Polar Coordinate System Research Articles

Mathematical modelling of couple stress fluid flow around a semi‐permeable sphere enclosing a solid core

AbstractThe focus of this article is to study theoretically the steady‐axisymmetric creeping flow dynamics of a couple stress fluid external to a semi‐permeable sphere containing a solid core. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behaviour often arises in addition to porous media effects. The non‐Newtonian Stokes’ couple stress fluid model features couple stresses and body couples that are absent in the classical Navier–Stokes viscous model. It provides a robust framework for simulating emulsions, complex suspensions and other liquids which possess microstructure. The physical regime is delineated into two zones – the interior of the semi‐permeable zone (region II) which engulfs the solid core and the external couple stress fluid zone (region I). The model is formulated using a spherical polar coordinate system in terms of the stream function ψ. The Brinkman‐extended Darcy model is deployed for the porous medium hydrodynamics and isotropic permeability is considered. Analytical expressions are derived for dimensionless pressure, tangential stress and the couple stress components using the method of separation of variables and Gegenbauer functions of the first kind. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the semi‐permeable zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the couple stress fluid on the semi‐permeable sphere and volumetric flow rate are also derived with corresponding expressions for the drag coefficient and non‐dimensional volumetric flow rate. The influence of permeability (k), separation parameter (l), couple stress viscosity coefficient (η), couple stress inverse length dependent parameter (λ = √(μ/η)) and couple stress viscosity ratio (τ = η/η/) on all key variables is studied graphically. Additionally, streamline contours are computed for a range of parameters including inverse permeability parameter (α = 1/k). The computations show that increasing couple stress inverse length dependent parameter (λ) greatly reduces the dimensionless volumetric flow rate in particular at high values of separation parameter (l). Flow rate is, however, markedly enhanced with permeability (k) and also couple stress viscosity parameter (η). In the presence of the solid core, a much greater drag coefficient is observed at all values of couple stress inverse length dependent parameter (λ) relative to the case without a solid core. A significant distortion in streamlines is computed with increasing separation parameter (l) with a dual vortex structure emanating. With greater couple stress viscosity parameter (τ), no tangible modification is observed in the streamline contours. The present work generalizes the earlier study of Krishnan and Shukla to consider a semi‐permeable (porous medium) outer sphere and furthermore presents detailed streamline visualizations of the creeping flow for a range of emerging parameters of relevance to non‐Newtonian chemical engineering processes including emulsion droplet dynamics in porous materials.

An inverse method for characterization of dynamic response of 2D structures under stochastic conditions

The reliable estimation of the wavenumber space (k-space) of the plates remains a long-term concern for acoustic modeling and structural dynamic behavior characterization. Most current analyses of wavenumber identification methods are based on the deterministic hypothesis. To this end, an inverse method is proposed for identifying wave propagation characteristics of two-dimensional structures under stochastic conditions, such as wavenumber space, dispersion curves, and band gaps. The proposed method is developed based on an algebraic identification scheme in the polar coordinate system framework, thus named Algebraic K-Space Identification (AKSI) technique. Additionally, a model order estimation strategy and a wavenumber filter are proposed to ensure that AKSI is successfully applied. The main benefit of AKSI is that it is a reliable and fast method under four stochastic conditions: (A) High level of signal noise; (B) Small perturbation caused by uncertainties in measurement points’ coordinates; (C) Non-periodic sampling; (D) Unknown structural periodicity. To validate the proposed method, we numerically benchmark AKSI and three other inverse methods to extract dispersion curves on three plates under stochastic conditions. One experiment is then performed on an isotropic steel plate. These investigations demonstrate that AKSI is a good in-situ k-space estimator under stochastic conditions.

An End-to-end Network for Rotary Motion Deblurring in the Polar Coordinate System

An End-to-end Network for Rotary Motion Deblurring in the Polar Coordinate System

Linear solution of the passive localization problem with combined single-station direction finding and two-station time difference information

Based on the main station’s direction measurement value and the time difference measurement value of the two stations, two definite solution equations can be listed in the polar coordinate system by directly utilizing the path difference equation and the cosine theorem, and the analytical solution can be obtained by solving the two definite solution equations jointly. Further comparison of the ranging errors shows that the ranging accuracy of the two-station directional and time-difference positioning system is only slightly worse than that of the three-station time-difference positioning system. The new results provide a better basis for the engineering design of two-station hybrid localization.

