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18 pages, 6987 KiB  
Article
Modeling of Measuring Transducers for Relay Protection Systems of Electrical Installations
by Iliya Iliev, Andrey Kryukov, Konstantin Suslov, Nikolay Kodolov, Aleksandr Kryukov, Ivan Beloev and Yulia Valeeva
Sensors 2025, 25(2), 344; https://doi.org/10.3390/s25020344 - 9 Jan 2025
Viewed by 219
Abstract
The process of establishing relay protection and automation (RPA) settings for electric power systems (EPSs) entails complex calculations of operating modes. Traditionally, these calculations are based on symmetrical components, which require the building of equivalent circuits of various sequences. This approach can lead [...] Read more.
The process of establishing relay protection and automation (RPA) settings for electric power systems (EPSs) entails complex calculations of operating modes. Traditionally, these calculations are based on symmetrical components, which require the building of equivalent circuits of various sequences. This approach can lead to errors both when identifying the operating modes and when modeling the RPA devices. Proper modeling of measuring transformers (MTs), symmetrical component filters (SCFs), and circuits connected to them effectively solves this problem, enabling the configuration of relay protection and automation systems. The methods of modeling the EPS in phase coordinates are proposed to simultaneously determine the operating modes of high-voltage networks and secondary circuits connected to the current and voltage transformers. The MT and SCF models are developed to concurrently identify the operating modes of secondary wiring circuits and calculate the power flow in the controlled EPS segments. This method is effective in addressing practical problems related to the configuration of the relay protection and automation systems. It can also be used when establishing cyber–physical power systems. For a comprehensive check of the adequacy of the MT models, 140 modes of the electric power system were determined which corresponded to time-varying traction loads. Based on the results of calculating the complexes of currents and voltages at the MT terminals, parametric identification of the power transmission line was performed. Based on this, the model of this transmission line was adjusted; repeated modeling was carried out, and errors were calculated. The modeling results showed a high accuracy when calculating the modules and phases of voltages using the identified model. The average error value for current modules was 0.6%, and for angles, it was 0.26°. Full article
(This article belongs to the Special Issue Mechanical Energy Harvesting and Self-Powered Sensors)
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Figure 1

Figure 1
<p>Circuits of voltage transformers (VTs) and current transformers (CTs): (<b>a</b>)—VT circuit; (<b>b</b>)—VT LEC; (<b>c</b>)—CT LEC; HV—high voltage; LV—low voltage.</p>
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<p>Circuit of the voltage transformer (<b>a</b>) and its model (<b>b</b>).</p>
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<p>CT model circuit.</p>
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<p>Original electrical circuit.</p>
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<p>The 220 kV electrical networks supplying traction substations of the main railway (<b>a</b>) and the CT and VT connection circuit (<b>b</b>).</p>
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<p>Errors in voltage magnitudes (<b>a</b>) and phases (<b>b</b>).</p>
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<p>Errors in current magnitudes (<b>a</b>) and phases (<b>b</b>).</p>
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<p>Diagrams of electrical network.</p>
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<p>Schedule of trains weighing 4000 tons (<b>a</b>) and traction current profiles for up (<b>b</b>) and down (<b>c</b>) trains.</p>
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<p>Change in phase voltages at the starting end of longitudinal power supply line.</p>
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<p>Change in magnitudes (<b>a</b>) and phases (<b>b</b>) of positive and negative sequence voltages at the F1 and F2 outputs; the magnitude of the negative sequence voltage is increased tenfold for visual clarity.</p>
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<p>Changes over time in the output voltage of the NSVF and the unbalance factor of the 10 kV busbars of TS 2; <math display="inline"><semantics> <mrow> <msubsup> <mi>k</mi> <mrow> <mn>2</mn> <mi>U</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> </semantics></math> is obtained using the PSVF and NSVF models; <math display="inline"><semantics> <mrow> <msubsup> <mi>k</mi> <mrow> <mn>2</mn> <mi>U</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>100</mn> <msub> <mi>U</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math>; <span class="html-italic">U</span><sub>1</sub> is the positive sequence voltage; <span class="html-italic">U</span><sub>2</sub> is negative sequence voltage.</p>
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<p>Graphs of relative (<b>a</b>) and absolute (<b>b</b>) errors in determining the negative sequence unbalance factor.</p>
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28 pages, 9647 KiB  
Article
Prioritized Decision Support System for Cybersecurity Selection Based on Extended Symmetrical Linear Diophantine Fuzzy Hamacher Aggregation Operators
by Muhammad Zeeshan Hanif and Naveed Yaqoob
Symmetry 2025, 17(1), 70; https://doi.org/10.3390/sym17010070 - 3 Jan 2025
Viewed by 476
Abstract
The symmetrical linear Diophantine fuzzy Hamacher aggregation operators play a fundamental role in many decision-making applications. The selection of a cyber security system is of paramount importance for maintaining digital assets. It necessitates a comprehensive review of threat landscapes, vulnerability assessments, and the [...] Read more.
The symmetrical linear Diophantine fuzzy Hamacher aggregation operators play a fundamental role in many decision-making applications. The selection of a cyber security system is of paramount importance for maintaining digital assets. It necessitates a comprehensive review of threat landscapes, vulnerability assessments, and the specific needs of the organization in order to ensure the implementation of effective security measures. Smart grid (SG) technology uses modern communication and monitoring technologies to enhance the management and regulation of electricity production and transmission. However, greater dependence on technology and connection creates new vulnerabilities, exposing SG communication networks to large-scale attacks. Unlike previous surveys, which often give broad overviews of SG design, our research goes a step further, giving a full architectural layout that includes major SG components and communication linkages. This in-depth review improves comprehension of possible cyber threats and allows SGs to analyze cyber risks more systematically. To determine the best cybersecurity strategies, this study introduces a multi-criteria group decision-making (MCGDM) approach using the linear Diophantine fuzzy Hamacher prioritized aggregation operator (LDFHPAO). In real-world applications, aggregation operators (AOs) are essential for information fusion. This research presents innovative prioritized AOs designed to address MCGDM problems in uncertain environments. We developed the LDF Hamacher prioritized weighted average (LDFHPWA) and LDF Hamacher prioritized weighted geometric (LDFHPWG) operators, which address the shortcomings of traditional operators and provide a more robust modeling approach for MCGDM challenges. This study also outlines key characteristics of these new prioritized AOs. An MCGDM approach incorporating these operators is proposed and demonstrated to be effective through an example that compares and selects the optimal cybersecurity. Full article
(This article belongs to the Special Issue Recent Developments on Fuzzy Sets Extensions)
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<p>A visual representation depicting the different types of cyberattacks.</p>
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<p>Graphical comparison of space in IFS, PFS, FFS, and q-ROFFS.</p>
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<p>LDF with <inline-formula><mml:math id="mm300"><mml:semantics><mml:mrow><mml:mfenced open="〈" close="〉" separators="|"><mml:mrow><mml:mi mathvariant="script">E</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">℘</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced open="〈" close="〉" separators="|"><mml:mrow><mml:mn>0.6,0.3</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
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<p>LDF with <inline-formula><mml:math id="mm301"><mml:semantics><mml:mrow><mml:mfenced open="〈" close="〉" separators="|"><mml:mrow><mml:mi mathvariant="script">E</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">℘</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced open="〈" close="〉" separators="|"><mml:mrow><mml:mn>0.9,0.01</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
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<p>LDF with <inline-formula><mml:math id="mm302"><mml:semantics><mml:mrow><mml:mfenced open="〈" close="〉" separators="|"><mml:mrow><mml:mi mathvariant="script">E</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="script">℘</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced open="〈" close="〉" separators="|"><mml:mrow><mml:mn>0.8,0.09</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
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<p>Different parameters of LDF space.</p>
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<p>Flow chart to determine best alternative.</p>
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<p>Graphical representation of comparisons.</p>
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<p>Visual depiction of the LDFHPWA operator with various parameters.</p>
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<p>Visual depiction of the LDFHPWG operator with various parameters.</p>
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20 pages, 269 KiB  
Article
Nonrelativistic Approximation in the Theory of a Spin-2 Particle with Anomalous Magnetic Moment
by Alina Ivashkevich, Viktor Red’kov and Artur Ishkhanyan
Axioms 2025, 14(1), 35; https://doi.org/10.3390/axioms14010035 - 3 Jan 2025
Viewed by 578
Abstract
We start with the 50-component relativistic matrix equation for a hypothetical spin-2 particle in the presence of external electromagnetic fields. This equation is hypothesized to describe a particle with an anomalous magnetic moment. The complete wave function consists of a two-rank symmetric tensor [...] Read more.
We start with the 50-component relativistic matrix equation for a hypothetical spin-2 particle in the presence of external electromagnetic fields. This equation is hypothesized to describe a particle with an anomalous magnetic moment. The complete wave function consists of a two-rank symmetric tensor and a three-rank tensor that is symmetric in two indices. We apply the general method for performing the nonrelativistic approximation, which is based on the structure of the 50×50 matrix Γ0 of the main equation. Using the 7th-order minimal equation for the matrix Γ0, we introduce three projective operators. These operators permit us to decompose the complete wave function into the sum of three parts: one large part and two smaller parts in the nonrelativistic approximation. We have found five independent large variables and 45 small ones. To simplify the task, by eliminating the variables related to the 3-rank tensor, we have derived a relativistic system of second-order equations for the 10 components related to the symmetric tensor. We then take into account the decomposition of these 10 variables into linear combinations of large and small ones. In accordance with the general method, we separate the rest energy in the wave function and specify the orders of smallness for different terms in the arising equations. Further, after performing the necessary calculations, we derive a system of five linked equations for the five large variables. This system is presented in matrix form, which has a nonrelativistic structure, where the term representing additional interaction with the external magnetic field through three spin projections is included. The multiplier before this interaction contains the basic magnetic moment and an additional term due to the anomalous magnetic moment. The latter characteristic is treated as a free parameter within the hypothesis. Full article
(This article belongs to the Special Issue Mathematical Aspects of Quantum Field Theory and Quantization)
57 pages, 31042 KiB  
Article
Development of a New Rubber Buckling-Restrained Brace System for Structures
by Nima Ostovar and Farzad Hejazi
Appl. Sci. 2025, 15(1), 276; https://doi.org/10.3390/app15010276 - 30 Dec 2024
Viewed by 537
Abstract
Buckling-Restrained Braces (BRBs) are widely utilized in structures as an anti-seismic system to enhance performance against lateral excitations. While BRBs are designed to yield symmetrically under both tension and compression without significant buckling, their effectiveness is often limited to moderate seismic events. During [...] Read more.
