Conjugate Problem Research Articles

Piezoelectric energy harvests by the usage of a laminar pulsating flow circulating in a rectangular channel

Abstract This work develops a theoretical study of a piezoelectric energy harvester, perturbed through a fully developed laminar flow with an oscillating pressure gradient. Considering a fully developed hydrodynamic flow, the electric energy generated in the piezoelectric element is due only to shear stresses yielded at the inner surfaces of the channel. In this manner, a fraction of the viscous forces is converted into unitary deformations at the piezoelectric, and the other fraction is transformed to induced electric piezoelectric energy. Using dimensionless analysis, the formulation of resulting dimensionless governing equations for the fluid and the corresponding deformation and electric potential fields for the piezoelectric material constitute a conjugate problem, solved by considering harmonic solutions. Although the dimensionless power output is a multi-parametric function, due to the large number of dimensionless parameters involved, we find that the main behavior of the electrical power depends very sensitively on two fundamental dimensionless parameters: the Womersley number, α, associated with the oscillating flow and a parameter that measures the physical importance of the electrical energy produced by the action of the velocity field. For the first case, it is seen that the power increases if the Womersley number is decreased while for the second case, the inverse behavior is predicted. Therefore, there is a clear operation of the physical system for which better conditions can be reached by selecting and varying appropriately the assumed values of different dimensionless parameters.

On a case of -linear conjugation problem on the unit circle

The R -linear conjugation problem on the unit circle with rational coefficients is solved. The problem is equivalently reduced to the vector–matrix C -linear conjugation problem for which the canonical matrix is effectively constructed. The applied method is based on the two-fold application of G.N.Chebotarev's approach to the factorization of rational matrix functions.

Transitions between equilibrium and nonequilibrium phenomena in the description of crystal growth

The close intertwining of equilibrium and nonequilibrium thermodynamic representations and transitions between the two limiting principles of thermodynamics: the second beginning and the principle of least coercion (minimum entropy production in the stationary regime) constitute the main content of phenomenological theories of crystal growth. The difference of basic postulates of two sections of thermodynamics forces to discuss problems of reversibility and irreversibility of time, scales of observed phenomena and rules of conjugation of thermodynamic forces and flows in theories of crystal growth. A variant of the solution of some conjugation problems is shown on the example of the fluctuation model of dislocation crystal growth, which is based on the stationary isothermal process of thermodynamic free energy fluctuations. In the case of the limiting mode of adsorption of impurities on the crystal face according to the Langmuir model, the free energy fluctuations possessing the absence of the memory effect allow us to identify three chemical potentials of building particles that determine the corresponding values of solution supersaturations realized at different scale levels at the growing crystal face containing a helical dislocation. The supersaturations control quasi-equilibrium and nonequilibrium thermodynamic processes that constitute a single dislocation mechanism of crystal growth.

Mass and Heat Exchange in Rotating Compressor Cavities With Variable Cob Separation

Abstract Next generation aeroengines will operate at ever-increasing pressure ratios with smaller cores, where the control of blade-tip clearances across the flight cycle is an emerging design challenge. Such clearances are affected by the thermal expansion of the compressor disks that hold the blades, where acute thermal stresses govern operating life. The cavities formed by corotating disks feature a heated shroud at high radius and cooler cobs at low radius. A three-dimensional, unsteady and unstable flow structure is induced by destabilizing buoyancy forces. The radial distribution of disk temperature is driven by a conjugate heat transfer at Grashof numbers of order 1013. Such flows are further influenced by the heat and mass exchange with an axial throughflow of cooling air at low radius, where the interaction depends on the Rossby number and separation of the disk cobs. This paper is the first to study the effect of cob separation ratio on mass and heat exchange for compressor cavities. A model is developed to predict the cavity-throughflow interaction, and disk and fluid-core temperatures. The judicious use of a physics-based methodology provides reliable, reduced-order solutions to the complex conjugate problem, thereby making it appropriate for practical engine thermo-mechanical design. The model is validated by detailed experimental measurements using the Bath Compressor Cavity Rig, where variable disk cob spacings were investigated over a range of engine-representative conditions. The unsteady pressure measurements collected in the frame of reference of the rotating disks reveal new insight into the fundamentally aperiodic nature of the flow structure. This new understanding of heat transfer informs an expedient reduced-order model and enables more efficient design of future high pressure-ratio aeroengines.

Numerical Solution of the Cauchy Problem for the Helmholtz Equation Using Nesterov’s Accelerated Method

In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented.

R-Linear Conjugation Problem on the Unit Circle in the Parabolic Case

R-Linear Conjugation Problem on the Unit Circle in the Parabolic Case

Hypersingular integro-differential equation with quadratic functions in coefficients

A new linear integro-differential equation of order n ≥ 3, given on a closed curve located in the complex plane, is investigated. Integrals in the equation are understood in the sense of the finite part according to Hadamard. A characteristic feature of the equation is the presence of quadratic functions of a special kind in its coefficients. The equation is reduced first to the boundary value problem of linear conjugation for analytical functions. In the case of its solvability, two linear differential equations should be further solved in the domains of the complex plane with some additional conditions for the solution. All conditions for the solvability of the original equation are explicitly specified. When they are executed, the desired solution is constructed in a closed form. An example is given.