Hybrid Long Short-Term Memory and Convolutional Neural Network Architecture for Electric Submersible Pump Condition Prediction and Diagnosis

Electric submersible pump (ESP) is one of the common artificial lift technologies in offshore production systems. ESP failures are the main cause of the decline in the production efficiency of oil wells. Early warning and diagnosis of ESP failures are crucial to improve well production efficiency. In this study, a hybrid model of long short-term memory neural network and convolutional neural network (LSTM-CNN) for accurate early warning and diagnosis of ESP faults is proposed, based on electrical data as the basis of analysis. Using hyper-parameters to optimize the LSTM neural network structure and highly fit the field electrical data so that it can be applied to anomaly prediction before ESP faults, the results show that the optimized LSTM model with R2 (test set) = 0.79, root mean square error (RMSE) (test set) = 0.89, which can predict the future electrical data more accurately, and the predicted data are plotted in the polar coordinate system to simulate the ammeter card as the validation set. The ammeter card data set is simulated by collecting electrical data from different working conditions, which is expanded using data expansion, and different CNN models are trained to fine-tune the parameters using transfer learning. The results show that the GoogLeNet model has a significant diagnostic accuracy of 97%, which is 2% and 18% better than VGG16 and ResNet34, respectively. The model evaluation shows that the GoogLeNet model has good recall, F1-Score, and confusion matrix. The trained GoogLeNet model was applied to the validation set, and the hybrid LSTM-GoogLeNet model was found to be well-suited for ESP warning and diagnosis.

Simplified Power Decoupling Control of Triple Active Bridge Converter in Polar Coordinate System for EV Application

Simplified Power Decoupling Control of Triple Active Bridge Converter in Polar Coordinate System for EV Application

Analysis of Effect of Hemodynamic Parameters on Two-layered Blood Flow in a Mild Stenosed Artery

Arterial stenosis is the lessening of the arterial wall due to the growth of aberrant tissues that prevent adequate blood flow in the human circulatory system and induces cardiovascular diseases. Mild stenosis, over time, can lead to serious and permanent damage if it remains uncured. Navier-Stokes equation in cylindrical polar coordinates system has been extended by two-layered blood flow along the axial direction with appropriate boundary conditions.A steady flow through a stenosed artery has been investigated extensively using the analytical approach in the case of two-layered blood flow. Mathematical expressions for two-layer hemodynamic parameters such as velocity profile, volumetric flow rate, and effect of stenosis progression on parameters with the variation of core and peripheral layered coefficient of viscosity are derived. Moreover, pressure drop, and shear stress have been calculated analytically in an artery with and without stenosis. Peripheral viscosity has less contribution to varying velocity distribution than the core and is proportional to stenosis size.The volumetric flow rate decreases with an increasing viscosity coefficient. Pressure drop and shear stress attain maximum value in the region stenosis occur maximum height in core layer.The present work could serve as a model in biomedical engineering for the cure of vascular-related diseases and has the potential in designing the devices of this field.

Analytical model of dynamic system "long cable - submersible induction motor with fixed rotor" in polar coordinate system

Relevance. The stock of oil wells in the Russian Federation, where production is carried out with the help of electric centrifugal pumps or electric-driven vane pumps, is expanding every year. This trend will continue for at least a couple of decades. In its turn, the number of wells with readily available oil products in the commissioned stock is decreasing. This leads to the need to improve production quality and reduce energy consumption to maintain profitability and production volumes. This can be achieved by implementing vector energy-efficient control systems for submersible motors based on observers of induction motor state variables. In this case one should take into account transients in the supply cable of considerable length, which transmits energy from a frequency converter to a motor. The observers are based on adjustable mathematical models that describe electrical, electromechanical and mechanical processes in electric drive and related process automation systems as reliably as possible. Using complex and cumbersome adjustable mathematical models in digital signal processors due to limited performance and memory causes objective difficulties in obtaining solutions of increased accuracy in real time. On this basis, the development of an analytical model of the dynamic system "long cable – submersible induction motor with fixed rotor" in the polar coordinate system is an urgent task. Aim. Development of analytical description of the mathematical model of the induction motor with a fixed rotor in the polar coordinate system taking into account the influence of a long cable. Methods. Electric drive theory, electric machine theory, mathematical models of submersible induction motors and reduced long cable model, Vandermonde's determinant, analytical methods for solving differential equations. Result and conclusions. The authors have developed an analytical model of the regulated induction motor with a fixed rotor in the polar coordinate system taking into account the supply cable of considerable length. The developed analytical model is recommended for use in vector energy-efficient electric drive with application of phase discrimination principles taking into account the influence of processes in the supply cable line. Such electric drive has a potential for increasing the profitability of oil production from low- and medium-debt wells.