Buckling-Restrained Braces (BRBs) are widely utilized in structures as an anti-seismic system to enhance performance against lateral excitations. While BRBs are designed to yield symmetrically under both tension and compression without significant buckling, their effectiveness is often limited to moderate seismic events. During high-intensity earthquakes, repetitive yielding can lead to core failure, resulting in the loss of BRB functionality and potentially causing structural collapse. This study proposes an innovative design for BRBs to improve energy dissipation capacity under severe seismic activity. The new design incorporates Ultra-High-Performance Fiber-Reinforced Concrete (UHPFRC) filler and hyper-elastic rubber components as primary load-bearing elements. Through extensive testing and simulation, the proposed Rubber Buckling-Restrained Brace (RBRB) was developed and manufactured by integrating hyper-elastic rubber between the concrete and core to enhance the device’s strength. Additionally, a prototype of the conventional BRB device was fabricated to serve as a benchmark for evaluating the performance of the RBRB. Experimental testing of both the conventional BRB and the proposed RBRB prototypes was conducted using a heavy-duty dynamic actuator to assess the RBRB’s performance under applied loads. Based on the experimental results, an analytical model of the proposed RBRB was formulated for use in finite element modeling and analysis. Furthermore, a specialized seismic design procedure for structures equipped with the RBRB was developed, according to the performance-based design method. This procedure was applied to the design of a seven-story steel structure, and the impact of the RBRB on the seismic response of the structure was investigated through finite element simulations. The analysis results demonstrated that the RBRB significantly improves the loading capacity and energy dissipation capabilities of structures, thereby enhancing their overall performance against earthquake excitations. Full article
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Figure 1
<p>(<b>a</b>) Procedure of fabrication of the RBRB. (<b>b</b>) Proposed RBRB after assembling. (<b>a</b>) Assembly steps. (<b>b</b>) Proposed RBRB.</p>
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<p>RBRB performance under compression.</p>
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<p>RBRB performance under tension.</p>
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<p>Steel plate sample cores.</p>
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<p>UTM testing device.</p>
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<p>Steel samples result.</p>
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<p>Rubber1 and Rubber2 testing molds.</p>
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<p>Rubber testing setup.</p>
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<p>Rubber testing setup under a compression testing device.</p>
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<p>Rubber samples’ load displacement curves.</p>
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<p>Dried concrete in the mixer.</p>
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<p>Fiber material used in the UHPFRC matrix.</p>
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<p>UHPFRC testing setup and result. (<b>a</b>) Compressive strength cube test. (<b>b</b>) Average strength of UHPFRC.</p>
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<p>RBRB fabrication steps.</p>
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<p>RBRB dimensions and model. (<b>a</b>) Side view of steel core and rubber component. (<b>b</b>) Side view of steel tube restrainer. (<b>c</b>) Top view of steel tube restrainer. (<b>d</b>) Assembly shape of steel core and rubber component in restrainer.</p>
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<p>RBRB steel components and covers.</p>
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<p>RBRB steel tube restrainer gap forming.</p>
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<p>RBRB steel tube restrained cast concrete.</p>
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<p>RBRB-leveled concrete in steel tube restrainer.</p>
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<p>Prepared RBRB prototype.</p>
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<p>RBRB installation in test setup.</p>
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<p>RBRB strain gauge location.</p>
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<p>Full setup of RBRB.</p>
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<p>RBRB LVDTs’ position.</p>
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<p>BRB size and dimensions. (<b>a</b>) Side view of steel core. (<b>b</b>) Top view of steel core. (<b>c</b>) Side view of steel tube restrainer. (<b>d</b>) Top view of tube restrainer and location hole for pouring concrete.</p>
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<p>Completed setup of BRB.</p>
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<p>Cyclic time-displacement amplitude used in the experiment.</p>
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<p>Local buckling of BRB core.</p>
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<p>Failure of BRB core after testing.</p>
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<p>Hysteresis curve of experimental BRB.</p>
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<p>Backbone curve of experimental BRB.</p>
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<p>Local buckling of RBRB core.</p>
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<p>Core failure of RBRB.</p>
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<p>Core failure and rubber components of RBRB.</p>
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<p>Hysteresis curve of experimental RBRB.</p>
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<p>Full backbone curve of experimental RBRB.</p>
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<p>Hysteresis curve in phase 1 of experimental RBRB.</p>
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<p>Hysteresis curve in phase 2 of experimental RBRB.</p>
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<p>Comparison of experimental results for full BRB and full RBRB behavior.</p>
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<p>Comparison of experimental results for BRB and RBRB.</p>
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<p>Effective stiffness equation parameters.</p>
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<p>BRB and RBRB effective stiffness.</p>
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<p>BRB and RBRB effective damping.</p>
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<p>BRB and RBRB energy dissipation.</p>
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<p>Design response spectra.</p>
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<p>Scaled seismic records.</p>
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<p>Scaled response spectra of seven selected accelerations.</p>
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<p>Schematic of implemented RBRB in the structure.</p>
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<p>(<b>a</b>) RBRBF under compression; (<b>b</b>) RBRBF under tension.</p>
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<p>(<b>a</b>–<b>d</b>) Bracing implementation in the building. (<b>e</b>) Bracing implementation in the building. (<b>a</b>) X-direction. (<b>b</b>) Y-direction. (<b>c</b>) Section sizes, elevation view. (<b>d</b>) Three-dimensional view. (<b>e</b>) Section sizes, plan view.</p>
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<p>(<b>a</b>–<b>d</b>) Bracing implementation in the building. (<b>e</b>) Bracing implementation in the building. (<b>a</b>) X-direction. (<b>b</b>) Y-direction. (<b>c</b>) Section sizes, elevation view. (<b>d</b>) Three-dimensional view. (<b>e</b>) Section sizes, plan view.</p>
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<p>BRB-6 properties.</p>
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<p>Maximum story displacement. (<b>A</b>) DBE TH Cape Mendocino in X-direction. (<b>B</b>) MCE TH Cape Mendocino in Y-direction.</p>
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<p>Maximum inter-story drift. (<b>A</b>) DBE TH Darfield in X-direction. (<b>B</b>) MCE TH Darfield in X-direction.</p>
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<p>Maximum base shear of MCE TH Imperial Valley1 in Y-direction.</p>
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<p>Top-story displacement of MCE TH Chi-Chi in X-direction.</p>
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<p>Dissipated energy under MCE in Chi-Chi TH.</p>
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<p>Plastic hinge formation. (<b>a</b>) Bare frame, (<b>b</b>) BRB frame, and (<b>c</b>) RBRB frame (MCE). Gray (A-IO), Green (IO-LS), Blue (LS-CP), Red CP.</p>
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<p>Plastic hinge status in DBE-hazard-level X-direction. (<b>a</b>) Percentage of plastic hinges formed in A-IO. (<b>b</b>) Percentage of plastic hinges formed in IO-LS. (<b>c</b>) Percentage of plastic hinges formed in LS-CP. (<b>d</b>) Percentage of plastic hinges formed in &gt;CP.</p>
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<p>Plastic hinge status in DBE-hazard-level Y-direction. (<b>a</b>) Percentage of plastic hinges formed in A-IO. (<b>b</b>) Percentage of plastic hinges formed in IO-LS. (<b>c</b>) Percentage of plastic hinges formed in LS-CP. (<b>d</b>) Percentage of plastic hinges formed in &gt;CP.</p>
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<p>Plastic hinge status in MCE-hazard-level X-direction. (<b>a</b>) Percentage of plastic hinges formed in A-IO. (<b>b</b>) Percentage of plastic hinges formed in IO-LS. (<b>c</b>) Percentage of plastic hinges formed in LS-CP. (<b>d</b>) Percentage of plastic hinges formed in &gt;CP.</p>
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<p>Plastic hinge status in MCE-hazard-level Y-direction. (<b>a</b>) Percentage of plastic hinges formed in A-IO. (<b>b</b>) Percentage of plastic hinges formed in IO-LS. (<b>c</b>) Percentage of plastic hinges formed in LS-CP. (<b>d</b>) Percentage of plastic hinges formed in &gt;CP.</p>
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<p>Plastic deformation of BRB.</p>
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17 pages, 3658 KiB  
Communication
Proteotranscriptomic Profiling of the Toxic Mucus of Kulikovia alborostrata (Pilidiophora, Nemertea)
by Vasiliy G. Kuznetsov, Daria I. Melnikova, Sergey V. Shabelnikov and Timur Yu. Magarlamov
Toxins 2025, 17(1), 5; https://doi.org/10.3390/toxins17010005 - 26 Dec 2024
Viewed by 343
Abstract
Nemertea is a phylum of bilaterally symmetrical, coelomate, unsegmented worms, also known as ribbon worms. Most species of the phylum Nemertea are marine predators that contain toxins in the single-celled glands of the proboscis and/or integument. Recent transcriptomic studies have shown that nemerteans [...] Read more.
