Radial Coordinate Research Articles

Influence of logarithmic radial inhomogeneity of the elastic modulus on the stress-strain state of a hollow cylinder

The paper is devoted to the formulation and solution of the boundary value problem for a hollow circular cylinder whose properties continuously depend on the radial coordinate. It is shown that the mathematical model of such a problem is a homogeneous second-order differential equation with variable coefficients and two boundary conditions. In the case where the Poisson's ratio is constant and the Young's modulus logarithmically depends on the radial coordinate, the resulting equation can be reduced to the Whittaker equation, the solution of which is a linear combination of confluent hypergeometric functions. Based on this, an analytical solution of the boundary value problem is constructed in this particular case. Functions describing displacements, strains, and stresses occurring in the cylinder are found.

Математичні моделі визначення температурних полів у теплоактивних елементах цифрових пристроїв з локальним внутрішнім нагріванням та із урахуванням термочутливості

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media subjected to internal local heat load, have been developed. In the case of a nonlinear boundary-value problem, the Kirchhoff transformation is applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions are linearized, and as a result, a linearized second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions is obtained. For the final linearization of the partially linearized boundary conditions, the approximation of the temperature by the radial spatial coordinate on the boundary surface of the thermosensitive medium was performed by a piecewise constant function, as a result of which the boundary value problem was obtained completely linearized. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters was performed. The influence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal heat load, software tools were developed, using which a geometric image of the temperature distribution depending on the spatial coordinates was made. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active environments with internal heating testify to their adequacy to a real physical process. They make it possible to analyze such environments for their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements, but also the entire structure.

Black hole solutions in the quadratic Weyl conformal geometric theory of gravity

We consider numerical black hole solutions in the Weyl conformal geometry and its associated conformally invariant Weyl quadratic gravity. In this model, Einstein gravity (with a positive cosmological constant) is recovered in the spontaneously broken phase of Weyl gravity after the Weyl gauge field (omega _{mu }) becomes massive through a Stueckelberg mechanism and it decouples. As a first step in our investigations, we write down the conformally invariant gravitational action, containing a scalar degree of freedom and the Weyl vector. The field equations are derived from the variational principle in the absence of matter. By adopting a static spherically symmetric geometry, the vacuum field equations for the gravitational, scalar, and Weyl fields are obtained. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, we obtain their solutions numerically. We detect the formation of a black hole from the presence of a Killing horizon for the timelike Killing vector in the metric tensor components, indicating the existence of the singularity in the metric. Several models corresponding to different functional forms of the Weyl vector are considered. An exact black hole model corresponding to a Weyl vector having only a radial spacelike component is also obtained. The thermodynamic properties of the Weyl geometric type black holes (horizon temperature, specific heat, entropy, and evaporation time due to Hawking luminosity) are also analyzed in detail.

Solution of Unsteady Flow Equations for Nanoscale Heat and Mass Transfer in Diverse Applications

Abstract A new innovative stable time-dependent compressible flow solution over the order of nanoseconds is provided here for wide-ranging critically important challenging applications. Specifically, a solution of the highly complex unsteady high speed oscillating compressible flow field inside a cylindrical tube, closed at one end with a piston oscillating at very high resonant frequency at the other end, has been obtained numerically, assuming one-dimensional, viscous, and heat-conducting flow, by solving the appropriate fluid dynamic and energy equations. An iterative implicit finite difference scheme is employed to obtain the solution. The scheme permits arbitrary boundary conditions at the piston and the end wall and allows assumptions for transport properties. In successfully predicting the time-dependent results/data, an innovative simple but stable solution of unsteady fluid dynamic and energy equations is provided here for wide ranging research, design, development, analysis, and industrial applications in solving a variety of complex fluid flow heat transfer problems. The method is directly applicable to pulsed or pulsating flow and wave motion thermal energy transport, fluid-structure interaction heat transfer enhancement, nanoscale heat and mass transfer, diverse range of advanced fluidics, biofluidics/bio-engineering, and fluidic pyrotechnic initiation devices. It can further be easily extended to cover muzzle blasts and high energy nuclear explosion blast wave propagations in one-dimensional and/or radial spherical coordinates with or without including energy generation/addition terms. No other solution exists for such applications.

