Hybrid Nanofluid Flow over a Stretching Curved Surface with Induced Magnetic Field and Homogeneous-Heterogeneous Reactions

This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering thermal radiation, suction, and magnetic boundary conditions. The nanofluid is made of water with copper and MWCNTs as nanoparticles. The equations are transformed into nonlinear ODEs and solved numerically. The model’s accuracy is confirmed by comparing it with published data. Results show that fluid velocity increases, temperature decreases, and concentration increases with the curvature radius parameter. The hybrid nanofluid is more sensitive to magnetic field changes in velocity, while the nanofluid is more sensitive to magnetic boundary coefficient changes. These insights can optimize heat and mass transfer in industrial processes like chemical reactors and wastewater treatment.

Magnetic Field Curves and Magnetic Equipotential Surfaces around Crossing Electrical Wires Replacing Classical Magnetic Field Lines

This article is based on a recent model specifically defining magnetic field values around electrical wires. With this model, calculations of field around parallel wires were obtained. Now, this model is extended with the new concept of magnetic equipotential surface to magnetic field curves around crossing wires. Cases of single, double, and triple wires are described. Subsequent article will be conducted for more general scenarios where wires are neither infinite nor parallel.

Programmable robotized‘transfer-and-jet’printing for large,3D curved electronics on complex surfaces

Large,3D curved electronics are a trend of the microelectronic industry due to their unique ability to conformally coexist with complex surfaces while retaining the electronic functions of 2D planar integrated circuit technologies.However,these curved electronics present great challenges to the fabrication processes.Here,we propose a reconfigurable,mask-free,conformal fabrication strategy with a robot-like system,called robotized‘transfer-and-jet’printing,to assemble diverse electronic devices on complex surfaces.This novel method is a ground-breaking advance with the unique capability to integrate rigid chips,flexible electronics,and conformal circuits on complex surfaces.Critically,each process,including transfer printing,inkjet printing,and plasma treating,are mask-free,digitalized,and programmable.The robotization techniques,including measurement,surface reconstruction and localization,and path programming,break through the fundamental constraints of 2D planar microfabrication in the context of geometric shape and size.The transfer printing begins with the laser lift-off of rigid chips or flexible electronics from donor substrates,which are then transferred onto a curved surface via a dexterous robotic palm.Then the robotic electrohydrodynamic printing directly writes submicrometer structures on the curved surface.Their permutation and combination allow versatile conformal microfabrication.Finally,robotized hybrid printing is utilized to successfully fabricate a conformal heater and antenna on a spherical surface and a flexible smart sensing skin on a winged model,where the curved circuit,flexible capacitive and piezoelectric sensor arrays,and rigid digital–analog conversion chips are assembled.Robotized hybrid printing is an innovative printing technology,enabling additive,noncontact and digital microfabrication for 3D curved electronics.

Magnetorheological finishing of low-gradient curved surfaces based on four-axis linkage technique

Based on the distribution characteristic of magnetic field along the polish wheel,the four-axis linkage technique is advanced to replace a standard five-axis one to figure low-gradient optical surfaces with a raster tool-path in magnetorheological finishing(MRF).After introducing the fundaments of such simplification,the figuring reachability of a four-axis system for the low-gradient optics was theoretically analyzed.Further validation including magnetic field intensity and influence function characteristic was performed to establish its application.To demonstrate the correctness,feasibility and applicability of such technique,a K4 spherical part was figured by two iterations of MRF with surface form error improved to 0.219λPV and 0.027λRMS.Meanwhile,the surface roughness was also improved a lot in MRF process.These theoretical analyses and experimental results both indicate that high form accuracy and excellent surface quality can be obtained by using the four-axis linkage technique in the process of figuring low-gradient optical elements,and the four-axis linkage system undoubtedly is much more easy to control and much more economical.