Nemertea is a phylum of bilaterally symmetrical, coelomate, unsegmented worms, also known as ribbon worms. Most species of the phylum Nemertea are marine predators that contain toxins in the single-celled glands of the proboscis and/or integument. Recent transcriptomic studies have shown that nemerteans from all taxonomic groups possess a wide range of putative protein and peptide toxins, while the proteomic data for these animals are highly limited. In this study, proteotranscriptomic analysis was used to investigate the major protein components of the poison of the nemertean Kulikovia alborostrata. We identified 146 transcripts of putative toxins in the transcriptome of K. alborostrata and five putative toxins among the secreted proteins and peptides of the mucus of the animal. The expression levels of cysteine-rich peptides found in the mucus with similarity to known toxins were evaluated in different parts of the body of the worm by quantitative real-time PCR. The high level of expression of investigated peptides in the integument indicate the protective function of these toxins. Overall, this supports the idea that the mucus of nemerteans is a valuable source of peptide and protein toxins. Full article
(This article belongs to the Special Issue Transcriptomic and Proteomic Study on Animal Venom: Looking Forward)
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Graphical abstract

Graphical abstract
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<p>Putative toxin families/domains identified in the transcriptome of <span class="html-italic">Kulikovia alborostrata</span>. The figure illustrates the proportional distribution of toxin family’s/domain’s transcripts in the transcriptome.</p>
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<p>Reversed-phase high-performance liquid chromatography plot of the mucus sample of <span class="html-italic">Kulikovia alborostrata</span>.</p>
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<p>Relative expression levels of three putative toxins identified in the mucus proteome of <span class="html-italic">Kulikovia alborostrata</span> in different parts of the body of the worm. Gene expression levels were quantified by quantitative real-time PCR using the 2<sup>−∆∆Ct</sup> method. Data represent the mean of three independent replicates ± SEM. Reference gene: 60S ribosomal protein L32. Calibrator sample: proboscis.</p>
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<p>Venn diagram showing the number of putative toxin gene families shared between <span class="html-italic">Kulikovia alborostrata</span> and all nemerteans (<b>a</b>) and <span class="html-italic">K</span>. <span class="html-italic">alborostrata</span> and pilidiophorans (<b>b</b>).</p>
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<p><span class="html-italic">Kulikovia alborostrata</span> (Takakura, 1898) live specimens and collection site. (<b>a</b>) Female, (<b>b</b>) male, (<b>c</b>) collection site (asterisk). The black arrowheads point to the head region of worms.</p>
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<p>Schematic diagram of the experiment design.</p>
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17 pages, 9141 KiB  
Article
Model-Free Generalized Super-Twisting Fast Terminal Sliding Mode Control for Permanent Magnet Synchronous Motors
by Xingyi Ma, Yu Xu, Lei Zhang and Jing Bai
Symmetry 2025, 17(1), 18; https://doi.org/10.3390/sym17010018 - 26 Dec 2024
Viewed by 343
Abstract
Permanent Magnet Synchronous Motors (PMSMs) are nonlinear, multi-parameter systems that exhibit structural symmetry but are susceptible to parameter variations and external disturbances. These challenges can disrupt the inherent symmetrical characteristics of PMSM dynamics during real-world operations, posing difficulties for achieving efficient control. To [...] Read more.
Permanent Magnet Synchronous Motors (PMSMs) are nonlinear, multi-parameter systems that exhibit structural symmetry but are susceptible to parameter variations and external disturbances. These challenges can disrupt the inherent symmetrical characteristics of PMSM dynamics during real-world operations, posing difficulties for achieving efficient control. To address this issue, this paper proposes a Model-Free Generalized Super-Twisting Algorithm Fast Terminal Sliding Mode Control (MFFTSMC-GSTA) method. First, a novel ultra-local model incorporating PMSM uncertainties is established, and the MFFTSMC-GSTA controller is designed to address the system’s complex dynamic behavior. By integrating the generalized super-twisting algorithm with the nonsingular fast terminal sliding mode algorithm, the proposed controller ensures finite-time convergence and effectively mitigates chattering. Second, an extended sliding mode disturbance observer is developed to estimate the unknown components of the ultra-local model and provide feedforward compensation, further enhancing system robustness and dynamic performance. The experimental results show that the total harmonic distortion (THD) value of the proposed control method is 1.38%, demonstrating significant improvements in response speed and robustness for motor speed control, and verifying the algorithm’s superior performance under complex operating conditions. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry in Motor Control, Drives and Power Electronics)
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<p>The PMSM Speed Control System Based on MFFTSMC-GSTA.</p>
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<p>Block diagram of MFSTFTSMC-GSTA algorithm.</p>
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<p>Block diagram of ESMDO algorithm.</p>
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<p>The speed response: (<b>a</b>) No-load startup; (<b>b</b>) Acceleration; (<b>c</b>) Acceleration; (<b>d</b>) Sudden increase of 0.5 N·m load; (<b>e</b>) Deceleration; (<b>f</b>) Sudden decrease of 0.5 N·m load; (<b>g</b>) Flux change; (<b>h</b>) Inertia changes.</p>
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<p>The A-axis current and THD: (<b>a</b>) A-axis current of the MFFTISMC; (<b>b</b>) THD of the MFFTISMC; (<b>c</b>) A-axis current of the MFFTSMC; (<b>d</b>) THD of the MFFTSMC; (<b>e</b>) A-axis current of the MFFTSMC-STA; (<b>f</b>) THD of the MFFTSMC-STA; (<b>g</b>) A-axis current of the MFFTSMC-GSTA; (<b>h</b>) THD of the MFFTSMC-GSTA.</p>
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<p>The A-axis current and THD: (<b>a</b>) A-axis current of the MFFTISMC; (<b>b</b>) THD of the MFFTISMC; (<b>c</b>) A-axis current of the MFFTSMC; (<b>d</b>) THD of the MFFTSMC; (<b>e</b>) A-axis current of the MFFTSMC-STA; (<b>f</b>) THD of the MFFTSMC-STA; (<b>g</b>) A-axis current of the MFFTSMC-GSTA; (<b>h</b>) THD of the MFFTSMC-GSTA.</p>
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<p>Experimental platform.</p>
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<p>No-load startup: (<b>a</b>) MFFTSMC-GSTA; (<b>b</b>) MFFTSMC-STA; (<b>c</b>) MFFTISMC; (<b>d</b>) MFFTSMC.</p>
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<p>Load application: (<b>a</b>) MFFTSMC-GSTA; (<b>b</b>) MFFTSMC-STA; (<b>c</b>) MFFTISMC; (<b>d</b>) MFFTSMC.</p>
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<p>Load application: (<b>a</b>) MFFTSMC-GSTA; (<b>b</b>) MFFTSMC-STA; (<b>c</b>) MFFTISMC; (<b>d</b>) MFFTSMC.</p>
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<p>Speed increase: (<b>a</b>) MFFTSMC-GSTA; (<b>b</b>) MFFTSMC-STA; (<b>c</b>)MFFTISMC; (<b>d</b>) MFFTSMC.</p>
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17 pages, 11037 KiB  
Article
Rapid Fluid Velocity Field Prediction in Microfluidic Mixers via Nine Grid Network Model
by Qian Li, Yuwei Chen, Taotao Sun and Junchao Wang
Micromachines 2025, 16(1), 5; https://doi.org/10.3390/mi16010005 - 24 Dec 2024
Viewed by 406
Abstract
The rapid advancement of artificial intelligence is transforming the computer-aided design of microfluidic chips. As a key component, microfluidic mixers are widely used in bioengineering, chemical experiments, and medical diagnostics due to their efficient mixing capabilities. Traditionally, the simulation of these mixers relies [...] Read more.
The rapid advancement of artificial intelligence is transforming the computer-aided design of microfluidic chips. As a key component, microfluidic mixers are widely used in bioengineering, chemical experiments, and medical diagnostics due to their efficient mixing capabilities. Traditionally, the simulation of these mixers relies on the finite element method (FEM), which, although effective, presents challenges due to its computational complexity and time-consuming nature. To address this, we propose a nine-grid network (NGN) model theory with a centrally symmetric structure.The NGN uses a symmetric structure similar to a 3 × 3 grid to partition the fluid space to be predicted. Using this theory, we developed and trained an artificial neural network (ANN) to predict the fluid dynamics within microfluidic mixers. This approach significantly reduces the time required for fluid evaluation. In this study, we designed a prototype microfluidic mixer and validated the reliability of our method by comparing it with predictions from traditional FEM software. The results show that our NGN model completes fluid predictions in just 40 s compared to approximately 10 min with FEM, with acceptable error margins. This technology achieves a 15-fold acceleration, greatly reducing the time and cost of microfluidic chip design. Full article
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Figure 1
<p>Introduction and usage example of nine-grid network model. (<b>A</b>) Meshing of microfluidic mixer. (<b>B</b>) ANN libarary. (<b>C</b>) Dataset structure. (<b>D</b>) Example input and output for the ANN_569 model. (<b>E</b>) An example of predicting the fluid field of the target system using the nine-gride network model.</p>
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<p>(<b>A</b>) The geometric structure diagram of a microfluidic mixer, which has two inlets, two outlets, and a 500 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m × 500 <math display="inline"><semantics> <mi mathvariant="sans-serif">μ</mi> </semantics></math>m reaction zone. (<b>B</b>) An example shows the fluid velocity field predicted by the FEM of a randomly generated mixer.</p>
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<p>The design process of a random microfluidic mixer. Within a month, 10,000 random microfluidic mixer chip designs were generated using MATLAB R2020b programs, and the performance of each chip was simulated using COMSOL <math display="inline"><semantics> <mrow> <mn>5.5</mn> </mrow> </semantics></math>. Finally, the results were saved in a MySQL <math display="inline"><semantics> <mrow> <mn>5.7</mn> <mo>.</mo> <mn>16</mn> </mrow> </semantics></math> database.</p>
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<p>(<b>A</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> of ANN_6 during 6000 epochs. (<b>B</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> of ANN_6 during 6000 epochs. (<b>C</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> of 185,337 items in the test set. (<b>D</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> of ANN_8 during 6000 epochs. (<b>E</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> of ANN_8 during 6000 epochs. (<b>F</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> of 185,337 items in the test set. (<b>G</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of ANN_9 during 6000 epochs. (<b>H</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of ANN_9 during 6000 epochs. (<b>I</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of 185,337 items in the test set.</p>
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<p>(<b>A</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> of ANN_69 during 8000 epochs. (<b>B</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> of ANN_69 during 8000 epochs. (<b>C</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>6</mn> </mrow> </msub> </semantics></math> of 157,763 items in the test set. (<b>D</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of ANN_69 during 8000 epochs. (<b>E</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of ANN_69 during 8000 epochs. (<b>F</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of 157,763 items in the test set.</p>
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<p>(<b>A</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> of ANN_89 during 8000 epochs. (<b>B</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> of ANN_89 during 8000 epochs. (<b>C</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> of 156,774 items in the test set. (<b>D</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of ANN_89 during 8000 epochs. (<b>E</b>) The training curve of the <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of ANN_89 during 8000 epochs. (<b>F</b>) The histogram of the absolute error of <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>v</mi> <mrow> <mi>y</mi> <mo>,</mo> <mn>9</mn> </mrow> </msub> </semantics></math> of 156,774 items in the test set.</p>
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<p>The step of predicting fluid velocity field of the microfluidic chip using ANN Tool.</p>
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<p>The velocity distribution in the reaction zone was predicted by COMSOL and ANN methods. The corresponding SSIM values between the two methods are listed for quantitative analysis.</p>
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<p>The COMSOL and ANN methods were used to predict the velocity distribution of 500 new microfluidic chip reaction zones, and the distribution of the corresponding SSIM values between the two methods was calculated.</p>
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<p>The SSIM for <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> is 0.7646, the SSIM for <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> is 0.7597, and the SSIM for the total velocity is 0.6459.</p>
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<p>The SSIM for <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> is 0.5494, the SSIM for <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> is 0.5489, and the SSIM for the total velocity is 0.5507.</p>
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<p>The SSIM for <math display="inline"><semantics> <msub> <mi>v</mi> <mi>x</mi> </msub> </semantics></math> is 0.5497, the SSIM for <math display="inline"><semantics> <msub> <mi>v</mi> <mi>y</mi> </msub> </semantics></math> is 0.5496, and the SSIM for the total velocity is 0.5227.</p>
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18 pages, 4423 KiB  
Article
Visualization of the 3D Structure of Subcritical Aqueous Ca(NO3)2 Solutions at 25~350 °C and 40 MPa by Raman and X-Ray Scattering Combined with Empirical Potential Structure Refinement Modeling
by Toshio Yamaguchi, Kousei Li, Yuki Matsumoto, Nami Fukuyama and Koji Yoshida
Liquids 2025, 5(1), 1; https://doi.org/10.3390/liquids5010001 - 24 Dec 2024
Viewed by 472
Abstract
Raman scattering measurements were performed on 1 mol dm−3 aqueous calcium nitrate (Ca(NO3)2) and sodium nitrate (NaNO3) solutions containing 4% (w/w) D2O in a temperature range from 25 to 350 [...] Read more.
