Axisymmetric Solutions Research Articles

Analytical Solutions for Axisymmetric Diffusion of Organic Pollutants in a Circular‐Shaped Cutoff Wall System

ABSTRACTThrough the variable substitution and separated variable methods, this study develops a two‐dimensional (2‐D) axisymmetric diffusion analytical solution for organic pollutants in a circular‐shaped cutoff wall (CCW) system for the first time, which can more precisely and reasonably simulate the diffusion behaviors in “circular‐shaped” vertical barriers. Then, the proposed analytical solution's reasonableness is verified by comparing it with an existing analytical solution and a corresponding finite‐difference solution. Meanwhile, the comparison suggests that this solution will degrade to a 2‐D diffusion analytical solution when the pollution source radius is large enough. Furthermore, the presented analytical solution can also be simplified to a one‐dimensional axisymmetric diffusion analytical solution, or to the axisymmetric diffusion analytical solutions in a single‐layered medium. These exact analytical solutions can not only be applied to study axisymmetric diffusion behaviors under specific scenarios, but also be used to validate other complex numerical models. Last, a case study is conducted to investigate the impacts of pollution source concentration distribution, CCW horizontal thickness, and defined equivalent diffusion coefficient on the barrier performance. Overall, the proposed analytical solutions and obtained diffusion laws in this study can provide guidance for the service effect assessment and the engineering design of cutoff walls.

Compressible ocean waves generated by sudden seabed rise near a step-type topography

A typical tsunami generation occurs through submarine earthquakes leading to large volume displacement. The corresponding mathematical problem involves modeling surface water waves generated by an arbitrary temporal motion of the ocean floor. The propagation of tsunami wave and the subsequent scattering from a sudden drop in bathymetry away from the ground motion is studied following linearized water wave theory and a weakly compressible ocean, including static oceanic background compression. The Fourier transformation and eigenfunction expansion techniques are employed to find the surface displacement and pressure profiles by leveraging appropriate matching conditions between regions of different depths. A novel energy balance relationship is derived by considering both the pure-gravity and acoustic-gravity modes. The model is validated in the limit that the depth difference approaches zero, showing a vanishing reflection contribution from the depth change. An efficient numerical code is developed that accurately captures the contribution of the cutoff frequencies of acoustic-gravity modes. Apart from the time-domain propagation of tsunami waves away from the origin, standing wave formations are observed within the shallow region, supported by significantly large pressure fluctuations in time. These standing waves or, equivalently, the pressure fluctuations sustain longer for larger ocean depth. The increase in tsunami speed in the deeper region is readily visible in the time-domain simulations. A three-dimensional axisymmetric solution is also developed, and results show a more gradual sloping tsunami wavefront compared to the equivalent two-dimensional solution for shallower depths. Animation movies corresponding to the two- and three-dimensional surface profiles are provided for better visualization.

Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis exclusively in physical-space. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis [13] through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin [63] in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.

Fine large-time asymptotics for the axisymmetric Navier–Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-based entropy methods.

Bifurcation tracking on moving meshes and with consideration of azimuthal symmetry breaking instabilities

We present a black-box method to numerically investigate the linear stability of arbitrary multi-physics problems. While the user just has to enter the system's residual in weak formulation, e.g. by a finite element method, all required discretized matrices are automatically assembled based on just-in-time generated and compiled C codes. Based on this method, entire phase diagrams in the parameter space can be obtained by bifurcation tracking and continuation at low computational costs. Particular focus is put on problems with moving domains, e.g. free surface problems in fluid dynamics, since a moving mesh introduces a plethora of complicated nonlinearities to the system. By symbolic differentiation before the code generation, however, these moving mesh problems are made accessible to bifurcation tracking methods. In a second step, our method is generalized to investigate symmetry-breaking instabilities of axisymmetric stationary solutions by effectively utilizing the symmetry of the base state. Each bifurcation type is validated on the basis of results reported in the literature on versatile fluid dynamics problems, for which we subsequently present novel results as well.