Problem of Linear Conjugation with Multipoint Conditions in the Case of Multiple Nodes for Higher-Order Strictly Hyperbolic Homogeneous Equations

Problem of Linear Conjugation with Multipoint Conditions in the Case of Multiple Nodes for Higher-Order Strictly Hyperbolic Homogeneous Equations

The method of determining optimal control of the thermoelastic state of piece-homogeneous body using a stationary temperature field

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

Conjugate mass transfer from a spherical moving droplet: Direct numerical simulations of enhancement by a chemical reaction

Chemical reactions can have a significant impact on the overall rate of mass transfer in fluid–fluid applications (such as gas absorption in a liquid or liquid–liquid extraction). However, as for non reacting systems, the rate at which a solute is exchanged between the two fluids is difficult to predict accurately. Yet from a theoretical point of view, most studies address uniform diffusion transport or creeping flow, i.e. conditions enabling the derivation of analytical or simple numerical solutions of the solute transport equation, therefore neglecting the effect of convection in the case of coupled mass transfer problems. On the other hand, experimental investigations available in the literature provided correlations corresponding to flow conditions, shape and contamination of the interface which are extremely difficult to control precisely.Direct numerical simulation (DNS) is therefore a powerful tool to investigate the coupling of hydrodynamics with transfer and reaction. It was recently applied to model the internal and external resistances (or Sherwood numbers, Shi and She) prevailing during the transfer of a non reacting solute, in the general case of conjugate mass transfer from a moving spherical droplet (Godé et al., 2023; 2024). This study is complemented by considering a first order reaction that consumes the transferred solute in the continuous phase. The results indicate that the double film model, based on these last correlations for Shi and She, with She corrected by an enhancement factor allows obtaining accurate predictions of the overall mass transfer coefficient in the case of a reactive conjugate problem, thus extending the work of Juncu (2002) to flows at moderate or even high Re, as long as the droplet remains spherical and the flow axisymmetric.

Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection

The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.

МЕТОДИЧЕСКОЕ ПОЛЕ ЦЕЛЕПОЛАГАНИЯ В ПРОГРАММНО-ЦЕЛЕВОМ УПРАВЛЕНИИ

Purpose: the purpose of the study is to analyze the scientific and practical toolkit of program-target management in terms of available target-setting practices to improve the quality and performance of the procedures of setting and managing goals in the framework of solving the problems of conjugation of state and regional management. Methods: research procedures were carried out with the use of general theoretical and formal-logical methods of research, which allowed us to describe, compare, generalize, identify positive characteristics and determine methodological difficulties of the techniques and algorithms considered in scientific sources, associated with the setting, coordination and management of goals in the implementation of program-target management, as well as to form conclusions on their application in changing conditions. Results of the work. During the study, an understanding of the “life cycle” of goal setting was formed and its structure was determined in the form of interrelated tasks. The methodological support of existing practices for solving each goal-setting problem is considered, the positive characteristics of methods for solving them are identified and the problems accompanying them are identified. For the task of constructing goal statements based on a generalization of discovered patterns and identified problems of goal setting identified in the process of studying the experience of implementing this task within the framework of program-target management, requirements for goal statements were formed and grouped and a content analysis was carried out to assess the possibility of implementing these requirements in the considered requirements. Conclusions. The results of the study make it possible to expand the scientific and methodological field of goal-setting procedures, increase the validity, effectiveness and quality of decisions made within the framework of the implementation of program-target management, especially in terms of the choice of approaches and methods for coordinating projects and programs that implement the target settings of federal and state bodies at the regional level management.

АППРОКСИМАЦИЯ СОПРЯЖЕНИЯ КРИВЫХ БЕЗЬЕ С СОХРАНЕНИЕМ ПОРЯДКА ГЛАДКОСТИ И ДОПОЛНИТЕЛЬНЫМИ ОГРАНИЧЕНИЯМИ

Bezier curves are a mandatory component of the geometric core of modern computer-aided design (CAD). The article proposes a mathematical approach that makes it possible to approximate the conjugation (connection) of Bezier curves of arbitrary degree, so that at the conjugation point the conditions of smoothness (continuity) are satisfied up to the order of an equal degree of the given Bezier curves. This approach helps to represent conjugate curves of a single Bezier curve with the degree equal to the degrees of the given curves. Additional restrictions can be imposed on the conjugate curves and the approximating curve in the form of a complete coincidence with one of the given curves, or the passage of the approximating curve through a given point and the equality of the derivatives to the given values at this point. To solve these problems, we introduced two difference metrics between the given curves and the approximating curve, and formulated optimization problems with constraints in the form of equalities. We applied the method of Lagrange multipliers which solves the corresponding system of linear algebraic equations. To represent Bezier curves, it is proposed to use the basic functions of B-splines, which allows you to use the software functions included in the geometric core of modern CAD systems. This greatly simplifies the derivation of all degree derivatives for curves and, without significant changes in the future, will make it possible to extend the results to conjugation problems of B-splines. The paper provides examples of approximations using various metrics and their limitations.