Raman scattering measurements were performed on 1 mol dm−3 aqueous calcium nitrate (Ca(NO3)2) and sodium nitrate (NaNO3) solutions containing 4% (w/w) D2O in a temperature range from 25 to 350 °C and pressure of 40 MPa. As the temperature increased, the N–O symmetric stretching vibrational band (ν1) of NO3 at 1045–1047 cm−1 shifted to a lower wavenumber by 5~6 cm−1. The band analysis using one Lorentzian component showed that the full-width at half maximum (FWHM) did not change significantly below 175 °C but increased rapidly above 200 °C for both solutions. The peak area for an aqueous Ca(NO3)2 solution showed a breakpoint between 225 and 250 °C, suggesting a change in the coordination shell of NO3 at 175~250 °C. The OD symmetric stretching vibrational band of HDO water was deconvoluted into two Gaussian components at 2530 and 2645 cm−1; the former component has high temperature dependence that is ascribed to the hydrogen bonds, whereas the latter one shows less temperature dependence due to the non-hydrogen bonds of water. X-ray scattering measurements were performed on a 1 mol dm−3 aqueous Ca(NO3)2 solution at 25 to 210 °C and 40 MPa. Empirical potential structure refinement (EPSR) modeling was used to analyze the X-ray scattering data. Ca2+ forms a rigid coordination shell consisting of about seven water molecules at 2.48 Å and one NO3 at 25~170 °C, with further water molecules substituted by NO3 at 210 °C. NO3 is surrounded by 13~14 water molecules at an N–Ow distance of 3.6~3.7 Å. The tetrahedral network structure of solvent water pertains from 25 to 170 °C but is transformed to a dense packing arrangement at 210 °C. Full article
(This article belongs to the Collection Feature Papers in Solutions and Liquid Mixtures Research)
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Graphical abstract

Graphical abstract
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<p>High-temperature and high-pressure Raman cell (Taiatsu Glass Ltd., Tokyo, Japan) (reproduced from ref. [<a href="#B30-liquids-05-00001" class="html-bibr">30</a>] with permission from the Japan Society of Analytical Chemistry, Copyright 2015).</p>
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<p>A high-temperature and high-pressure cell assembly used for EDXD.</p>
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<p>(<b>a</b>,<b>b</b>) Raman Spectra of the <span class="html-italic">ν</span><sub>1</sub> (NO) band of NO<sub>3</sub><sup>−</sup> in 1 mol dm<sup>−3</sup> Ca(NO<sub>3</sub>)<sub>2</sub> and 1 mol dm<sup>−3</sup> NaNO<sub>3</sub> aqueous solutions at selected thermodynamic states; the dots represent the experimental values, the green peak area, and the red line show the fitted spectra. (<b>c</b>–<b>e</b>) plot the position, FWHM, and area of the <span class="html-italic">ν</span><sub>1</sub>(NO) peak as a function of temperature obtained by least-squares fitting.</p>
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<p>(<b>a</b>,<b>b</b>) The Raman spectra of symmetric stretching vibration of the O–D band (<span class="html-italic">ν</span><sub>1</sub>) of HDO molecules at selected thermodynamic conditions. The solid lines show the experimental values, and the green and pink areas and red solid lines represent the components of the hydrogen bonds, non-hydrogen bonds, and their sum, respectively, at selected thermodynamic conditions. (<b>c</b>–<b>e</b>) The temperature dependence of the peak position, FWHM, and area of the <span class="html-italic">ν</span><sub>1</sub> (OD) band. The pink circles and green triangles correspond to the aqueous Ca(NO<sub>3</sub>)<sub>2</sub> and NaNO<sub>3</sub> solutions. The filled and unfilled circles and triangles correspond to hydrogen bonds and non-hydrogen bonds, respectively.</p>
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<p>(<b>a</b>) <span class="html-italic">Q</span> weighted interference functions <span class="html-italic">i</span>(<span class="html-italic">Q</span>) and (<b>b</b>) the corresponding radial distribution functions (RDFs) in the form of <span class="html-italic">D</span>(<span class="html-italic">r</span>)-4π<span class="html-italic">r</span><sup>2</sup><span class="html-italic">ρ</span><sub>0</sub> of a 1 mol dm<sup>−3</sup> Ca(NO<sub>3</sub>)<sub>2</sub> aqueous solution over a temperature range from 25 at 0.1 MPa to 210 °C at 40 MPa. The data obtained by ADXD are included for comparison with those obtained by EDXD.</p>
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<p>(<b>a</b>) Pair distribution functions, (<b>b</b>) coordination number distributions, and (<b>c</b>) angle distributions of Ow–Ow pairs. (<b>d</b>) Spatial density functions of the water oxygen atoms in the first-neighbor (red), second-neighbor (blue), and third-neighbor (yellow) shells around a central water molecule (a water oxygen atom in red and two water hydrogen atoms in white) obtained by EPSR modeling for 1 mol dm<sup>−3</sup> Ca(NO<sub>3</sub>)<sub>2</sub> aqueous solution in various thermodynamic states.</p>
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<p>(<b>a</b>) Ca–Ow pair distribution functions, (<b>b</b>) Ca–Ow coordination number distributions, and (<b>c</b>) the angle distributions of ∠Ow–Ca–Ow for the Ca<sup>2+</sup> solvation in 1 molar Ca(NO<sub>3</sub>)<sub>2</sub> aqueous solutions at 100, 170, and 210 °C and 40 MPa, together with those at ambient condition (25 °C, 0.1 MPa) (color online).</p>
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<p>(<b>a</b>) Ca–O<sub>N</sub> pair distribution functions, (<b>b</b>) Ca–N pair distribution functions, (<b>c</b>) Ca–O<sub>N</sub> coordination number distributions, (<b>d</b>) Ca–N coordination number distributions, and (<b>e</b>) the angle distributions of ∠Ca–O–N for Ca<sup>2+</sup>–NO<sub>3</sub><sup>−</sup> interactions in a 1 mol dm<sup>−3</sup> Ca(NO<sub>3</sub>)<sub>2</sub> aqueous solution at 100, 170, and 210 °C and 40 MPa, together with those at ambient conditions (25 °C, 0.1 MPa) (color online).</p>
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<p>(<b>a</b>) N-Ow pair distribution functions, (<b>b</b>) N-Ow coordination number distributions, (<b>c</b>) the angle distributions of ∠Ow-N-Ow for the NO<sub>3</sub><sup>−</sup> solvation, and (<b>d</b>) the spatial density functions in a 1 mol dm<sup>−3</sup> Ca(NO<sub>3</sub>)<sub>2</sub> aqueous solution at 100, 170, and 210 °C and 40 MPa, together with those at ambient conditions (25 °C, 0.1 MPa). In (<b>d</b>), the yellow lobes represent the nearest-neighbor water oxygen atoms around a central NO<sub>3</sub><sup>−</sup>. The black and white balls show the nitrogen and oxygen atoms of NO<sub>3</sub><sup>−</sup>, respectively (color online).</p>
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14 pages, 6409 KiB  
Article
Research on CFRP Defects Recognition and Localization Based on Metamaterial Sensors
by Zhaoxuan Zhu and Rui Han
Symmetry 2024, 16(12), 1706; https://doi.org/10.3390/sym16121706 - 23 Dec 2024
Viewed by 387
Abstract
In the paper, the concept of symmetry is utilized to detect internal defects in Carbon fiber reinforced polymer (CFRP), that is, the reconstruction and localization methods for internal defects in CFRP are symmetrical. CFRP is widely used in industrial, biological and other fields. [...] Read more.
In the paper, the concept of symmetry is utilized to detect internal defects in Carbon fiber reinforced polymer (CFRP), that is, the reconstruction and localization methods for internal defects in CFRP are symmetrical. CFRP is widely used in industrial, biological and other fields. When there are defects inside the composite materials, its dielectric constant, magnetic permeability, etc. change. Therefore, metamaterial sensors are widely used in non-destructive testing of CFRP Defects. This paper proposes a defect identification and location method based on principal component analysis (PCA) and support vector machine (SVM). The trained model is used to classify the dimensionally reduced data, and the reconstructed defect binary image is obtained. Simulation and physical experiment results show that the method used in this article can effectively identify and locate defects in carbon fiber composite materials. Full article
(This article belongs to the Section Engineering and Materials)
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<p>Simulation model of carbon fiber composite materials.</p>
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<p>Carbon fiber composite overall simulation model.</p>
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<p>Schematic diagram of resonance sensor detection.</p>
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<p>(<b>a</b>) The contribution rate of each principal component; (<b>b</b>) Principal component cumulative contribution rate based on contribution rate arrangement.</p>
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<p>Decision result.</p>
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<p>(<b>a</b>) Resonant sensing structure; (<b>b</b>) Structural parameter settings.</p>
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<p>Relationship between resonant frequency shift and defect size.</p>
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<p>Relationship between energy attenuation and depth.</p>
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<p>(<b>a</b>) Defect top view; (<b>b</b>) Defect side view.</p>
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<p>Scanning process and distribution of bubble defects.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> parameter frequency-amplitude plot of 441 scan points near internal bubble defects.</p>
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<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> resonant frequency offset size at each scan position; (<b>b</b>) 3D plot of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> resonant frequency shift at each scan position.</p>
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<p>Reconstructed image of internal bubble defects.</p>
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<p>Reconstructed image of internal fiber breakage defects.</p>
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<p>Aircraft skin. (<b>a</b>) Top view; (<b>b</b>) Cross section diagram.</p>
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<p>Testing experimental platform.</p>
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<p>Frequency amplitude plot of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> parameter for all scanning positions.</p>
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<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> resonant frequency offset size at each scan position; (<b>b</b>) 3D plot of <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </semantics></math> resonant frequency shift at each scan position; (<b>c</b>) Reconstructed images of aircraft skin composite defects.</p>
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22 pages, 2765 KiB  
Article
New Combined Metric for Full-Reference Image Quality Assessment
by Mariusz Frackiewicz, Łukasz Machalica and Henryk Palus
Symmetry 2024, 16(12), 1622; https://doi.org/10.3390/sym16121622 - 7 Dec 2024
Viewed by 667
Abstract
In recent years, many new metrics highly correlated with the Mean Opinion Score (MOS) have been proposed for assessing image quality through Full-Reference Image Quality Assessment (FR-IQA) methods, such as MDSI, HPSI, and GMSD. Eight of these selected metrics, which compare reference and [...] Read more.