Stability of the Global Weak Axisymmetric Solution to the Quantum Euler System with Vorticity in Dimension d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d=2$\\end{document}

We consider the stability of the global weak solution of the Quantum Euler system in two space dimensions. More precisely, we establish compactness properties of global finite energy weak solution for large initial data provided that these are axisymmetric. The main novelty is that the initial velocity is not necessary irrotational when the density is not vanishing, our main argument is based on the Madelung transform which enables us to prove new Kato estimates on the irrotational part of the velocity.

Fully extremal black holes: A black hole graveyard?

While the standard point of view is that the ultimate endpoint of black hole evolution is determined by Hawking evaporation, there is a growing evidence that classical and semi-classical instabilities affect both black holes with inner horizons as well as their ultra-compact counterparts. In this paper we start from this evidence pointing towards extremal black holes as stable endpoints of the gravitational collapse, and develop a general class of spherical and axisymmetric solutions with multiple extremal horizons. Excluding more exotic possibilities, entailing regular cores supporting wormhole throats, we argue that these configurations could be the asymptotic graveyard, the end-point, of dynamical black hole evolution — albeit the timescale of such evolution are still unclear and possibly long and compatible with current astrophysical observations.

An exact stationary axisymmetric vacuum solution within a metric-affine bumblebee gravity

Within the framework of the spontaneous Lorentz symmetry breaking (LSB), we consider a metric-affine generalization of the gravitational sector of the Standard Model Extension (SME), including the Lorentz-violating (LV) coefficients u and sμν . In this model, we derive the modified Einstein field equations in order to obtain a new axisymmetric vacuum spinning solution for a particular bumblebee's profile. Such a solution has the remarkable property of incorporating the effects of LSB through the LV dimensionless parameter X = ξb 2, with ξ is the nonminimal coupling constant, and b 2 = bμbμ , with bμ is the vacuum expectation value of the bumblebee field; as the LSB is turned off, X = 0, we recover the well-established result, the Kerr solution, as expected. Afterwards, we calculate the geodesics, the radial acceleration and thermodynamic quantities for this new metric. We also estimate an upper bound for X by using astrophysical data of the advance of Mercury's perihelion.

Premixed flames in narrow heated channels of circular cross-section: Steady-state solutions, their linear stability analysis and the multiplicity of stable dynamic modes

Premixed flames in narrow heated circular channels subjected to a Poiseuille flow are investigated within the constant density model for various Lewis numbers using irreversible one-step Arrhenius kinetics. A global stability analysis of steady-state axisymmetric solutions is carried out, together with time-dependent direct numerical simulations. The analysis reveals the criteria for the appearance of oscillatory and three-dimensional cellular flame structures. The problem is also studied separately within the framework of the narrow-channel approximation.Among the results obtained, the following can be singled out as the main ones. First, the multiplicity of stable dynamic modes, oscillatory and steady-state, taking place for the same set of parameters for flames with Le<1 is demonstrated. The actual occurrence of one mode or another depends on the initial conditions. Second, the appearance of chaotic regimes is shown for flames with Le>1. The chaotic dynamics occurs in a narrow range of values of the flow rate, with Feigenbaum period-doubling cascades taking place both before and after this interval. The results of this study could be useful in the development and use of small-scale combustion devices.Novelty and significance statement: A systematic study of premixed flames in narrow heated circular channels in the presence of Poiseuille flow is carried out for various Lewis numbers using irreversible one-step Arrhenius kinetics. One of the novelties presented in the paper is the linear global stability analysis of steady state axisymmetric solutions of this problem, which has not been reported before. Also, for the first time, the existence of multiple stable dynamic modes, oscillatory and time-independent, which occurs at the same parameter values, is demonstrated for flames with Lewis number smaller than one. For flames with a Lewis number greater than one, cases with chaotic dynamics are found that manifest themselves in a narrow range of the flow rate. Finally, it is demonstrated that the Feigenbaum cycle-doubling cascade can appear before and after this interval of chaotic dynamics. Such analysis has not been reported before for this problem of flames in partially heated channels.

