Singular integral equation for an electric dipole taking into account the finite metal conductivity from which it is made

A singular integral equation for an electric dipole has been obtained, which makes it possible to take into account the finite conductivity of the metal from which it is made. The derivation of the singular integral equation is based on the application of the Greens function for free space, written in a cylindrical coordinate system, taking into account the absence of the dependence of the field on the azimuthal coordinate, on a point source located on the surface of an electric dipole. Methods for its solution are proposed. In contrast to the well-known mathematical models of an electric dipole, built in the approximation of an ideal conductor, the use of the singular integral equation obtained in this work makes it possible to take into account heat losses and calculate the efficiency.

SIMPLIFIED MATHEMATICAL MODEL OF A MAGNETIC SYSTEM WITH A CIRCULAR MAGNETIC CONDUCTOR

A simplified design of a magnetic system with a circular magnetic core is presented and its mathematical model is developed to determine the magnetic flux. Transition from a cylindrical coordinate system to a rectangular coordinate system.

DEVELOPMENT OF A MATHEMATICAL MODEL FOR RESEARCHING THE PROCESS OF REDUCTION OF TUNGSTEN HEXAFLUORIDE BY GASEOUS HYDROGEN

В работе рассмотрена математическая модель, описывающая движение течения стационарной, ламинарной, вязкой, несжимаемой смеси газа в трехмерном осесимметричном канале. Математическая модель, описывающая этот процесс, состоит из уравнений Навье – Стокса, уравнения неразрывности и массообмена, которые записаны в безразмерной форме с учетом осесимметричности в цилиндрической системе координат. Решение уравнений осуществляется в физических переменных «скорость – давление» на разнесенной разностной сетке. Показано влияние характерных параметров на распределение концентрации смеси газа гексафторида вольфрама и водорода в канале. Полученная математическая модель позволяет проводить численные исследования по выбору оптимальных условий осуществления процесса восстановления гексафторида вольфрама водородом. The paper considers a mathematical model describing the flow motion of a stationary, laminar, viscous, incompressible gas mixture in a three-dimensional axisymmetric channel. The mathematical model describing this process consists of the Navier-Stokes equations, the continuity and mass transfer equations, which are written in dimensionless form taking into account axisymmetry in a cylindrical coordinate system. The equations are solved in the physical variables "velocity - pressure" on a spaced difference grid. The influence of characteristic parameters on the concentration distribution of a mixture of tungsten hexafluoride gas and hydrogen in the channel is shown. The obtained mathematical model makes it possible to conduct numerical studies on the choice of optimal conditions for the process of reduction of tungsten hexafluoride with hydrogen.

Spatially Highly Resolved Solar-wind-induced Magnetic Field on Venus

The current work investigates the Venusian solar-wind-induced magnetosphere at a high spatial resolution using all Venus Express (VEX) magnetic observations through an unbiased statistical method. We first evaluate the predictability of the interplanetary magnetic field (IMF) during VEX’s Venusian magnetospheric transits and then map the induced field in a cylindrical coordinate system under different IMF conditions. Our mapping resolves structures on various scales, ranging from the ionopause to the classical IMF draping. We also resolve two recently reported structures, a low-ionosphere magnetization over the terminator, and a global “looping” structure in the near magnetotail. In contrast to the reported IMF-independent cylindrical magnetic field of both structures, our results illustrate their IMF dependence. In both structures, the cylindrical magnetic component is more intense in the hemisphere with an upward solar wind electric field (E SW) than in the opposite hemisphere. Under downward E SW, the looping structure even breaks, which is attributable to an additional draped magnetic field structure wrapping toward −E SW. In addition, our results suggest that these two structures are spatially separate. The low-ionosphere magnetization occurs in a very narrow region, at about 88°–95° solar zenith angle and 185–210 km altitude. A least-squares fit reveals that this structure is attributable to an antisunward line current with 191.1 A intensity at 179 ± 10 km altitude, developed potentially in a Cowling channel.