In recent years, many new metrics highly correlated with the Mean Opinion Score (MOS) have been proposed for assessing image quality through Full-Reference Image Quality Assessment (FR-IQA) methods, such as MDSI, HPSI, and GMSD. Eight of these selected metrics, which compare reference and distorted images in a symmetrical manner, are briefly described in this article, and their performance is evaluated using correlation criteria (PLCC, SROCC, and KROCC), as well as RMSE. The aim of this paper is to develop a new, efficient quality index based on a combination of several high-performance metrics already utilized in the field of Image Quality Assessment (IQA). The study was conducted on four benchmark image databases (TID2008, TID2013, KADID-10k, and PIPAL) and identified the three best-performing metrics for each database. The paper introduces a New Combined Metric (NCM), which is a weighted sum of three component metrics, and demonstrates its superiority over each of its component metrics across all the examined databases. An optimization method for determining the weights of the NCM is also presented. Additionally, an alternative version of the combined metric, based on the fastest metrics and employing symmetric calculations for pairs of compared images, is discussed. This version also demonstrates strong performance. Full article
(This article belongs to the Section Computer)
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<p>Reference images of the TID2008 and TID2013 databases [<a href="#B21-symmetry-16-01622" class="html-bibr">21</a>].</p>
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<p>Reference images of the KADID-10k database [<a href="#B23-symmetry-16-01622" class="html-bibr">23</a>].</p>
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<p>Examples of reference images from the PIPAL database [<a href="#B24-symmetry-16-01622" class="html-bibr">24</a>].</p>
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<p>Scatter plots of subjective MOS against IQA metrics obtained from the TID2008 database.</p>
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<p>Scatter plots of subjective MOS against IQA metrics obtained from the TID2013 database.</p>
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<p>Scatter plots of subjective MOS against IQA metrics obtained from the KADID-10k database.</p>
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<p>Scatter plots of subjective MOS against IQA metrics obtained from the PIPAL database.</p>
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25 pages, 13748 KiB  
Article
Research on Stability of Removal Function in Figuring Process of Mandrel of X-Ray-Focusing Mirror with Variable Curvature
by Jiadai Xue, Yuhao Li, Mingyang Gao, Dongyun Gu, Yanlin Wu, Yanwen Liu, Yuxin Fan, Peng Zheng, Wentao Chen, Zhigao Chen, Zheng Qiao, Yuan Jin, Fei Ding, Yangong Wu and Bo Wang
Micromachines 2024, 15(12), 1415; https://doi.org/10.3390/mi15121415 - 25 Nov 2024
Viewed by 540
Abstract
Over the past 30 years, researchers have developed X-ray-focusing telescopes by employing the principle of total reflection in thin metal films. The Wolter-I focusing mirror with variable-curvature surfaces demands high precision. However, there has been limited investigation into the removal mechanisms for variable-curvature [...] Read more.
Over the past 30 years, researchers have developed X-ray-focusing telescopes by employing the principle of total reflection in thin metal films. The Wolter-I focusing mirror with variable-curvature surfaces demands high precision. However, there has been limited investigation into the removal mechanisms for variable-curvature X-ray mandrels, which are crucial for achieving the desired surface roughness and form accuracy, especially in reducing mid-spatial frequency (MSF) errors. It is essential to incorporate flexible control in deterministic small-tool polishing to improve the tool’s adaptability to curvature variations and achieve stable, Gaussian-like tool influence functions (TIFs). In this paper, we introduce a curvature-adaptive prediction model for compliance figuring, based on the Preston hypothesis, using a compliant shaping tool with high slurry absorption and retention capabilities. This model predicts the compliance figuring process of variable-curvature symmetrical mandrels for X-ray grazing incidence mirrors by utilizing planar tool influence functions. Initially, a variable-curvature pressure model was developed to account for the parabolic and hyperbolic optical surfaces’ curvature characteristics. By introducing time-varying removal functions for material removal, the model establishes a variable-curvature factor function, which correlates actual downward pressure with parameters such as contact radius and contact angle, thus linking the variable-curvature surface with a planar reference. Subsequently, through analysis of the residence time distribution across different TIF models, hierarchical filtering, and PSD distribution, real-time correction of the TIFs was achieved to enable customized variable-curvature polishing. Furthermore, by applying a time-varying deconvolution algorithm, multiple rounds of flexible polishing iterations were conducted on the mandrels of a rotationally symmetric variable-curvature optical component, and the experimental results demonstrate a significant improvement in form accuracy, surface quality, and the optical performance of the mirror. Full article
(This article belongs to the Special Issue Advanced Optical Manufacturing Technologies and Applications)
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Figure 1

Figure 1
<p>Optical characteristics of mandrel with variable curvature: (<b>a</b>,<b>b</b>) Geometric analysis of hyperboloid and paraboloid of mandrel; (<b>c</b>–<b>e</b>) spatial wavelength of different sizes; (<b>f</b>) influence of spatial wavelength amplitude change on PSD curve.</p>
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<p>Variable-curvature optical surface polishing mechanism: (<b>a</b>) Variable-curvature polishing system; (<b>b</b>) variable-curvature polishing principle; (<b>c</b>) stable removal function for material removal; (<b>d</b>) time-varying removal function for material removal.</p>
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<p>Matching flexible tools with variable curvature: (<b>a</b>,<b>b</b>) TIF; (<b>c</b>–<b>e</b>) performance of polishing tools on planar, convex, and concave workpieces with large and small radii of curvature matched with different radii of curvature under different pressures.</p>
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<p>Pressure model with variable curvature: (<b>a</b>) Schematic diagram of pressure distribution of convex and concave workpieces; (<b>b</b>) variation in key parameters of plane pressure under limit conditions; (<b>c</b>,<b>d</b>) influence of distribution of plane pressure and tool curvature radius on contact radius and edge contact angle.</p>
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<p>CFP diagram of a certain frequency band with the same workpiece curvature radius.</p>
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<p>Simulation of pressure distribution on surfaces with variable curvature. (<b>a</b>,<b>b</b>) Pressure simulation results of convex and concave workpieces; (<b>c</b>,<b>d</b>) changes in pressure distribution of convex and concave workpieces with change in the workpiece curvature radius. <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math> (65 and 90 mm) and tool curvature radius <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>ρ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> </semantics></math> (25 and 40 mm).</p>
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<p>Different removal functions of four TIFs with measured and Gaussian fit in conditions of rotation speed 1400 rpm, offset 0.6 mm, dwell time 12 min, and process angles 10°, 15°, 20°, and 25°, respectively.</p>
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<p>Four TIFs with low-, medium-, and high-order filtering for variable-curvature polishing. (<b>a</b>) <span class="html-italic">TIF1</span>; (<b>b</b>) <span class="html-italic">TIF2</span>; (<b>c</b>) <span class="html-italic">TIF3</span>; (<b>d</b>) <span class="html-italic">TIF4</span>.</p>
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<p>PSD results of four TIFs: (<b>a</b>) a convex part; (<b>b</b>) a concave part.</p>
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<p>Flowchart of the nonlinear method for dwell time calculation.</p>
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<p>The mandrel figuring experiment. (<b>a</b>) Principle of figuring variable-curvature mandrel on CFP1000. (<b>b</b>) An ideal initial elliptic Gaussian removal function. (<b>c</b>) Initial profile of #24 mandrel and twice figuring with calculation and prediction. (<b>d</b>) Profile accuracy improving results for paraboloid and hyperboloid.</p>
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<p>The iterative figuring of the variable-curvature optical surface: (<b>a</b>) The enhancement of the form accuracy; (<b>b</b>) the improvement of surface roughness with the star-point method; (<b>c</b>) the promotion of the optical performance with the visible light observation approach; (<b>d</b>) the PSF image and the encircled energy curve before iterative figuring; (<b>e</b>) the PSF image and the encircled energy curve after iterative figuring.</p>
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<p>The iterative figuring of the variable-curvature optical surface: (<b>a</b>) The enhancement of the form accuracy; (<b>b</b>) the improvement of surface roughness with the star-point method; (<b>c</b>) the promotion of the optical performance with the visible light observation approach; (<b>d</b>) the PSF image and the encircled energy curve before iterative figuring; (<b>e</b>) the PSF image and the encircled energy curve after iterative figuring.</p>
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17 pages, 2970 KiB  
Article
Synthesis and Characterization of New Organoammonium, Thiazolium, and Pyridinium Triiodide Salts: Crystal Structures, Polymorphism, and Thermal Stability
by Madhushi Bandara, Khadijatul Kobra, Spencer R. Watts, Logan Grady, Connor Hudson, Claudina Veas, Timothy W. Hanks, Rakesh Sachdeva, Jorge Barroso, Colin D. McMillen and William T. Pennington
Crystals 2024, 14(12), 1020; https://doi.org/10.3390/cryst14121020 - 25 Nov 2024
Viewed by 616
Abstract
Triiodide salts are of interest for a variety of applications, including but not limited to electrochemical and photochemical devices, as antimicrobials and disinfectants, in supramolecular chemistry and crystal engineering, and in ionic liquids and deep eutectic solvents. Our work has focused on the [...] Read more.