On the Structure of Axisymmetric Helical Solutions to the Incompressible Navier–Stokes System

On the Structure of Axisymmetric Helical Solutions to the Incompressible Navier–Stokes System

Sustainable heating and cooling for residential buildings: coaxial ground heat exchangers

As the world fights climate change challenges and the soaring demand for energy-efficient solutions, the need for sustainable heating and cooling systems in residential buildings has become paramount. Traditional systems rely on fossil fuels and contribute significantly to greenhouse gas emissions. Ground source heat pumps (GSHPs) have emerged as a promising technology, offering a sustainable alternative to conventional space conditioning systems. Central to the success of GSHPs is the efficient and reliable operation of their ground heat exchangers, with lower installation costs, as a prospective solution for future residential applications. This paper presents a numerical axisymmetric (r-z) finite volume method solution of the thermal performance of coaxial ground heat exchangers (CGHE) that incorporate phase change material (PCM) in the ground for GSHP applications. An analysis of the CGHE with PCM grout is presented, highlighting the advantages of heat exchange performance over normal grouts. The temperature comparison on three different points in the grout with different grouting materials was carried out to analyse the behaviour of different grouting materials during the heat exchange process. The investigation from a temperature point of view showed that the inclusion of PCM in the grouting material improves the thermal performance of the CGHE.

Observational signatures of rotating black holes in the semiclassical gravity with trace anomaly* *Partly Supported by the National Natural Science Foundation of China (12275004, 12205013, 11873044). MG is also endorsed by ”the Fundamental Research Funds for the Central Universities” (2021NTST13)

In a recent work by Fernandes [Phys. Rev. D 108(6), L061502 (2023)], an exact stationary and axisymmetric solution was discovered in semiclassical gravity with type-A trace anomaly. This was identified as a quantum-corrected version of the Kerr black hole. In this study, we explore the observational signatures of this black hole solution. Our investigation reveals that prograde and retrograde light rings exist, whose radii increase monotonically with the coupling parameter α. When α is negative, the shadow area for the quantum-corrected black hole is smaller than that of the Kerr black hole, whereas when α is positive, the area is larger. For a near-extremal black hole, its high-spin feature (the NHEKline) is found to be highly susceptible to disruption by α. Furthermore, we discuss the images of the quantum-corrected black hole in the presence of a thin accretion disk and compare them to those of the Kerr black hole. Our study highlights the importance of near-horizon emission sources in detecting the effects of quantum corrections by black hole images.

Exact non-Abelian supertubes

Supertubes are supersymmetric configurations in string theory in which branes are extending along a closed curve. For a supertube of codimension two, its dipole charge is characterized by the duality monodromy around the closed curve. When multiple codimension-2 supertubes are present, the monodromies around different supertubes can be non-commuting, namely non-Abelian. Non-Abelian configurations of supertubes are expected to play an important role in non-perturbative physics of string theory, especially black holes. In this paper, in the framework of five-dimensional supergravity, we construct exact solutions describing codimension-2 supertubes in three-dimensional space. We use an extension formula to construct a three-dimensional solution from a two-dimensional seed solution. The two-dimensional seed is an F-theory like configuration in which a torus is nontrivially fibered over a complex plane. In the first example, there is a stack of circular supertubes around which there is a non-trivial monodromy. In some cases this can be thought of as a microstate of a black hole in AdS2 × S2. The second example is an axi-symmetric solution with two stacks of circular supertubes with non-Abelian monodromies. In addition, there is a continuous distribution of charges on the symmetry axis.

A falling fluid droplet in an oscillating flow field

We examine the flow in and around a falling fluid droplet in a vertically oscillating flow. We assume axisymmetric Stokes flow, and for small deformations to the droplet, the governing equations can be linearized leading to an infinite system of linear ordinary differential equations. In this study, we have analytically solved the problem in the small-capillary limit. We note that the solution locally breaks down at the poles of the droplet. The drag and center of the mass were also obtained. In the case when only odd modes are present, the droplet shows three-dimensional axisymmetric heart-shaped solutions oscillating vertically in time. When only even modes are present, the droplet exhibits axisymmetric stretching and squeezing.

