Conformal representation of Kerr space–time poloidal sub-manifolds

Chantry, L.; Cayatte, V.; Sauty, C.

doi:10.1088/1361-6382/ab8399

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…Due to the continuous movement of boundaries, it is extremely difficult to obtain the electric field intensity of the electrolyte domain (Ω). To exactly calculate the analytical solution of electric field intensity, the conformal transformation is used to simplify the original electric field model [33][34][35][36]. As shown in Figure 2b, two non-concentric circular electrodes can be mapped to two concentric circular electrodes.…”

Section: Mathematical Model 21 Electric Field Modelmentioning

confidence: 99%

Evolution of the Interelectrode Gap during Co-Rotating Electrochemical Machining

Zhou,

Wang,

et al. 2023

Metals

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A new co-rotating electrochemical machining method is presented to machine the complex structure inside annular parts such as flame tubes and aero-engine casings. Due to the unique shape and motion of electrodes, it is difficult to accurately compute the electric field intensity in the machining area. In this paper, the complex electric field model is simplified by conformal transformation, and the analytical solution of electric field intensity is exactly calculated. A material removal model is built on the basis of the electric field model, and the dynamic simulation of the material removal process is realized. The effects of the cathode radius, applied voltage, feed rate and initial interelectrode gap on the interelectrode gap (IEG) and material removal rate (MRR) are analyzed. The simulation results indicate that the MRR is always slightly less than the feed rate in a quasi-equilibrium state, resulting in a slow reduction in IEG. In addition, the final machining state is not affected by the initial IEG, and the MRR in a quasi-equilibrium state is determined by the feed rate. Several comparative experiments were carried out using the optimized processing parameters, in which the MRR and IEG were measured. The convex structures were successfully machined inside the annular workpiece with optimum machining parameters. The experimental results are in good agreement with the theoretical results, indicating that the established model can effectively predict the evolution process of MRR and IEG.

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Section: Mathematical Model 21 Electric Field Modelmentioning

confidence: 99%

Evolution of the Interelectrode Gap during Co-Rotating Electrochemical Machining

Zhou,

Wang,

et al. 2023

Metals

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“…Instead of using a simple Cartesian version of the Boyer-Lindquist coordinates, or what is called pseudo-Cartesian coordinates, we opted for the use of so-called quasi-isotropic coordinates. InChantry et al (2020) we discussed in details their properties. This choice of coordinates presents two main advantages.…”

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confidence: 99%

Double flows anchored in a Kerr black hole horizon – I. Meridionally self-similar MHD models with loading terms

Chantry

Cayatte

Sauty

et al. 2022

Monthly Notices of the Royal Astronomical Society

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Recent observations of supermassive black holes have brought us new information on their magnetospheres. In this study we attempt a theoretical modelling of the coupling of black holes with their jets and discs, via three innovations. First, we propose a semi-analytical MHD description of a steady relativistic inflow-outflow structure characteristic to the extraction of the hole rotational energy. The mass-loading is ensured in a thin layer, the stagnation surface, by a two-photon pair production originating to a gamma-ray emission from the surrounding disc. The double flow is described near the polar axis by an axisymmetric meridionally self-similar MHD model. Second, the inflow and outflow solutions are crossing the MHD critical points and are matched at the stagnation surface. Knowledge of the MHD field on the horizon give us the angular momentum and energy extracted from the black hole. Finally, we illustrate the model with three specific examples of double-flow solutions by varying the energetic interaction between the MHD field and the rotating black hole. When the isorotation frequency is half of the black hole one, the extracted Poynting flux is comparable to the one obtained using the force-free assumption. In two of the presented solutions, the Penrose process dominates at large colatitudes, while the third is Poynting flux dominated at mid colatitudes. Mass injection rate estimations, from disk luminosity and inner radius, give an upper limit just above the values obtained for two solutions. This model is pertinent to describe the flows near the polar axis where pair production is more efficient.

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“…In a previous study (Chantry et al 2020), we have introduced different ways for obtaining a conformal representation, i.e. producing an accurate representation of the angles between curves of the same manifold.…”

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confidence: 99%

Quasi-isometric embedding of Kerr poloidal submanifolds

Chantry¹,

Dauvergne²,

Temmam³

et al. 2021

Class. Quantum Grav.

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We propose two approaches to obtain an isometric embedding of the poloidal Kerr submanifold. The first one relies on the convex integration process using the corrugation from a primitive embedding. This allows us to obtain one parameter family of embeddings reaching the limits of an isometric embedding. The second one consists in consecutive numerical resolutions of the Gauss-Codazzi-Mainardi and frame equations. This method requires geometric assumptions near the equatorial axis of the poloidal submanifold to get initial and boundary conditions. The second approach allows to understand some physical properties in the vicinity of a Kerr black hole, in particular the fast increasing ergoregion extent with angular momentum.

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Conformal representation of Kerr space–time poloidal sub-manifolds

Cited by 4 publications

References 13 publications

Evolution of the Interelectrode Gap during Co-Rotating Electrochemical Machining

Evolution of the Interelectrode Gap during Co-Rotating Electrochemical Machining

Double flows anchored in a Kerr black hole horizon – I. Meridionally self-similar MHD models with loading terms

Quasi-isometric embedding of Kerr poloidal submanifolds

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