Simulation of the hexagonal 3D braiding process for stent preforms

Due to the complex braiding process and long development cycle of the hexagonal three-dimensional braided stent, a MatLab based computer-aided braiding method for a stent is proposed to speed up the development process. First, an oblique coordinate system for the chassis and a polar coordinate system for the chassis unit are constructed, respectively, precisely to coordinate the carrier’s movements on the chassis. Subsequently, an iterative formula delineating the trajectory of the carrier is introduced. The formula effectively translates the entire braiding process into the positional coordinates of the carrier on the chassis and the yarn heights on the mandrel during different stages. Based on the specific characteristics of the braiding process and the stent’s structure, the stent is divided into pressing and twisting sections. The interwoven pattern for both the pressing and twisting sections is determined by establishing the basic tiling form, solving the yarn interwoven sequence, judging the interwoven type and computing the number of kinks. Finally, while considering the stent's dimensional parameters and the interwoven pattern of the yarn, the spatial curve equations for the yarns in both the pressing and the twisting sections are formulated. By concatenating these equations for each section, the three-dimensional trajectory equations and a comprehensive solid model of the stent are successfully derived. Through a rigorous comparative analysis of dimensions and the braiding pattern between the three-dimensional solid model and the physical stent preform, the accuracy and fidelity of the model generated through the implementation of computer-aided braiding technology are verified.

Quaternionic metamonogenic functions in the unit disk

We construct a set of quaternionic metamonogenic functions (that is, in Ker(D+λ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Ker}\\,}}(D+\\lambda )$$\\end{document} for diverse real λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}) in the unit disk, such that every metamonogenic function is approximable in the quaternionic Hilbert module L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} of the disk. The set is orthogonal except for the small subspace of elements of orders zero and one. These functions are used to express time-dependent solutions of the imaginary-time wave equation in the polar coordinate system.

Transformer-based stroke relation encoding for online handwriting and sketches

Stroke classification for online handwriting and sketches, aimed at grouping strokes into different semantic categories, has drawn considerable attention due to its wide applications. This task is challenging since individual strokes look similar and are easily confused with each other. The key is to consider both the individual strokes and the contextual information jointly for making prediction. Previous methods are insufficient in modeling and exploiting complex contextual information of strokes. To overcome this limitation, we propose a Transformer-based model for Online Handwriting and Sketches (T-OHS), with novel relation encoding schemes to take advantage of temporal and spatial information in stroke sequence. Particularly, we introduce a coarse-to-fine hierarchical encoding approach based on the polar coordinate system for precisely modeling spatial relations between strokes. Experiments on three types of handwriting data, including online handwritten documents, diagrams, and sketches, demonstrate that our method is universal and provides state-of-the-art performance.

Investigation of Web-Based Eye-Tracking System Performance under Different Lighting Conditions for Neuromarketing

The increasing popularity of neuromarketing has led to the emergence of various measurement methods, such as webcam-based eye-tracking technology. Webcam-based eye-tracking technology is noteworthy not only for its use in laboratories but also for its ability to be applied to participants online in their natural environments through a link. However, the complexity of e-commerce interfaces necessitates high performance in eye-tracking methods. This complexity and the applicability of webcam-based eye-tracking technology in various environments have raised research questions about how its performance changes depending on the type and location of lighting. To answer these questions, experiments were conducted with 30 users in two different experimental environments illuminated by artificial and natural methods, with the lighting from the left, right, and front. Participants were asked to focus on targets located in specially prepared graphics for the experiment. In the heatmaps obtained in the eye-tracking tests, the distance and angular difference between the focal point and the target point were measured using the polar coordinate system. The findings indicate that measurements taken with lighting coming from the center were more efficient in both natural and artificial lighting types and measurements taken under natural lighting performed 24% better than artificial ones. Web camera-based eye-tracking technology is a promising method. However, detailed statistical analyses have demonstrated that for complex interfaces like e-commerce, the position and type of lighting are crucial parameters.

Generalized hydrodynamic coefficients of twin connected circular cylinders in finite water depth

In this work the generalized hydrodynamic coefficients of connected twin circular cylinders are studied. Firstly, a matched boundary element method with eigen-function expansion is applied to study the generalized hydrodynamic coefficients of floating and submerged tunnels with twin horizontal cylinders in finite water depth. By numerical examination it is found that gap resonance problem can be neglected for the submerged floating tunnels with a certain submergence depth, and the axial wave number play an important role for the added mass. Then, analytic solutions are derived for the high frequency approximations of the generalized added masses of twin connected cylinders by the eigenfunction expansion in the polar coordinate system with aid of their images about the x-axis in the infinite fluid domain. The high frequency approximations of generalized added masses decrease with increasing vibration wave number. When T/a and B/a are larger than 3 (where T is the cylinder submergence, B is the spacing distance between the twin cylinders and a is the radius of the cross-section of the cylinder), the hydrodynamic interference between the two cylinders can be neglected and the generalized added mass coefficients of connected twin cylinders can be derived from these of single isolated cylinders.