The fuzzy BTZ

We introduce a model of a noncommutative BTZ black hole, obtained by quantisation of Poincaré coordinates together with a moving frame. The fuzzy BTZ black hole carries a covariant differential calculus, satisfies Einstein’s equations and has a constant negative curvature. The construction passes through a larger space, the fuzzy anti-de Sitter, and implements discrete BTZ identifications as conjugations by a unitary operator. We derive the spectrum of the suitably regularised radial coordinate: it consists of a continuum of scattering states outside the horizon r+ and an infinite discrete set of bound states inside.

Аналіз теплового режиму електропровідного трубчастого елемента з тонким зовнішнім електропровідним покриттям за дії неусталеного електромагнітного поля

An electrically conductive tubular element with a thin external electri-cally conductive coating underthe action of an unsteady electromagnetic field of the radio frequency range is considered. To analyze the thermal re-gime, the initial-boundary problem of electrodynamics for a two-layer conductive non-ferromagnetic hollow cylinder under the action of an ex-ternal non-stationary electromagnetic field is formulated. The electromag-netic field is given by the values of the axial component of the magnetic field intensity vector on the inner and outer surfaces of the cylinder. To construct the general solution of the formulated initial-boundary value problem, a quadratic approximation of the key function (the axial compo-nent of the magnetic field intensity vector) along the radial coordinate in each constituent layer (base and cover) of the hollow cylinder was used. As a result, the original initial-boundary value problem for the key function is reduced to the Cauchy problem in terms of the time variable for the charac-teristics of the key function integral over the radial coordinate. The coeffi-cients of the approximation polynomials are presented through the integral characteristics of the key function and its given values on the inner and outer surfaces of the cylinder as a function of time. The solution of the Cauchy problem is obtained using the integral Laplacetransform. Expres-sions of the key function in each component layer of the cylinder are rec-orded in the form of quadratic polynomials, the coefficients of which are convolutions of functions describing the given limit values of the key func-tion on the inner and outer surfaces of the cylinder and homogeneous solu-tions of the Cauchy problem on integral characteristics. Expressions of the key function and Joule heat in the base and coating of the considered tubu-lar element under the action of an unsteady electromagnetic field in the mode with a pulsed modulating signal were obtained on the basis of gen-eral solutions for homogeneous non-stationary electromagnetic action. A computer analysis of the change in time and thickness of the constituent layers of this tubular element of the axial component of the vector of mag-netic field intensity and Joule heat was carried out.

Vortex-ring quantum droplets in a radially-periodic potential

We establish stability and characteristics of two-dimensional (2D) vortex ring-shaped quantum droplets (QDs) formed by binary Bose–Einstein condensates. The system is modeled by the Gross–Pitaevskii (GP) equation with the cubic term multiplied by a logarithmic factor (as produced by the Lee-Huang-Yang correction to the mean-field theory) and a potential which is a periodic function of the radial coordinate. Narrow vortex rings with high values of the topological charge, trapped in particular circular troughs of the radial potential, are produced. These results suggest an experimentally relevant method for the creation of vortical QDs (thus far, only zero-vorticity ones have been reported). The 2D GP equation for the narrow rings is approximately reduced to the one-dimensional form, which makes it possible to study the modulational stability of the rings against azimuthal perturbations. Full stability areas are delineated for these modes. The trapping capacity of the circular troughs is identified for the vortex rings with different winding numbers (WNs). Stable compound states in the form of mutually nested concentric multiple rings are constructed too, including ones with opposite signs of the WNs. Other robust compound states combine a modulationally stable narrow ring in one circular potential trough and an azimuthal soliton performing orbital motion in an adjacent one. The results may be used to design a device employing coexisting ring-shaped modes with different WNs for data storage.

Dual geometry of entanglement entropy via deep learning

For a given entanglement entropy of quantum field theory, we investigate how to reconstruct its dual geometry by applying the Ryu-Takayanagi formula and the deep learning method. In the holographic setup, the radial coordinate is identified with the energy scale of the dual quantum field theory. Therefore, the holographic dual geometry can describe how physical properties of a quantum field theory change along the renormalization group flow. Intriguingly, we show that the reconstructed geometry only from the entanglement entropy data can give us more information about other physical properties like thermodynamic quantities in the infrared region.