Triiodide salts are of interest for a variety of applications, including but not limited to electrochemical and photochemical devices, as antimicrobials and disinfectants, in supramolecular chemistry and crystal engineering, and in ionic liquids and deep eutectic solvents. Our work has focused on the design of salt–solvate cocrystals and deep eutectic solvents in which the triiodide anion interacts as a halogen bond acceptor with organoiodine molecules. To understand structure–property relationships in these hybrid materials, it is essential to have benchmark structural and physical data for the parent triiodide salt component. Herein, we report the structure and thermal properties of eight new triiodide salts, three of which exhibit polymorphism: tetrapentylammonium triiodide (1a and 1b), tetrahexylammonium triiodide (2), trimethylphenylammonium triiodide (3), trimethylbenzylammonium triiodide (4), triethylbenzylammonium triiodide (5), tri-n-butylbenzylammonium triiodide (6), 3-methylbenzothizolium triiodide (7a and 7b), and 2-chloro-1-methylpyridinium triiodide (8a and 8b). The structural features of the triiodide anion, Raman spectroscopic analysis, and melting and thermal decomposition behavior of the salts, as well as a computational analysis of the polymorphs, are discussed. The polymorphic pairs here are distinguished by symmetric versus asymmetric triiodide anions, as well as different packing patterns. Computational analyses revealed more subtle differences in their isosurface plots. Importantly, this study provides reference data for these new triiodide salts for comparison to hybrid cocrystals and deep eutectic solvents formed from their combination with various organoiodines. Full article
(This article belongs to the Special Issue Crystalline Materials: Polymorphism)
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Figure 1
<p>Organoammonium, thiazolium, and pyridinium cations prepared as triiodide salts in the present study, shown with shorthand notations and numbers used in this study.</p>
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<p>Crystal structures of triiodides <b>1</b>–<b>8</b>, shown as 50% probability ellipsoids. Hydrogen atoms are omitted for clarity. Note that extended asymmetric units are shown for triiodides <b>1b</b>, <b>2</b>, <b>7b</b>, and <b>8b</b> in order to show complete molecules. In these cases, the central atom of the triiodide anions sits on a special position, with the remaining two iodine atoms produced from only one unique iodine atom at a general position. In this way, there are two unique half-triiodides in the asymmetric unit of <b>7b</b> and <b>8b</b>, both expanded to their full triiodide geometry but shown with only one cation.</p>
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<p>Packing diagrams of NPe<sub>4</sub>I<sub>3</sub> polymorphs <b>1a</b> and <b>1b</b>, viewed along the <span class="html-italic">a</span>-axis and <span class="html-italic">b</span>-axis, respectively. Unit cell axis designations are <span class="html-italic">a</span>-axis = red, <span class="html-italic">b</span>-axis = green, and <span class="html-italic">c</span>-axis = blue. Iodine atoms are shown in dark purple, nitrogen atoms in light purple, carbon atoms in gray, and hydrogen atoms in white.</p>
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<p>Packing diagrams (top) of NMeBenzoSI<sub>3</sub> polymorphs <b>7a</b> and <b>7b</b>, viewed along the <span class="html-italic">b</span>-axis, and I···S/S···I interactions (bottom, blue dashed lines) occurring in polymorphs <b>7a</b> and <b>7b</b>. Unit cell axis designations are <span class="html-italic">a</span>-axis = red, <span class="html-italic">b</span>-axis = green, and <span class="html-italic">c</span>-axis = blue. Iodine atoms are shown in dark purple, nitrogen atoms in light purple, sulfur atoms in yellow, carbon atoms in gray, and hydrogen atoms in white.</p>
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<p>Packing diagrams (top) of ClMePyrI<sub>3</sub> polymorphs <b>8a</b> and <b>8b</b>, viewed along the <span class="html-italic">a</span>-axis and <span class="html-italic">b</span>-axis, respectively, and selected intermolecular interactions (bottom, blue dashed lines) occurring in polymorphs <b>8a</b> and <b>8b</b>. Unit cell axis designations are <span class="html-italic">a</span>-axis = red, <span class="html-italic">b</span>-axis = green, and <span class="html-italic">c</span>-axis = blue. Iodine atoms are shown in dark purple, nitrogen atoms in light purple, chlorine atoms in green, carbon atoms in gray, and hydrogen atoms in white. Only the major component of the cation disorder in <b>8b</b> is shown.</p>
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<p>Non-covalent interactions (NCIs) plot of the triiodide polymorphs. Green isosurfaces represent weak van der Waals interactions. The corresponding isovalues are 0.3 a.u.</p>
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28 pages, 10407 KiB  
Article
On the Viscous Ringed Disk Evolution in the Kerr Black Hole Spacetime
by Daniela Pugliese, Zdenek Stuchlík and Vladimir Karas
Universe 2024, 10(12), 435; https://doi.org/10.3390/universe10120435 - 22 Nov 2024
Viewed by 529
Abstract
Supermassive black holes (SMBHs) are observed in active galactic nuclei interacting with their environments, where chaotical, discontinuous accretion episodes may leave matter remnants orbiting the central attractor in the form of sequences of orbiting toroidal structures, with strongly different features as different rotation [...] Read more.
Supermassive black holes (SMBHs) are observed in active galactic nuclei interacting with their environments, where chaotical, discontinuous accretion episodes may leave matter remnants orbiting the central attractor in the form of sequences of orbiting toroidal structures, with strongly different features as different rotation orientations with respect to the central Kerr BH. Such ringed structures can be characterized by peculiar internal dynamics, where co-rotating and counter-rotating accretion stages can be mixed and distinguished by tori interaction, drying–feeding processes, screening effects, and inter-disk jet emission. A ringed accretion disk (RAD) is a full general relativistic model of a cluster of toroidal disks, an aggregate of axi-symmetric co-rotating and counter-rotating disks orbiting in the equatorial plane of a single central Kerr SMBH. In this work, we discuss the time evolution of a ringed disk. Our analysis is a detailed numerical study of the evolving RAD properties formed by relativistic thin disks, using a thin disk model and solving a diffusion-like evolution equation for an RAD in the Kerr spacetime, adopting an initial wavy (ringed) density profile. The RAD reaches a single-disk phase, building accretion to the inner edge regulated by the inner edge boundary conditions. The mass flux, the radial drift, and the disk mass of the ringed disk are evaluated and compared to each of its disk components. During early inter-disk interaction, the ring components spread, destroying the internal ringed structure and quickly forming a single disk with timescales governed by ring viscosity prescriptions. Different viscosities and boundary conditions have been tested. We propose that a system of viscously spreading accretion rings can originate as a product of tidal disruption of a multiple stellar system that comes too close to an SMBH. Full article
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Figure 1
<p>Geodesic structure of the Kerr spacetime. Spin <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>x</mi> </mrow> </msub> <mo>≡</mo> <mn>0.37258</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>γ</mi> </mrow> <mi>I</mi> </msubsup> <mo>≡</mo> <mn>0.3137</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>a</mi> <mrow> <mi>γ</mi> </mrow> <mrow> <mi>I</mi> <mi>I</mi> </mrow> </msubsup> <mo>≡</mo> <mn>0.6383</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> are defined in Section Constraints from <b>RAD</b> Systems. The black region is <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>&lt;</mo> <msub> <mi>r</mi> <mo>+</mo> </msub> </mrow> </semantics></math>, with <math display="inline"><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math> being the outer horizon of the Kerr geometry, the gray region is <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>&lt;</mo> <msubsup> <mi>r</mi> <mrow> <mi>ϵ</mi> </mrow> <mo>+</mo> </msubsup> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>ϵ</mi> </mrow> <mo>+</mo> </msubsup> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> is the outer ergosurface on the attractor equatorial plane. Radius <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>s</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> </semantics></math> is the marginally stable orbit, <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mi>γ</mi> </mrow> <mo>±</mo> </msubsup> </semantics></math> is the marginally circular orbit, and <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>b</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> </semantics></math> is the marginally bounded orbit for counter-rotating and co-rotating particles, respectively. Radii <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mrow> <mo>[</mo> <mi>m</mi> <mi>b</mi> <mi>o</mi> <mo>]</mo> </mrow> <mo>±</mo> </msubsup> <mo>,</mo> <msubsup> <mi>r</mi> <mrow> <mo>[</mo> <mi>γ</mi> <mo>]</mo> </mrow> <mo>±</mo> </msubsup> </mrow> </semantics></math> are in <a href="#universe-10-00435-t001" class="html-table">Table 1</a>. <b>The upper-left</b> (<b>center</b>) panel shows the situation for the co-rotating (counter-rotating) orbits. <b>The upper-right</b> panel shows the co-rotating and counter-rotating geodesic structures, and <b>the left-bottom</b> panel is a close-up view. Colored stripes in the panels are the regions locating the disk’s inner edges <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mrow> <mo>×</mo> </mrow> <mo>±</mo> </msubsup> <mo>∈</mo> <mrow> <mo>]</mo> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>b</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> <mo>,</mo> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>s</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> <mo>]</mo> </mrow> </mrow> </semantics></math>. The bottom-center panel shows the co-rotating and counter-rotating geodesic structures where dotted stripes in the panels are the regions locating the disk’s inner edges <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mrow> <mo>×</mo> </mrow> <mo>±</mo> </msubsup> <mo>∈</mo> <mrow> <mo>]</mo> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>b</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> <mo>,</mo> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>s</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> <mo>]</mo> </mrow> </mrow> </semantics></math>, and colored stripes locate the disk’s centers <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>c</mi> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> <mo>±</mo> </msubsup> <mo>∈</mo> <mrow> <mo>[</mo> <msubsup> <mi>r</mi> <mrow> <mo>[</mo> <mi>m</mi> <mi>s</mi> <mi>o</mi> <mo>]</mo> </mrow> <mo>±</mo> </msubsup> <mo>,</mo> <msubsup> <mi>r</mi> <mrow> <mo>[</mo> <mi>m</mi> <mi>b</mi> <mi>o</mi> <mo>]</mo> </mrow> <mo>±</mo> </msubsup> <mo>[</mo> </mrow> </mrow> </semantics></math>. <b>The bottom-right</b> panel shows the co-rotating and counter-rotating geodesic structures for <b>BHs</b> spins <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math>. All quantities are dimensionless.</p>
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<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> for the <span class="html-italic">ℓ</span>co-rotating rings orbiting in the Kerr spacetime with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>∓</mo> <mn>0.9</mn> </mrow> </semantics></math> for counter-rotating <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>+</mo> <mo>)</mo> </mrow> </semantics></math> and co-rotating <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>−</mo> <mo>)</mo> </mrow> </semantics></math> flows, respectively, at different times <span class="html-italic">t</span> signed on the panel (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">1</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, model <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </semantics></math> (<b>upper-right</b> panel) is composed of one counter-rotating ring). All the quantities are dimensionless.</p>
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<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f002" class="html-fig">Figure 2</a> for the Kerr spacetime with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.9</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). Dimensionless time values for the different stages of evolution are signed on the panel. All the quantities are dimensionless. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">1</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, system <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>+</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </semantics></math> is an <span class="html-italic">ℓ</span>co-rotating triplet of counter-rotating tori <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. Notation <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>±</mo> <mo>)</mo> </mrow> </semantics></math> is for counter-rotating/co-rotating fluids, respectively. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>+</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>+</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>+</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings: <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f002" class="html-fig">Figure 2</a>.</p>
Full article ">Figure 4
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>∓</mo> <mn>0.9</mn> </mrow> </semantics></math> at different times <span class="html-italic">t</span> for counter-rotating and co-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All the quantities are dimensionless. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>), with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">2</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient.</p>
Full article ">Figure 5
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two clusters of <span class="html-italic">ℓ</span>co-rotating disks from the integration in <a href="#universe-10-00435-f004" class="html-fig">Figure 4</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>), with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">2</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>+</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>+</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>+</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f004" class="html-fig">Figure 4</a>.</p>
Full article ">Figure 6
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.9</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively, at different times <span class="html-italic">t</span> signed on the panels (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All the quantities are dimensionless. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>), with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">3</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient.</p>
Full article ">Figure 7