A General Axisymmetric Elastic-Plastic Solution for an Arbitrary Isotropic Yield Criterion under Plane Stress

A General Axisymmetric Elastic-Plastic Solution for an Arbitrary Isotropic Yield Criterion under Plane Stress

Global weighted regularity for the 3D axisymmetric non-resistive MHD system

We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in $ \mathbb{R}^{3} $. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time $ T $ if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.

The global stability of 3-D axisymmetric solutions to compressible viscous and heat-conductive fluids

The global stability of 3-D axisymmetric solutions to compressible viscous and heat-conductive fluids

Pile field numerical analysis at the stage of long-term impacts

The authors propose a combined approach for definition of the shear rigidity of the multilayered soil which is cut through by a pile. The solution for the vertical direction is presented in the view of an axisymmetric problem. As to the horizontal direction, the solution is presented in view of a beam on elastic subsoil with genetically non-linear transition to equivalent horizontal rigidity of the wide pile field in condition of dynamic forces action. The axisymmetric solution provides visual clarity in the analysis of the stress-strain state of the pile and near-pile soil in comparison with the approved analytical methods. To speed up calculations at the stage of the main combination of constant and long-term impacts, the vertical rigidity of the base under the foot of the pile can be calculated analytically as for a stamp on an elastic-plastic base. The horizontal rigidity is considered as for a discrete single bent pile in the medium of an elastic layered half-space at the stage of formation of the stress-strain state of the system under the main combination of static loads. These methods of numerical modeling of deformations of a single pile make it possible to expand the algorithm of analytical calculation of a large pile field, which in turn is modified by the authors by excluding deformations of the pile body due to its natural consideration in the finite element formulation, as well as by introducing the parameter of the limiting radius of the influence of a single pile on the settlement of neighboring piles. The considered numerical approaches to the calculation of the pile field are applicable in a complex calculation taking into account the history of loading at the stage of the main combination of permanent and long-term impacts. At the stages of short-term or special dynamic impacts an integral rigidity of pile cells is proposed, which will be considered in the next publication of the authors.

Pile field numerical analysis at the stage of dynamic impacts

The authors propose a combined approach for definition of the shear rigidity of the multilayered soil which is cut through by a pile. The static solution for the vertical direction is presented in the view of an axisymmetric problem. As to the horizontal direction, the solution is presented in view of a beam on elastic subsoil with genetically non-linear transition to equivalent horizontal dynamic rigidity of the wide pile field. At the stage of formation of the stress-strain state of the system under the main combination of static loads еhe axisymmetric solution provides visual clarity in the analysis of the stress-strain state of the pile and near-pile soil in comparison with the approved analytical methods. The horizontal rigidity is considered as for a discrete single bent pile in the medium of an elastic layered half-space. These methods of numerical modeling of deformations of a single pile make it possible to expand the algorithm of analytical calculation of a large pile field, which in turn is modified by the authors by excluding deformations of the pile body due to its natural consideration in the finite element formulation, as well as by the parameter of the limiting radius of the influence of a single pile on the settlement of neighboring piles. In this publication, the authors describe a methodological approach to constructing a model of a large-size pile field at the second stage of short-term or special dynamic impacts. A transition to the integral rigidity of pile groups is proposed, taking into account the spatial dynamic model of a large-size pile foundation. The numerical problems of modeling a spatial soil array, a wave method for determining the discreteness of the model, methods for replacing the underlying half-space with a contact model are considered.

Exact rotating black hole solutions for f(R) gravity by modified Newman Janis algorithm

We show that the f(R)-gravity theories with constant Ricci scalar in the Jordan/Einstein frame can be described by Einstein or Einstein–Maxwell gravity with a cosmological term and a modified gravitational constant. To obtain the rotating axisymmetric solutions for the Einstein/Einstein–Maxwell gravity with a cosmological constant, we also propose a modified Newmann–Janis algorithm which involves the non-complexification of the radial coordinate and a complexification of the polar coordinate. Using the duality between the two gravity theories we show that the stationary or static solutions for the Einstein/Einstein–Maxwell gravity with a cosmological constant will also be the solutions for the dual f(R)-gravity with constant Ricci scalar.