Convergence of the Fefferman-Graham expansion and complex black hole anatomy

Given a set of sources and one-point function data for a Lorentzian holographic QFT, does the Fefferman-Graham expansion converge? If it does, what sets the radius of convergence, and how much of the interior of the spacetime can be reconstructed using this expansion? As a step towards answering these questions we consider real analytic conformal field theory data, where in the absence of logarithms, the radius is set by singularities of the complex metric reached by analytically continuing the Fefferman-Graham radial coordinate. With the conformal boundary at the origin of the complex radial plane, real Lorentzian submanifolds appear as piecewise paths built from radial rays and arcs of circles centred on the origin. This allows singularities of Fefferman-Graham metric functions to be identified with gauge-invariant singularities of maximally extended black hole spacetimes, thereby clarifying the physical cause of the limited radius of convergence in such cases. We find black holes with spacelike singularities can give a radius of convergence equal to the horizon radius, however for black holes with timelike singularities the radius is smaller. We prove that a finite radius of convergence does not necessarily follow from the existence of an event horizon, a spacetime singularity, nor from caustics of the Fefferman-Graham gauge, by providing explicit examples of spacetimes with an infinite radius of convergence which contain such features.

The influence of full drifts on density shoulder formation at the midplane by numerical modeling

The density shoulder at the midplane may influence core plasma confinement during H-mode discharge, thus affecting long-pulse steady-state discharge. Drifts in the edge plasma play a remarkable role in plasma transport and the divertor operation regime, which determine density shoulder formation (DSF). In this work, the SOLPS-ITER code package is used to evaluate the influence of full drifts on DSF in poloidal and radial coordinates. An open divertor of DIII-D-like geometry with weak neutral compression is chosen for the modeling. Cases without drifts, with only E × B drifts in forward B t and with full drifts in both forward and reversed B t are simulated for comparison. It is confirmed that the high upstream density promotes DSF when the drift is not considered, which has also been observed in various investigations. When the drifts are taken into account, the divertor in/out asymmetry (or upstream ionization source) is determined by the direction of B t due to the variation of particle transport, thus the shoulder can be facilitated or suppressed. Two mechanisms of DSF with full drifts are elucidated: (1) E × B and B × ∇B drifts promote DSF at the inner midplane (IMP) by raising the ionization source (at IMP) in forward B t; (2) the drifts contribute to DSF at the outer midplane by enhancing the particle transport loss in reversed B t. In a high-recycling regime, ionization is the dominant term for DSF, while in the low-recycling regime enhanced particle transport loss plays a more important role. Comprehensively understanding the mechanisms of DSF is of great importance for the improvement of core–edge compatibility in fusion reactors.

Static-fluid black hole and wormhole in three-dimensions

We introduce the so-called static-fluid solutions in 2+1-dimensional spacetime. We assume the spacetime to be static and circular symmetric and for the perfect fluid we consider an equation of state (EoS) of the form where r is the radial coordinate. Since, unlike the 3+1-dimensions, in 2+1-dimensions there is no vacuum solution other than the flat spacetime, our approach is not exactly the same as the 3+1-dimensions such that the static-fluid is accompanied by a cosmological constant. In addition to that, at first, we present a static solution supported by a static-fluid of constant energy-momentum tensor with a general EoS. Then, in the nonconstant energy-momentum tensor case, we introduce a large class of solutions. Depending on the values of the integration constants it implies black holes, wormholes, or cosmological solutions. Some of these solutions are closed in two-space which are considered for the first time in 2+1-dimensions. In both cases i.e. the constant and nonconstant energy-momentum tensor we imposed the satisfaction of conservation of the energy-momentum i.e. .

Analytical analysis for non-homogeneous two-layer functionally graded material

Abstract In this study, the nonlinear analytical analysis of a two-layer geometry made of functionally graded materials (FGMs) is examined. FGMs can be used in various engineering applications, such as building materials in civil engineering, due to the advantages of smoothly varying properties. The equations of stresses and displacements in the radial and circumferential directions (r, θ ) have been found by extracting the governing equations and defining them in the form of power-exponential functions. In the present paper, modulus of elasticity and heat conductivity coefficient (except for Poisson’s coefficient) are assumed to be expressed by power-exponential functions in radial and circumferential coordinates. The temperature distribution is also considered as a function of radius (r) and angle (θ). The analysis is implemented based on the theory of small elastic deformations and with the assumption of a very large length in plane strain mode. To analyze the governing equations, first, the heat transfer equations are obtained, and then the Navier’s equations are derived by combining the stress–strain, strain–displacement, and stress equilibrium equations. Then, the displacement equations and stress equations are obtained by solving the Navier’s equations. A direct method is presented to solve these equations analytically.