Research on Planar Development of Curved Surfaces

Development of curved surface is a useful tool in CAD(computer aided design) and CAGD(computer aided geometric design).This paper presents the algorithms for developing (flattening) a smooth continuous curved surface embedded in three dimensional space into a planar shape. First the definition of planar development of a curved surface is presented, and the distortions (at length and area) of development are discussed in this paper. Then several planar flattening methods of curved surface, and their advantages and disadvantages are analyzed in detail. For NURBS(non uniform rational basic spline) surfaces, which are broadly used in CAGD, we put forward a new planar developing algorithm, i.e. hybrid developing, and present the steps of the algorithm. At last, some examples are used to show the effectiveness of the algorithm.

Shape gradient and classical gradient of curvatures:driving forces on micro/nano curved surfaces

Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity of the conical surfaces. Aimed at this important phenomenon, a gen- eral theoretical explanation is provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces, we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed. The following results are derived： （a） Whatever the exponents in the pair potentials may be, the potential of the particle/hard curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature and the Gaussian curvature of the curved surface. （b） On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard curved surfaces may be realized. （c） Similar to the extrinsic mechanics on micro/nano soft curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradi- ents form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the driving forces on the curved spaces. （d） The direction of the driving force induced by the hard curved surface is independent of the hydrophilieity and hydrophobicity of the curved surface, explaining the experimental phenomenon of the spontaneous and directional motion.

High-Order Method with Moving Frames to Compute the Covariant Derivatives of Vectors on General 2D Curved Surfaces

The covariant derivative is a generalization of differentiating vectors.The Euclidean derivative is a special case of the covariant derivative in Euclidean space.The covariant derivative gathers broad attention,particularly when computing vector derivatives on curved surfaces and volumes in various applications.Covariant derivatives have been computed using the metric tensor from the analytically known curved axes.However,deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional(2D)surface.Consequently,computing the covariant derivative has been difficult or even impossible.A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface.A set of orthonormal vectors,known as moving frames,expand vectors to compute accurately covariant derivatives on 2D curved surfaces.The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols.The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames.Consequently,the proposed scheme can be extended to the general 2D curved surface.As an application,the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

Finding Intersection Points of Curves and Surfaces for Solid Modeling in Curved-Surface Environment

Intersecting is an important factor which influences the effociency androbustness of Boolean algorithms in solid modeling based on surved-surfaces,andintersecting algorithms are closely related to geometric representations of curved-surfaces.Although surfaces can be commonly represented with NURBS,unnecessary complexitiesare caused in the intersecting of quadric surfaces.Quadrics are frequently used to des-cribe geometric features of shafts,holes and grooves etc.in mechanical part designing,therefore;their intersection algorithms are required to have higher accuracy,higher efficiency and higher robustness.For this reason,a practical representation ofquadric surfaces is studied in detail,and on the basis of that,algorithms of intersectingpoints are developed between quadric suraces and their boundaies,i.e.,conics,quarticnonplanar space curves.

A new method for specular curved surface defect inspection based on reflected pattern integrity

Defect inspection of specular curved surface is a challenging job. Taking steel balls for example, a new method based on reflected pattern integrity recognition is put forward. The specular steel ball surface will totally reflect the patterns when it is placed inside a dome-shaped light source, whose inner wall is modified by patterns with certain regular. Distortion or intermittence of reflected pattern will occur at the defective part, which indicates the pattern has lost its integrity. Based on the integrity analysis of reflected pattern images？ surface defects can be revealed. In this paper, a set of concentric circles are used as the pattern and an image processing algorithm is customized to extract the surface defects. Results show that the proposed method is effective for the specular curved surface defect inspection

The Experimental Research on a New Multiple Curved Surfaces Compound Focus Solar Trough Collector

A new trough imaging solar collector with multiple compounding curved surfaces has been designed. Its working principle and design parameters have been introduced. The experimental curve of temperature rising of the system with time under the real weather has been given. The system efficiency and the relation between efficiency and temperature have been calculated. The test result shows that the system has the advantages of high collecting temperature and not obvious variety of the collecting efficiency with the operating temperature. Therefore, this collector is a quite ideal medium temperature solar collector.

STUDY ON THE METHOD OF CONSTRUCTING RATIONAL CUBIC CURVES AND CURVED SURFACE INCLUDING CONTROLLING PARAMETERS

By adopting the method of controlling parameters this paper describes the construction of various kinds of cubic curve segment and curved surface fragment with rational and non rational parameters, and discusses the relationship between controlling parameters, weighted factors and types, kinds and characteristics of curve segments and curved surface fragments. A mathematical method is provided for CAGD with abundant connotations, broad covering region, convenience, flexibility and direct simplicity.