Analytical model of a T-core coil above a multi-layer conductor with hidden hole using the TREE method for nondestructive evaluation

Purpose In eddy current nondestructive testing, a probe with a ferrite core such as an E-core coil is usually used to detect and locate defects such as cracks and corrosion in conductive material. However, the E-core coil has some disadvantages, such as large volume and difficulty in the process of winding the coils. This paper aims to present a novel T-core probe and its analytical model used for evaluating hidden holes in a multi- layer conductor. Design/methodology/approach By using a cylindrical coordinate system, the solution domain is truncated in the radial direction. The magnetic vector potential of each region excited by a filamentary coil is derived, and the expansion coefficients of the solutions are obtained by matching the boundary and interface conditions between the regions. By using the truncated region eigenfunction expansion method, the final expression in closed form for the impedance of the multi-turn coil is worked out, and the impedance calculation is performed in Mathematica. For frequencies ranging from 100 Hz to 100 kHz, both the impedance changes of the T-core coil above the multi-layer conductor without a hidden hole and in the absence of the layered conductor were calculated, and the influence of a hidden hole in the multi-layer conducting structure on the impedance change was investigated. Findings The correctness of the analytical model of the T-core coil was verified by the finite element method and experiments. The proposed T-core coil has higher sensitivity than an air-core coil, and similar sensitivity and smaller size than an E-core coil. Originality/value A new T-core coil probe and its accurate theoretical model for defect evaluation of conductor were presented; probe and analytical model can be used in probe design, detection process simulation or can be directly used in defect evaluation of multi-layer conductor.

Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolution

Swirling laminar axisymmetric flows of viscous incompressible fluids in a potential field of body forces are considered. The study of flows is carried out in a cylindrical coordinate system. In the flows, the regions in which the axial derivative of the circumferential velocity cannot take on zero value in some open neighborhood (essentially swirling flows) and the regions in which this derivative is equal to zero (the region with layered swirl) are considered separately. It is shown that a well-known method (the method of viscous vortex domains) developed for non-swirling flows can be used for regions with layered swirling. For substantially swirling flows, a formula is obtained for calculating the radial-axial velocity of an imaginary fluid through the circumferential vorticity component, the circumferential circulation of a real fluid, and the partial derivatives of these functions. The particles of this imaginary fluid “transfer” vortex tubes of the radial-axial vorticity component while maintaining the intensity of these tubes, and also “transfer” the circumferential circulation and the product of the circular vorticity component by some function of the distance to the axis of symmetry. A non-integral method for reconstructing the velocity field from the vorticity field is proposed. It is reduced to solving a system of linear algebraic equations in two variables. The obtained result is proposed to be used to extend the method of viscous vortex domains to swirling axisymmetric flows.

Stability Analysis for Complex Rotational Flow

Based on the earlier developed mathematical model of the complex flow due to the double rotations in two perpendicular directions, the stability analysis is performed in the paper. The Navier-Stokes equations are derived in the coordinate system rotating around the two perpendicular different axes, the vertical one of them is arranged on some distance from the other axis of rotation, the horizontal axis is directed along the tangential line to the circle of the vertical rotation. The two centrifugal and Coriolis forces create the unique features in high oscillating flow, with localities of the stretched liquid, due to their action varying by the circumferential cylindrical coordinate in the channel flow. Stability analysis for the complex rotational flow under double rotations creating strongly varying mass forces and stretching of the liquid is considered at first

Bending of an elastoplastic circular sandwich plate on an elastic foundation in a temperature field

Today, the development of the general theory of quasi-static deformation of three-layer structural elements, including plates, is not yet complete and is being intensively studied. Mathematical models of deformation under complex thermo-force and thermo-irradiation loads are created. The problems of strength, stability, and dynamic behaviour are considered. In strength calculations of three-layer structural elements, it is necessary to take kinematic hypotheses for each layer separately, which complicates the mathematical side of the problem but leads to significant refinement of the stress-strain state. The reaction of an elastic foundation is described by the Winkler model. The use of variational methods allows one to obtain a refined system of three differential equations of equilibrium in internal forces. The thermo-force bending of an elastoplastic circular sandwich plate with a light core connected to an elastic foundation is considered. The polyline normal hypotheses are used to describe the kinematics of a plate package that is not symmetric in thickness. In thin base layers, the Kirchhoff-Love hypotheses are accepted. In a light relatively thick core, the Timoshenko hypothesis is true, while the normal remains rectilinear, but rotates at some additional angle, the radial displacements change linearly in thickness. The differential equations of equilibrium are obtained using the Lagrange variation method. The statement of the boundary value problem in displacements is given in a cylindrical coordinate system. Numerical results for circular metal-polymer sandwich plates are presented.