Analysis of Cavity Disk Heat Transfer by Solving Inverse Heat Transfer Problem

Abstract The flow and heat transfer within compressor rotor cavities of aero-engines is a conjugate problem. The operating conditions buoyancy forces, caused by radial temperature difference between the cold throughflow and the hotter shroud, can influence the amount of entrained air significantly. By this, the heat transfer depends on the radial temperature gradient of the cavity walls, and in turn, the disk temperatures are dependent on the heat transfer. In this article, disk Nusselt numbers are calculated in reference to the air inlet temperature and in comparison to a modeled local air temperature inside the cavity. The local disk heat flux is determined from measured steady-state surface temperatures by solving the inverse heat transfer problem in an iterative procedure. The conduction equation is solved on a 2D mesh using a validated finite element approach, and the heat flux confidence intervals are calculated with a stratified Monte Carlo approach. An estimate for the amount of air entering into the cavity is calculated by a simplified heat balance. The major influences on the Nusselt number were found to be the mass flowrate entering the cavity and the density of the fluid inside the cavity.

Ob asimptotike spektra integral'nogo operatora s logarifmicheskim yadrom spetsial'nogo vida

We study the asymptotic behavior of the spectrum of an integral operator similarto an integral operator with a logarithmic kernel depending on the sum of arguments. Bya simple change of variables, the corresponding equation is reduced to an integral equation ofconvolution type defined on a finite interval (as is well known, such equations in the general casecannot be solved by quadratures). Next, using the Fourier transform, the equation is reduced toa conjugation problem for analytic functions and then to an infinite system of linear algebraicequations, the isolation of the main terms in which allows deriving a relation that determinesthe spectrum of the original problem.

Influence of Nonisolated Singularities in a Lower-Order Coefficient of the Bitsadze Equation on the Statement of Boundary Value Problems

We study the influence of nonisolated singularities (i.e., singularities along closed lines lying inside the domain) in the lower-order coefficients of the Bitsadze equation on the statement of boundary value problems. We discover that the conditions on the boundary of the domain in the Riemann–Hilbert problem are not sufficient for the solution; therefore, we consider a problem that combines elements of the Riemann–Hilbert problem on the boundary of the domain and the linear conjugation problem on the circles that support the singularities of the coefficients lying inside the domain. Using an appropriate refinement of Kellogg’s theorem on the conformal mapping of this domain onto a disk, we study the solvability of the problem

Результаты совместной интерпретации гравитационного и теплового полей Уральского региона

A method for solving conjugation problems for the Poisson equation is implemented, which makes it possible to interpret potential fields (gravitational and stationary thermal fields) based on unified grid algorithms. A numerical algorithm has been developed for recalculating the mantle component of the heat flow from the level of the earth’s surface to the “crust–mantle” boundary through a layered medium inhomogeneous in thermal conductivity. Paleoclimate corrections were introduced into the measured values of the temperature gradient and heat flux for the northern territories. In the axial part of the Ural geosyncline, it was possible to exclude the negative depression of the mantle component of the heat flow, obtained from the results of geothermal modeling; taking into account the Pleistocene-Holocene warming of the preceding interglacial paleoclimate cycle leads to positive values of the heat flow recalculated to the “crust–mantle” boundary. It is shown that the use of heat flow data can significantly increase the geological information content of gravity modeling.

Conjugate Modeling of a Closed Co-Rotating Compressor Cavity

Abstract Robust methods to predict heat transfer are vital to accurately control the blade-tip clearance in compressors and the radial growth of the disks to which these blades are attached. Fundamentally, the flow in the cavity between the co-rotating disks is a conjugate problem: the temperature gradient across this cavity drives large-scale buoyant structures in the core that rotate asynchronously to the disks, which in turn governs the heat transfer and temperature distributions in the disks. The practical engine designer requires expedient computational methods and low-order modeling. A conjugate heat transfer (CHT) methodology that can be used as a predictive tool is introduced here. Most simulations for rotating cavities only consider the fluid domain in isolation and typically require known disk temperature distributions as the boundary condition for the solution. This paper presents a novel coupling strategy for the conjugate problem, where unsteady Reynolds averaged Navier–Stokes (URANS) simulations for the fluid are combined with a series of steady simulations for the solid domain in an iterative approach. This strategy overcomes the limitations due to the difference in thermal inertia between fluid and solid; the method retains the unsteady flow features but allows a prediction of the disk temperature distributions, rather than using them as a boundary condition. This approach has been validated on the fundamental flow configuration of a closed co-rotating cavity. Metal temperatures and heat transfer correlations predicted by the simulation are compared to those measured experimentally for a range of engine-relevant conditions.

Теоретический анализ задач магнитной маскировки с использованием эллиптических метаматериалов

The conjugation problem and control problems for the 3D model of magnetostatics are considered. These problems are related to the design of three-dimensional magnetic cloaking shells. An elliptical metamaterial is chosen as a cloaking medium that fills a region which is topologically equivalent to a spherical layer. The solvability of boundary and control problems is proved, an optimality system is derived that describes the necessary conditions for an extremum, some properties of optimal solutions are established.

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.