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating ring couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f006" class="html-fig">Figure 6</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">3</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>+</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">E</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>+</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">H</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>+</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">F</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">G</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f006" class="html-fig">Figure 6</a>.</p>
Full article ">Figure 8
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) at different times <span class="html-italic">t</span> signed on the panels in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>∓</mo> <mn>0.9</mn> </mrow> </semantics></math> (for counter-rotating and co-rotating fluids, respectively) (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, model <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </semantics></math> (<b>upper-right</b> panel) is composed of one counter-rotating ring).</p>
Full article ">Figure 9
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) at different times <span class="html-italic">t</span> signed on the panels in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.9</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">5</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, model <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </semantics></math> (<b>upper-right</b> panel) is composed of one counter-rotating ring).</p>
Full article ">Figure 10
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating ring couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f008" class="html-fig">Figure 8</a>. The initial density profiles are combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, system <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>+</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </semantics></math> is a <span class="html-italic">ℓ</span>co-rotating triplet of counter-rotating tori <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. Notation <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>±</mo> <mo>)</mo> </mrow> </semantics></math> is for counter-rotating/co-rotating fluids, respectively). System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>+</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>+</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>+</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings: <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f008" class="html-fig">Figure 8</a>.</p>
Full article ">Figure 11
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed of two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f009" class="html-fig">Figure 9</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">5</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, system <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>+</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </semantics></math> is an <span class="html-italic">ℓ</span>co-rotating triplet of counter-rotating tori <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">A</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. Notation <math display="inline"><semantics> <mrow> <mo>(</mo> <mo>±</mo> <mo>)</mo> </mrow> </semantics></math> is for counter-rotating/co-rotating fluids, respectively). System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>+</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>+</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">B</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>+</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings: <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">D</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">C</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f009" class="html-fig">Figure 9</a>.</p>
Full article ">Figure 12
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All the quantities are dimensionless. Dimensionless time values for the different stages of evolution are signed on the panel. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">1</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, model <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math> (<b>upper-right</b> panel) is composed of one counter-rotating ring).</p>
Full article ">Figure 13
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f012" class="html-fig">Figure 12</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">1</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>+</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating triplet of counter-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>+</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>+</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>+</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f012" class="html-fig">Figure 12</a>.</p>
Full article ">Figure 14
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively, at different times <span class="html-italic">r</span> signed on the panels (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">O</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">2</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient.</p>
Full article ">Figure 15
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f014" class="html-fig">Figure 14</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">O</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">1</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">O</mi> <mo>+</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>+</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>+</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f014" class="html-fig">Figure 14</a>.</p>
Full article ">Figure 16
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively, at different times <span class="html-italic">t</span> signed on the panels (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">O</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">3</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient.</p>
Full article ">Figure 17
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f016" class="html-fig">Figure 16</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">O</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">3</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">O</mi> <mo>+</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>+</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">P</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>+</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating quadruplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">Q</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f016" class="html-fig">Figure 16</a>.</p>
Full article ">Figure 18
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All the quantities are dimensionless. Dimensionless time values for the different stages of evolution are signed on the panel. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, model <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math> (upper-right panel) is composed of one counter-rotating ring).</p>
Full article ">Figure 19
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f018" class="html-fig">Figure 18</a>. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>+</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating triplet of counter-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>+</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>+</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>+</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f018" class="html-fig">Figure 18</a>.</p>
Full article ">Figure 20
<p>The mass flux <math display="inline"><semantics> <mi mathvariant="script">F</mi> </semantics></math> of Equation (<a href="#FD11-universe-10-00435" class="html-disp-formula">11</a>) for the <span class="html-italic">ℓ</span>co-rotating rings couples in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All the quantities are dimensionless. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Density evolution is shown in <a href="#universe-10-00435-f018" class="html-fig">Figure 18</a>.</p>
Full article ">Figure 21
<p>Radial drift of the fluid density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math>. Solution <math display="inline"><semantics> <mrow> <msup> <mo>Σ</mo> <mo>′</mo> </msup> <mo>≡</mo> <msub> <mo>∂</mo> <mi>r</mi> </msub> <mo>Σ</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> in the plane <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>−</mo> <mi>r</mi> </mrow> </semantics></math> for co-rotating and counter-rotating fluids orbiting the <b>BH</b> spacetime with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math>, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All quantities are dimensionless. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Density evolution is shown in <a href="#universe-10-00435-f018" class="html-fig">Figure 18</a>. The integration ranges consider the <b>RAD</b> inner <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math>, center <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">c</mi> <mo>)</mo> </mrow> </semantics></math>, and outer <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> tori. The radial ranges distinguishing the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">c</mi> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> components are defined by the radii <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>IN</mi> </msub> <mo>&lt;</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>≤</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>&lt;</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> </mrow> </semantics></math>. For the <span class="html-italic">ℓ</span>co-rotating seeds <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math>, the inner torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math> is defined in the range <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mi>IN</mi> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>]</mo> </mrow> </semantics></math> and the outer torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="normal">o</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> <mo>]</mo> </mrow> </semantics></math>, where the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </semantics></math> seed there is <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math>, and the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math> seed there is <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>22</mn> </mrow> </semantics></math>. For the combined independent system <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>+</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math>, there is <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>=</mo> <mn>22</mn> <mo>,</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>28</mn> <mo>)</mo> </mrow> </semantics></math>, where the inner torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mi>IN</mi> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math>; the center torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">c</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>]</mo> </mrow> </semantics></math>; the outer torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> <mo>]</mo> </mrow> </semantics></math>. The systems <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> <mo>+</mo> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics></math> for two general models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> consider the sums two evolutions apart, which is the solution of <math display="inline"><semantics> <mrow> <msup> <mo>Σ</mo> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mo>Σ</mo> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, while <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>+</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics></math> is the solution of <math display="inline"><semantics> <mrow> <msup> <mo>Σ</mo> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mo>Σ</mo> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 22
<p>Evolution of the disk mass as a function of the dimensionless time. For large times, the curves decrease approximately with a power law of <math display="inline"><semantics> <msup> <mi>t</mi> <mi>s</mi> </msup> </semantics></math> for co-rotating and counter-rotating fluids orbiting the <b>BH</b> spacetime with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math>, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All quantities are dimensionless. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">4</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Density evolution is shown in <a href="#universe-10-00435-f018" class="html-fig">Figure 18</a>. The integration ranges consider the <b>RAD</b> inner <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math>, center <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">c</mi> <mo>)</mo> </mrow> </semantics></math>, and outer <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> tori. The radial ranges distinguishing the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">c</mi> <mo>)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> components are defined by the radii <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mi>IN</mi> </msub> <mo>&lt;</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>≤</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>&lt;</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> </mrow> </semantics></math>. For the <span class="html-italic">ℓ</span>co-rotating seeds <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math>, the inner torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math> is defined in the range <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mi>IN</mi> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>]</mo> </mrow> </semantics></math>, and the outer torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> is defined as <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> <mo>]</mo> </mrow> </semantics></math>, where for the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </semantics></math> seed, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math>, and for the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math> seed, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>22</mn> </mrow> </semantics></math>. For the combined independent system <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>+</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>=</mo> <mn>22</mn> <mo>,</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>28</mn> <mo>)</mo> </mrow> </semantics></math>, where the inner torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">i</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mi>IN</mi> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math>; the center torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">c</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>]</mo> </mrow> </semantics></math>; the outer torus <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">o</mi> <mo>)</mo> </mrow> </semantics></math> is defined in <math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi>r</mi> <mo>*</mo> </msub> <mo>,</mo> <msub> <mi>r</mi> <mo>∞</mo> </msub> <mo>]</mo> </mrow> </semantics></math>. The systems <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> <mo>+</mo> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics></math> for two general models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> consider the sums of the two evolutions apart, which is the sum of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>π</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>π</mi> </mrow> </semantics></math>, while <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>+</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics></math> is the mass of the composed system formed by <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="script">Q</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics></math> (the two evaluations clearly are coincident).</p>
