforated formations, in the annular space between the casing string and the cement ring is considered. The distribution curves of the cross-sectional average temperature in the casing string body in the induction heating interval at different flow rates in the casing cross-flow channel are plotted. It is shown that with an increase in the volume flow in the overflow channel, the maximum heating of the column decreases due to more intense heat transfer to the flow in the overflow channel. It has been established that the temperature curves show a “pull” of temperature (temperature disturbances in the body of the casing string propagate in the direction of flow in the overflow channel), the value of which increases with the flow rate in the overflow channel. It is shown that the temperature “pull” (the distance over which the thermal disturbance propagates) during upward flow exceeds the “pull” during downward flow, which is due to the influence of natural thermal convection in the fluid inside the casing string. On the example of the modeling conditions adopted in the work, it was found that up and down flows of more than 0.5 m3/day can be reliably determined from temperature measurements during the induction effect.

High-frequency homogenization of nonstationary periodic equations

ABSTRACT In L 2 ( R ) , we consider an elliptic differential operator A ϵ , ϵ > 0 , of the form A ϵ = − d d x g ( x / ϵ ) d d x + ϵ − 2 V ( x / ϵ ) with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian A ϵ and for the hyperbolic equation with the operator A ϵ , analogs of homogenization problems, related to the edges of the spectral bands of the operator A ϵ , are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 ( R ) -norm for small ε are obtained.

Study of the model of the phase transition envelope taking into account the process of thermal storage under natural draft and by air injection

The study proposed mathematical modeling of the enclosing structure of the building, which includes an energy-active panel accumulating solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear heat conduction equation describing the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel, taking into account the process of heat accumulation under natural and forced traction by air injection. In comparison of the calculation results, it was found. that the efficiency of the use of forced convection relative to natural was 33–44% in the question of the selection of useful heat, depending on the time (second or first day), and the value of heat accumulation showed almost the same results. Taking into account the research carried out by the authors, it is shown by calculation that enclosing structures in which a heat-accumulating material with a phase transition is used to accumulate solar energy are more energy efficient than simple heat accumulation in the carrier layer. The introduction of highly conductive inclusions into the energy-active panel and the use of forced convection instead of natural traction during heat extraction also significantly increase the energy efficiency of the structure, which can subsequently be used as an option for external fencing under appropriate climatic conditions.

Second-order partitioned method and adaptive time step algorithms for the nonstationary Stokes-Darcy equations

Second-order partitioned method and adaptive time step algorithms for the nonstationary Stokes-Darcy equations

A fully-discrete virtual element method for the nonstationary Boussinesq equations in stream-function form

In the present work we propose and analyze a fully-coupled virtual element method of high order for solving the two dimensional nonstationary Boussinesq system in terms of the stream-function and temperature fields. The discretization for the spatial variables is based on the coupling C1- and C0-conforming virtual element approaches, while a backward Euler scheme is employed for the temporal variable. Well-posedness and unconditional stability of the fully-discrete problem are provided. Moreover, error estimates in H2- and H1-norms are derived for the stream-function and temperature, respectively. Finally, a set of benchmark tests are reported to confirm the theoretical error bounds and illustrate the behavior of the fully-discrete scheme.

Simulation of Low-Temperature Oxidation and Combustion of N-Dodecane Droplets under Microgravity Conditions

Fires are considered among the most dangerous accidents on manned spacecraft. That is why several programs of combustion experiments were implemented at the International Space Station (ISS) since 2008. In the experiments with n-heptane and n-dodecane droplet combustion, a new phenomenon was discovered, namely, the phenomenon of the radiative extinction of a burning droplet with subsequent multiple flashes of flame. In this paper, n-dodecane droplet ignition, combustion, radiative extinction, and subsequent low-temperature oxidation with multiple flashes of cool, blue, and hot flames under microgravity conditions are studied computationally. The mathematical model takes into account multiple elementary chemical reactions in the vicinity of a droplet in combination with heat and mass transfer in liquid and gas, heat release, convection, soot formation, and heat removal by radiation. The model is based on the non-stationary one-dimensional differential equations of the conservation of mass and energy in liquid and gas phases with variable thermophysical properties within the multicomponent diffusion concept in the gas phase. Calculations confirm the important role of the soot shell formed around the droplet and low-temperature reactions in the phenomenon of droplet radiative extinction with multiple flame flashes in the space experiment at the ISS. Calculations reveal the decisive role of the blue flame, arising due to the decomposition of hydrogen peroxide, in the multiple flame flashes. Calculations with forced ignition of the droplet reveal the effect of the ignition procedure on droplet evolution in terms of the timing and the number of cool, blue, and hot flame flashes, as well as in terms of the combustion rate constant of the droplet. Calculations with droplet self-ignition reveal the possible existence of new modes of low-temperature oxidation of droplets with the main reaction zone located very close to the droplet surface and with only partial conversion of fuel vapor in it.