Laparoscopic Image-based Critical Action Recognition and Anticipation with Explainable Features.

Surgical workflow analysis integrates perception, comprehension, and prediction of the surgical workflow, which helps real-time surgical support systems provide proper guidance and assistance for surgeons. This paper promotes the idea of critical actions, which refer to the essential surgical actions that progress towards the fulfillment of the operation. Fine-grained workflow analysis involves recognizing current critical actions and previewing the moving tendency of instruments in the early stage of critical actions. Aiming at this, we propose a framework that incorporates operational experience to improve the robustness and interpretability of action recognition in in-vivo situations. High-dimensional images are mapped into an experience-based explainable feature space with low-dimension to achieve critical action recognition through a hierarchical classification structure. To forecast the instrument's motion tendency, we model the motion primitives in the polar coordinate system (PCS) to represent patterns of complex trajectories. Given the laparoscopy variance, the adaptive pattern recognition (APR) method, which adapts to uncertain trajectories by modifying model parameters, is designed to improve prediction accuracy. The in-vivo dataset validations show that our framework fulfilled the surgical awareness tasks with exceptional accuracy and real-time performance.

Optimization of Solar-Sail Control When a Vehicle Moves along Cyclic Heliocentric Trajectories

This paper considers the problem of constructing time-optimal trajectories for a spacecraft (SC) with a solar sail. The trajectories under consideration consist of repeated cycles of spacecraft movement to the target heliocentric orbit and back to the initial one. A model of a perfectly reflecting sail is used, which allows using the programs for optimal control of the sail angle, obtained on the basis of the Pontryagin maximum principle. The heliocentric motion is modeled in a flat polar coordinate system, and the spacecraft itself makes cyclic flights between two terrestrial planets along a closed trajectory. A boundary-value problem is formulated, in the solution of which the approach of the spacecraft to the target planet with the equalization of velocities is ensured (the encounter problem). Simulations of four cycles of Earth–Mercury–Earth and Earth–Mars–Earth motion with a characteristic acceleration of the solar sail of 0.25 mm/s2 have been carried out, for which the duration of one cycle is on average 2000 and 2341 days, respectively. Optimal sail-orientation control programs are obtained for a wide range of launch dates, and methods of searching for and choosing the initial values of conjugate variables are shown. The obtained results demonstrate the ability of a spacecraft with a solar sail to implement controlled motion along closed trajectories with a minimum duration of individual Earth–destination planet–Earth flights.

Nonlinear vibrational analysis and linearization error investigation in size-dependent resonant pressure microsensors

Dynamic and vibrational behavior of nonlinear resonant pressure micro sensors is studied in this paper using a proposed dynamic model. The basic part of the device is a microplate deflects due to external pressure in presence of electrostatic actuation. The frequency changes because of the microplate deformation is used to estimate the external pressure. For modeling of the micro sensor, the nonlinear governing equation of size dependent microplate in polar coordinate system and based on the modified couple stress theory is presented. Assuming specified number of Taylor expansion terms for electrostatic force, the linear and nonlinear ODE is obtained and nonlinear equation is solved both with multiple time scales method (MTS) and finite difference numerical method. Results indicate that considering finite terms of Taylor expansion (1 term for linear and three terms for MTS method) is the source of error in estimation of pressure or system stability. Using sufficient terms of Taylor Expansion, the error in linear and MTS solutions is vanished. Therefore, the limited values of microplate vibrational amplitude is extracted to increase the accuracy of the sensor in linear action. The effect of size dependency, applied voltage on nonlinear behavior and linearization error is also studied.