Extremal surfaces and thin-shell wormholes

We study extremal surfaces in a traversable wormhole geometry that connects two locally ${\mathrm{AdS}}_{5}$ asymptotic regions. In the context of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, we use these to compute the holographic entanglement entropy for different configurations: First, we consider an extremal surface anchored at the boundary on a spatial 2-sphere of radius $R$. The other scenario is a slab configuration which extends in two of the boundary spacelike directions while having a finite size $L$ in the third one. We show that in both cases the divergent and the finite pieces of the holographic entanglement entropy give results consistent with the holographic picture and this is used to explore the phase transitions that the dual theory undergoes. The geometries we consider here are stable thin-shell wormholes with flat codimension-one hypersurfaces at fixed radial coordinate. They appear as electrovacuum solutions of higher-curvature gravity theories coupled to Abelian gauge fields. The presence of the thin shells produces a refraction of the extremal surfaces in the bulk, leading to the presence of cusps in the phase space diagram. Further, the traversable wormhole captures a phase transition for the subsystems made up of a union of disconnected regions in different boundaries. We discuss these and other features of the phase diagram.

On Fluxbrane Polynomials for Generalized Melvin-like Solutions Associated with Rank 5 Lie Algebras

We consider generalized Melvin-like solutions corresponding to Lie algebras of rank 5 (A5, B5, C5, D5). The solutions take place in a D-dimensional gravitational model with five Abelian two-forms and five scalar fields. They are governed by five moduli functions Hs(z) (s=1,...,5) of squared radial coordinates z=ρ2, which obey five differential master equations. The moduli functions are polynomials of powers (n1,n2,n3,n4,n5)=(5,8,9,8,5),(10,18,24,28,15),(9,16,21,24,25),(8,14,18,10,10) for Lie algebras A5, B5, C5, D5, respectively. The asymptotic behavior for the polynomials at large distances is governed by some integer-valued 5×5 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A5 and D5 cases) with the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances.

Flow of a Viscous Incompressible Fluid from a Moving Point Source

The flow of a viscous incompressible fluid outflowing from a uniformly moving point source is considered. An exact solution to the problem is found in the way that the velocity decreases inversely with the radial coordinate. It is shown that a spherical volume of fluid is carried away by the source, the radius of which is inversely proportional with respect to the velocity of motion. In this case, a cylindrical discontinuity arises in the region of forming a wake behind the body, the dimensions of which are determined by the magnitude of the external pressure and do not depend on the velocity of the source. The obtained solutions are governed by hydrodynamical fields of flows which can be recognized as special invariants at symmetry reduction.

Needle Tip Pose Estimation for Ultrasound-Guided Steerable Flexible Needle With a Complicated Trajectory in Soft Tissue

Visualization of the surgical needle is critical for an image guided insertion. However, it is difficult to obtain a 3D tip pose of a steerable flexible needle under 2D US images. In this letter, an image processing method is used to extract the needle shaft radial cross-sectional centroid coordinates (NSRCCCs). Based on the NSRCCCs, two methods are proposed to estimate the needle tip pose in real-time, which are based on Rodrigues’ rotation formula, and Fourier expansion with three-terms and linear interpolation algorithm (FEWT&LIA), respectively. In order to validate the methods proposed in this letter, the insertion experiments with different depths are conducted in phantom tissues, and the results show that the mean errors are 0.68 mm and 0.65 mm with using the Rodrigues’ rotation formula based method and the FEWT&LIA based method, respectively, showing clinically acceptable accuracy.