Unrestricted Douglas-Rachford algorithms for solving convex feasibility problems in Hilbert space

In this work we focus on the convex feasibility problem (CFP) in Hilbert space. A specific method in this area that has gained a lot of interest in recent years is the Douglas-Rachford (DR) algorithm. This algorithm was originally introduced in 1956 for solving stationary and non-stationary heat equations. Then in 1979, Lions and Mercier adjusted and extended the algorithm with the aim of solving CFPs and even more general problems, such as finding zeros of the sum of two maximally monotone operators. Many developments which implement various concepts concerning this algorithm have occurred during the last decade. We introduce an unrestricted DR algorithm, which provides a general framework for such concepts. Using unrestricted products of a finite number of strongly nonexpansive operators, we apply this framework to provide new iterative methods, where, inter alia, such operators may be interlaced between the operators used in the scheme of our unrestricted DR algorithm.

Calculation of river deformation under conditions of regulated flow of Amu-Darya

In this paper, the authors present the deformation equation in the form of a system of nonlinear non-stationary differential equations. They are presented in partial products describing the deformations occurring in the river. This paper presents a mathematical model and an algorithm for calculating channel deformations in the inflowless section of channel processes. The system of non-stationary differential equations has been provided in the work. The presentation concerns the construction of hydroelectric power plants on the Amu-Darya river.

Global existence of strong solutions to the kinetic Cucker-Smale model coupled with the two dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>In this paper, we investigate existence of global-in-time strong solutions to the Cauchy problem of the kinetic Cucker–Smale model coupled with the incompressible Navier–Stokes equations in the two dimensional space. By introducing a weighted Sobolev space and using the maximal regularity estimate on the linear non-stationary Stokes equations, we present a complete analysis on existence of global-in-time strong solutions to the coupled model, without any smallness assumptions on initial data.</p>

Adaptive step size controllers based on Runge-Kutta and linear-neighbor methods for solving the non-stationary heat conduction equation

<abstract><p>We systematically test families of explicit adaptive step size controllers for solving the diffusion or heat equation. After discretizing the space variables as in the conventional method of lines, we are left with a system of ordinary differential equations (ODEs). Different methods for estimating the local error and techniques for changing the step size when solving a system of ODEs were suggested previously by researchers. In this paper, those local error estimators and techniques are used to generate different types of adaptive step size controllers. Those controllers are applied to a system of ODEs resulting from discretizing diffusion equations. The performances of the controllers were compared in the cases of three different experiments. The first and the second system are heat conduction in homogeneous and inhomogeneous media, while the third one contains a moving heat source that can correspond to a welding process.</p></abstract>

Локальное лазерное окисление фотонных кристаллов на основе пористого кремния

The process of thermal oxidation of porous silicon photonic crystals by laser treatment has been studied experimentally and theoretically. The temperature of laser heating was calculated by solving of the non-stationary thermal conductivity equation. The degree of silicon oxidation was estimated experimentally by measuring the spectral shift of the reflection curve of photonic crystals. We demonstrate the controllable shift of the photonic band gap of porous silicon photonic crystals made by means of laser irradiation.

Quantization of the Kinetic Energy of Deterministic Chaos

In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic scalar dynamic structures, and the theoretical and experimental deterministic vector dynamic structures have been developed to compute the exact solution for deterministic chaos of the exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. To explore properties of the kinetic energy, rectangular, diagonal, and triangular summations of a matrix of the kinetic energy and general terms of various sums have been used in the current paper to develop quantization of the kinetic energy of deterministic chaos. Nested structures of a cumulative energy pulson, an energy pulson of propagation, an internal energy oscillon, a diagonal energy oscillon, and an external energy oscillon have been established. In turn, the energy pulsons and oscillons include group pulsons of propagation, internal group oscillons, diagonal group oscillons, and external group oscillons. Sequentially, the group pulsons and oscillons contain wave pulsons of propagation, internal wave oscillons, diagonal wave oscillons, and external wave oscillons. Consecutively, the wave pulsons and oscillons are composed of elementary pulsons of propagation, internal elementary oscillons, diagonal elementary oscillons, and external elementary oscillons. Topology, periodicity, and integral properties of the exponential pulsons and oscillons have been studied using the novel method of the inhomogeneous Fourier expansions via eigenfunctions in coordinates and time. Symbolic computations of the exact expansions have been performed using the experimental and theoretical programming in Maple. Results of the symbolic computations have been justified by probe visualizations.

Numerical study of flow around flat plate using higher-order accuracy scheme

Analytical methods exist to solve the problems of hydromechanics and heat transfer, but it is not possible to obtain the solution to some inhomogeneous and nonlinear problems of hydromechanics and heat transfer by analytical methods. The solution to such problems is carried out using numerical methods. Currently, there are many textbooks and monographs on numerical methods for solving problems of hydromechanics, thermal conductivity, heat and mass transfer, and others. The article presents the results of a numerical study of the flow structure in the flow around a flat plate. The calculations are based on the numerical solution of a system of nonstationary equations using a two-fluid turbulence model. For the numerical solution of these problems, schemes of the second and fourth order of accuracy were applied. The control volume method was used for the difference approximation of the initial equations, and the relationship between velocities and pressure was found using the SIMPLE procedure. To confirm the correctness of the numerical results, comparisons were made with each other and experimental data.

Difference schemes of the finite element method of increased accuracy for solving nonstationary equations

Difference schemes of the finite element method of increased accuracy for solving nonstationary equations