Ring artifacts correction for computed tomography image using unsupervised contrastive learning

Objective. Computed tomography (CT) is a widely employed imaging technology for disease detection. However, CT images often suffer from ring artifacts, which may result from hardware defects and other factors. These artifacts compromise image quality and impede diagnosis. To address this challenge, we propose a novel method based on dual contrast learning image style transformation network model (DCLGAN) that effectively eliminates ring artifacts from CT images while preserving texture details. Approach. Our method involves simulating ring artifacts on real CT data to generate the uncorrected CT (uCT) data and transforming them into strip artifacts. Subsequently, the DCLGAN synthetic network is applied in the polar coordinate system to remove the strip artifacts and generate a synthetic CT (sCT). We compare the uCT and sCT images to obtain a residual image, which is then filtered to extract the strip artifacts. An inverse polar transformation is performed to obtain the ring artifacts, which are subtracted from the original CT image to produce a corrected image. Main results. To validate the effectiveness of our approach, we tested it using real CT data, simulated data, and cone beam computed tomography images of the patient’s brain. The corrected CT images showed a reduction in mean absolute error by 12.36 Hounsfield units (HU), a decrease in root mean square error by 18.94 HU, an increase in peak signal-to-noise ratio by 3.53 decibels (dB), and an improvement in structural similarity index by 9.24%. Significance. These results demonstrate the efficacy of our method in eliminating ring artifacts and preserving image details, making it a valuable tool for CT imaging.

Oblique collisions of three wet spheres

Oblique collisions of three solid spheres coated with thin viscous layers are simulated, both to elucidate the interesting physics of the collision outcomes and to lay the groundwork for a new approach to modeling flows of many wet particles. Included in the analysis are fluid viscous and capillary forces, as well as solid contact and friction forces. A novel approach is developed based on a rotating polar coordinate system for each particle pair in near contact, including the possibility that a given particle is in simultaneous contact with both other particles. As the Stokes number (a dimensionless ratio of particle inertia and viscous forces) is increased, the collision outcome progresses from full agglomeration (all three particles sticking together due to viscous and capillary forces) to partial agglomeration (two particles sticking together while the third one separates) to full separation (all three particles separating post-collision). The results are also sensitive to various physical and geometrical properties, such as the ratio of fluid film thickness to particle diameter, the coefficient of friction, and the collision angles.

Research on modeling joining and joining modification method of hybrid FE-SEA model

Since the connection method at the boundary of the hybrid FE-SEA (finite element-statistical energy analysis, FE-SEA) model affects the overall calculation accuracy of the model, the missing or inaccurate connection relation will lead to a large error in the model calculation. In order to effectively solve the calculation error caused by the connection problem of hybrid FE-SEA model, this paper starts from the modeling methods and connection relations of “hybrid-point” connection, “hybrid-line” connection and “hybrid-surface” connection. In order to solve the modeling method and correction problems of the “hybrid-point” connection in the hybrid FE-SEA model, the “out-of-plane” wave motion equation at the “hybrid-point” connection was established by using the superposition principle of plane waveforms in polar coordinate system. The “hybrid-point” connection and wave-number relation in Cartesian coordinate system are studied. The radiation radius correction method of “hybrid-point” connection is proposed. An example is given to verify the effectiveness of the method. In order to solve the modeling and correction problem of “hybrid-line” in hybrid model, a “hybrid-line” connected triangular waveform function model was established by using the method of linear difference. The direct field dynamic stiffness matrix of “hybrid-line” connection in node coordinate system is studied. According to the shape function of the “hybrid-line” connection wavenumber space, a method to correct the shape function of the “hybrid-line” connection is proposed, and the validity of the method is verified.

Regular reaction dynamics in analytical form in the vicinity of symmetrical transition states. Central barrier crossings in SN2 reactions.

When an activated complex, as defined in transition state theory (TST), has a polyhedral shape, its kinetic energy is found to be diagonal in a system of spherical polar coordinates. If, in addition, the polyhedron is characterized by a high symmetry, then its dynamics considerably simplifies. An application of this approach to the most symmetrical TS known to date, i.e., that which controls the Cl- + CH3Cl → ClCH3 + Cl- SN2 nucleophilic substitution, is presented and an analytical expression of its potential energy surface is provided. In a substantial range around the saddle point, approximate equations of motion for the two components of the reaction coordinate, i.e., the antisymmetrical stretching motion of the ClCCl core and the wagging motion of the hydrogen triad, can be derived in an analytical form. During an extensive period of time, the main component of the reaction coordinate is governed by an unexpectedly simple equation of motion that depends on a single initial condition, irrespective of the other ones and of the internal energy. Reactive trajectories are observed to form a perfectly collimated bundle characterized by undetectable dispersion, thereby giving a spectacular example of regular dynamics in an anharmonic potential. Regularity and collimation are brought about by local symmetry, which is a widespread feature of potential energy surfaces. Anharmonicity is observed to influence the dynamics only at a late stage. As energy increases, trajectories tend to fan out and to deviate from the analytical equation. For the wagging motion, chaos sets in at much lower energies.