Conformally Schwarzschild cosmological black holes

We thoroughly investigate conformally Schwarzschild spacetimes in different coordinate systems to seek for physically reasonable models of a cosmological black hole. We assume that a conformal factor depends only on the time coordinate and that the spacetime is asymptotically flat Friedmann–Lemaître–Robertson–Walker Universe filled by a perfect fluid obeying a linear equation state p = wρ with w > −1/3. In this class of spacetimes, the McClure–Dyer spacetime, constructed in terms of the isotropic coordinates, and the Thakurta spacetime, constructed in terms of the standard Schwarzschild coordinates, are identical and do not describe a cosmological black hole. In contrast, the Sultana–Dyer and Culetu classes of spacetimes, constructed in terms of the Kerr–Schild and Painlevé–Gullstrand coordinates, respectively, describe a cosmological black hole. In the Sultana–Dyer case, the corresponding matter field in general relativity can be interpreted as a combination of a homogeneous perfect fluid and an inhomogeneous null fluid, which is valid everywhere in the spacetime unlike Sultana and Dyer’s interpretation. In the Culetu case, the matter field can be interpreted as a combination of a homogeneous perfect fluid and an inhomogeneous anisotropic fluid. However, in both cases, the total energy–momentum tensor violates all the standard energy conditions at a finite value of the radial coordinate in late times. As a consequence, the Sultana–Dyer and Culetu black holes for −1/3 < w ⩽ 1 cannot describe the evolution of a primordial black hole after its horizon entry.

Image of Bonnor black dihole with a thin accretion disk and its polarization information

We have studied the image of Bonnor black dihole surrounded by a thin accretion disk where the electromagnetic emission is assumed to be dominated respectively by black body radiation and synchrotron radiation. Our results show that the intensity of Bonnor black dihole image increases with the magnetic parameter and the inclination angle in both radiation models. The image of Bonnor black dihole in the synchrotron radiation model is one order of magnitude brighter than that in the black body radiation model, but its intensity in the former decreases more rapidly with the radial coordinate. Especially, for the synchrotron radiation model, the intensity of the secondary image is stronger than that of the direct image at certain an inclination angle. We also present the polarization patterns for the images of Bonnor black dihole arising from the synchrotron radiation, which depend sharply on the magnetic parameter and inclination angle. Finally, we make a comparison between the polarimetric images of Bonnor black dihole and M87*. Our result further confirms that the image of black hole depends on the black hole’s properties itself, the matter around black hole and the corresponding radiation occurred in the accretion disk.

Spanwise distribution of the loads on a hydrofoil working in the wake of an upstream propeller

The spanwise distribution of the loads acting on a hydrofoil working in the wake of a propeller is analyzed both as a function of the propeller load and the incidence of the hydrofoil. The analysis is based on data from Large Eddy Simulations, conducted on cylindrical grids consisting of O(109) points. Although very large coherent structures are shed by the propeller at outer and inner radii (tip and hub vortices), the average loads on the hydrofoil are characterized by peaks correlating with the radial coordinate of maximum load of the propeller blades, equivalent to about 70% of their outer radius. The azimuthal velocity of the propeller wake causes a displacement of the stagnation point in front of the hydrofoil, affecting substantially the pressure fields on its port and starboard sides and, as a result, the loads it experiences. This flow physics suggests that twisted geometries, able to decrease the local incidence angle, can produce substantial benefits on the stresses experienced by rudders operating in the wake of upstream propellers.

A new look at black holes via thermal dimensions and the complex coordinates/ temperature vectors correspondence

The action of active diffeomorphisms (diffs) [Formula: see text] on the Schwarzschild metric leads to metrics which are also static spherically symmetric solutions of the Einstein vacuum field equations. It is shown how in a [Formula: see text] case it allows to introduce a deformation of the manifold such that [Formula: see text], and [Formula: see text] corresponding, respectively, to the spacelike singularity and horizon of the Schwarzschild metric. In doing so, one ends up with a spherical [Formula: see text] surrounding the singularity at [Formula: see text]. In order to explore the “interior” region of this void, we introduce complex radial coordinates whose imaginary components have a direct link to the inverse Hawking temperature, and which furnish a path that provides access to interior region. In addition, we show that the black hole entropy [Formula: see text] (in Planck units) is equal to the [Formula: see text] of a rectangular strip in the [Formula: see text] radial-coordinate plane associated to this path. The gist of the physical interpretation behind this construction is that there is an emergence of thermal dimensions which unfolds as one plunges into the interior void region via the use of complex coordinates, and whose imaginary components capture the span of the thermal dimensions. Namely, the filling of the void leads to an [Formula: see text] internal/ thermal dimension via the imaginary part [Formula: see text] of the complex radial variable [Formula: see text].