Full article ">Figure 23
<p>Evolution of the surface density <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> (<span class="html-italic">ℓ</span>co-rotating rings) in the Kerr metric with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>∓</mo> <mn>0.2</mn> </mrow> </semantics></math> for counter-rotating and co-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). All the quantities are dimensionless. Dimensionless time values for the different stages of evolution are signed on the panel. The initial density profiles are the models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> of <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with the boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">5</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. Note, model <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math> (<b>upper-right</b> panel) is composed of one counter-rotating ring).</p>
Full article ">Figure 24
<p>Combined (independent) evolution of the surface densities <math display="inline"><semantics> <mo>Σ</mo> </semantics></math> of the <span class="html-italic">ℓ</span>counter-rotating rings couples composed by two sets of <span class="html-italic">ℓ</span>co-rotating rings from the integration in <a href="#universe-10-00435-f023" class="html-fig">Figure 23</a> for the Kerr spacetime with spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>±</mo> <mn>0.2</mn> </mrow> </semantics></math> for co-rotating and counter-rotating fluids, respectively (see <a href="#universe-10-00435-t003" class="html-table">Table 3</a>). Dimensionless time values for the different stages of evolution are signed on the panel. All the quantities are dimensionless. The initial density profiles are the combinations of models <math display="inline"><semantics> <mrow> <mo>{</mo> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> <mo>}</mo> </mrow> </semantics></math> defined in <a href="#universe-10-00435-t002" class="html-table">Table 2</a>, with boundary condition <math display="inline"><semantics> <mrow> <mo>[</mo> <mn mathvariant="bold">5</mn> <mo>]</mo> </mrow> </semantics></math> of Equation (<a href="#FD16-universe-10-00435" class="html-disp-formula">16</a>). <math display="inline"><semantics> <mi>ν</mi> </semantics></math> is the viscosity coefficient. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>+</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating triplet of counter-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>+</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>+</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>co-rotating quadruplet of co-rotating rings <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">M</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. System <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>+</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </semantics></math> is the <span class="html-italic">ℓ</span>counter-rotating triplet <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>+</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">L</mi> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi mathvariant="normal">C</mi> <mo>−</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">N</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The radial range has been adapted according to the combination of the independent integrations ranges. The viscosity coefficients <math display="inline"><semantics> <mi>ν</mi> </semantics></math> are fixed according to <a href="#universe-10-00435-f023" class="html-fig">Figure 23</a>.</p>
Full article ">Figure A1
<p>On the null flux <math display="inline"><semantics> <mi mathvariant="script">F</mi> </semantics></math> condition: <a href="#app1-universe-10-00435" class="html-app">Appendix A</a>. Radii <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mo>⊙</mo> </mrow> <mo>±</mo> </msubsup> </semantics></math> for counter-rotating and co-rotating fluids, respectively, are plotted as functions of the central Kerr <b>BH</b> spin <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math>. All quantities are dimensionless. On <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mo>⊙</mo> </mrow> <mo>±</mo> </msubsup> </semantics></math>, there is <math display="inline"><semantics> <mrow> <mi mathvariant="script">F</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msup> <mo>Σ</mo> <mo>′</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The black region is <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>&lt;</mo> <msub> <mi>r</mi> <mo>+</mo> </msub> </mrow> </semantics></math>, with <math display="inline"><semantics> <msub> <mi>r</mi> <mo>+</mo> </msub> </semantics></math> being the outer horizon of the Kerr geometry, and the gray region is <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>&lt;</mo> <msubsup> <mi>r</mi> <mrow> <mi>ϵ</mi> </mrow> <mo>+</mo> </msubsup> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>ϵ</mi> </mrow> <mo>+</mo> </msubsup> <mo>=</mo> <mn>2</mn> <mi>M</mi> </mrow> </semantics></math> is the outer ergosurface on the attractor equatorial plane. The geodesic structure of the Kerr spacetime is also plotted: radius <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>s</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> </semantics></math> is the marginally stable orbit, <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mi>γ</mi> </mrow> <mo>±</mo> </msubsup> </semantics></math> is the marginally circular orbit, and <math display="inline"><semantics> <msubsup> <mi>r</mi> <mrow> <mi>m</mi> <mi>b</mi> <mi>o</mi> </mrow> <mo>±</mo> </msubsup> </semantics></math> is the marginally bounded orbit for counter-rotating and co-rotating particles, respectively. At <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mo>∝</mo> </msub> <mo>=</mo> <mn>0.8117</mn> </mrow> </semantics></math>, there is <math display="inline"><semantics> <mrow> <msubsup> <mi>r</mi> <mo>⊙</mo> <mo>−</mo> </msubsup> <mo>=</mo> <msubsup> <mi>r</mi> <mrow> <mi>ϵ</mi> </mrow> <mo>+</mo> </msubsup> </mrow> </semantics></math>.</p>
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20 pages, 3466 KiB  
Article
Symmetric Tridiagonal Eigenvalue Solver Across CPU Graphics Processing Unit (GPU) Nodes
by Erika Hernández-Rubio, Alberto Estrella-Cruz, Amilcar Meneses-Viveros, Jorge Alberto Rivera-Rivera, Liliana Ibeth Barbosa-Santillán and Sergio Víctor Chapa-Vergara
Appl. Sci. 2024, 14(22), 10716; https://doi.org/10.3390/app142210716 - 19 Nov 2024
Viewed by 636
Abstract
In this work, an improved and scalable implementation of Cuppen’s algorithm for diagonalizing symmetric tridiagonal matrices is presented. This approach uses a hybrid-heterogeneous parallelization technique, taking advantage of GPU and CPU in a distributed hardware architecture. Cuppen’s algorithm is a theoretical concept and [...] Read more.
In this work, an improved and scalable implementation of Cuppen’s algorithm for diagonalizing symmetric tridiagonal matrices is presented. This approach uses a hybrid-heterogeneous parallelization technique, taking advantage of GPU and CPU in a distributed hardware architecture. Cuppen’s algorithm is a theoretical concept and a powerful tool in various scientific and engineering applications. It is a key player in matrix diagonalization, finding its use in Functional Density Theory (FDT) and Spectral Clustering. This highly efficient and numerically stable algorithm computes eigenvalues and eigenvectors of symmetric tridiagonal matrices, making it a crucial component in many computational methods. One of the challenges in parallelizing algorithms for GPUs is their limited memory capacity. However, we overcome this limitation by utilizing multiple nodes with both CPUs and GPUs. This enables us to solve subproblems that fit within the memory of each device in parallel and subsequently combine these subproblems to obtain the complete solution. The hybrid-heterogeneous approach proposed in this work outperforms the state-of-the-art libraries and also maintains a high degree of accuracy in terms of orthogonality and quality of eigenvectors. Furthermore, the sequential version of the algorithm with our approach in this work demonstrates superior performance and potential for practical use. In the experiments carried out, it was possible to verify that the performance of the implementation that was carried out scales by 2× using two graphic cards in the same node. Notably, Symmetric Tridiagonal Eigenvalue Solvers are fundamental to solving more general eigenvalue problems. Additionally, the divide-and-conquer approach employed in this implementation can be extended to singular value solvers. Given the wide range of eigenvalue problems encountered in scientific and engineering domains, this work is essential in advancing computational methods for efficient and accurate matrix diagonalization. Full article
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<p>Heterogeneous parallel architecture.</p>
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<p>Process flow.</p>
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<p>Time-symmetric tridiagonal eigensystem.</p>
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<p>Eigenpairs error <math display="inline"><semantics> <msub> <mrow> <mo>∥</mo> <mi>A</mi> <mi>Q</mi> <mo>−</mo> <mi>Q</mi> <mo>Λ</mo> <mo>∥</mo> </mrow> <mi>F</mi> </msub> </semantics></math>.</p>
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<p>Orthogonality error <math display="inline"><semantics> <mrow> <mrow> <mo>∥</mo> <mi>Q</mi> </mrow> <msup> <mi>Q</mi> <mi>T</mi> </msup> <msub> <mrow> <mo>−</mo> <mi>I</mi> <mo>∥</mo> </mrow> <mi>F</mi> </msub> </mrow> </semantics></math>.</p>
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18 pages, 6972 KiB  
Article
The Design and Experimental Research on a High-Frequency Rotary Directional Valve
by Shunming Hua, Siqiang Liu, Zhuo Qiu, Xiaojun Wang, Xuechang Zhang and Huijuan Zhang
Processes 2024, 12(11), 2600; https://doi.org/10.3390/pr12112600 - 19 Nov 2024
Viewed by 475
Abstract
A directional valve is a core component of the electro-hydraulic shakers in fatigue testing machines, controlling the cylinder or motor that drives the piston for reciprocating linear or rotary motion. In this article, a high-speed rotating directional valve with a symmetrical flow channel [...] Read more.
A directional valve is a core component of the electro-hydraulic shakers in fatigue testing machines, controlling the cylinder or motor that drives the piston for reciprocating linear or rotary motion. In this article, a high-speed rotating directional valve with a symmetrical flow channel layout is designed to optimize the force on the valve core of the directional valve. A comparative analysis is conducted on the flow capacity of valve ports with different shapes. A steady-state hydrodynamic mathematical model is established for the valve core, the theoretical analysis results of which are verified through a Computational Fluid Dynamics (CFD) simulation of the fluid domain inside the directional valve. A prototype of the rotatory directional valve is designed and manufactured, and an experimental platform is built to measure the hydraulic force acting on the valve core to verify the performance of the valve. The displacement curves at different commutation frequencies are also obtained. The experimental results show that the symmetrical flow channel layout can significantly optimize the hydraulic force during the movement of the valve core. Under a pressure of 1 MPa, the hydraulic cylinder driven by the prototype can achieve a sinusoidal curve output with a maximum frequency of 60 Hz and an amplitude of 2.5 mm. The innovation of this design lies in the creation of a directional valve with a symmetric flow channel layout. The feasibility of the design is verified through modeling, simulation, and experimentation, and it significantly optimizes the hydraulic forces acting on the spool. It provides us with the possibility to further improve the switching frequency of hydraulic valves and has important value in engineering applications. Full article
(This article belongs to the Section Manufacturing Processes and Systems)
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<p>Structure of rotary directional valve.</p>
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<p>Valve port shapes.</p>
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<p>Relationship between circular orifice area and valve core angle.</p>
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<p>Relationship between triangle orifice area and valve core angle.</p>
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<p>Relationship between square orifice area and valve core angle.</p>
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<p>The average flow area of the valve port and its ratio to the maximum flow area.</p>
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<p>Fluid domain division.</p>
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<p>Mesh quality test results.</p>
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<p>Cloud diagrams of the valve port velocity with different port openings. (<b>a</b>) The inlet I is used as the pressure inlet. (<b>b</b>) The inlet II is used as the pressure inlet.</p>
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<p>Vector diagrams of liquid flow at different valve openings. (<b>a</b>) The inlet I is used as the pressure inlet. (<b>b</b>) The inlet II is used as the pressure inlet.</p>
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<p>Jet angle and spool torque at different rotation angles. (<b>a</b>) Jet angle curves. (<b>b</b>) Spool torque curves.</p>
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<p>Hydraulic simulation system diagram.</p>
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<p>System flow rate and cylinder amplitude at different commutation frequencies.</p>
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<p>Flow rate and cylinder amplitude at different pressures.</p>
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<p>Flow rate and cylinder amplitude at different loads.</p>
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<p>Test platform. (<b>a</b>) Overview of experimental system. (<b>b</b>) Prototype and sensors.</p>
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<p>Hydraulic cylinder displacement curves at different commutation frequencies.</p>
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<p>The relationship between the steady-state hydraulic torque and the spool rotation angle.</p>
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<p>The torque of the spool at different pressures.</p>
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