Study on curved surface fitting model using GPS and leveling in local area

The height anomaly surface is fitt and the quasi-geoid can be obtained when the height anomaly is determined with the geometric analytic method. Therefore, some mathematical models to fit height anomaly surface using GPS, leveling and terrain data in a local area, including the polynomial fitting model, the multi-surface function fitting model, the motion surface fitting model and the fitting model of little flexibility deformation of thin board, are given. Then the digital characteristics are analyzed with the curved surface theory. The General curvature and the mean curvature of surface are concluded. The advantage, disadvantage and application of the above models are discussed. The effect of terrain undulation on height anomaly is considered in the surface fitting models. The practical case indicates that these models are of validity and practicability. It is concluded that the above models can give the good fitting results at the centimeter level. But the polynomial fitting model is worse than the other models.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

Conformal manufacturing of soft deformable sensors on the curved surface

Health monitoring of structures and people requires the integration of sensors and devices on various 3D curvilinear,hierarchically structured,and even dynamically changing surfaces.Therefore,it is highly desirable to explore conformal manufacturing techniques to fabricate and integrate soft deformable devices on complex 3D curvilinear surfaces.Although planar fabrication methods are not directly suitable to manufacture conformal devices on 3D curvilinear surfaces,they can be combined with stretchable structures and the use of transfer printing or assembly methods to enable the device integration on 3D surfaces.Combined with functional nanomaterials,various direct printing and writing methods have also been developed to fabricate conformal electronics on curved surfaces with intimate contact even over a large area.After a brief summary of the recent advancement of the recent conformal manufacturing techniques,we also discuss the challenges and potential opportunities for future development in this burgeoning field of conformal electronics on complex 3D surfaces.

Effective dynamics for a spin-1/2 particle constrained to a curved layer with inhomogeneous thickness

We derive an effective Hamiltonian for a spin-1/2 particle confined within a curved thin layer with non-uniform thickness using the confining potential approach.Our analysis reveals the presence of a pseudo-magnetic field and effective spin–orbit interaction(SOI)arising from the curvature,as well as an effective scalar potential resulting from variations in thickness.Importantly,we demonstrate that the physical effect of additional SOI from thickness fluctuations vanishes in low-dimensional systems,thus guaranteeing the robustness of spin interference measurements to thickness imperfection.Furthermore,we establish the applicability of the effective Hamiltonian in both symmetric and asymmetric confinement scenarios,which is crucial for its utilization in one-side etching systems.

Curved surface effect and emission on silicon nanostructures

The curved surface （CS） effect on nanosilicon plays a main role in the activation for emission and photonic manipulation. The CS effect breaks the symmetrical shape of nanosilicon on which some bonds can produce localized electron states in the band gap. The investigation in calculation and experiment demonstrates that the different curvatures can form the characteristic electron states for some special bonding on the nanosilicon surface, which are related to a series of peaks in photoluminecience （PL）, such as LN, LNO, Lo1, and Lo2 lines in PL spectra due to Si-N, Si-NO, Si=O, and Si-O-Si bonds on curved surface, respectively. Si-Yb bond on curved surface of Si nanostructures can provide the localized states in the band gap deeply and manipulate the emission wavelength into the window of optical communication by the CS effect, which is marked as the Lyb line of electroluminescence （EL） emission.

Residual strain evaluation of curved surface by grating-transferring technique and GPA

This paper investigates an advanced grating-transferring technique combined with geometric phase analysis (GPA) for residual strain evaluation of curved surface.A standard holographic grating is first transferred to a pre-produced epoxy resin film and then consolidated to a test region of curved surface.With a rubber mold and silicone rubber the deformed grating is replicated to a sheet metal after hole-drilling for release of residual stress.After that the grating is transferred from the sheet metal to the glass plate,which would be served as an analyzer grating (specimen grating).By GPA the local strain distributions related to the phase difference between the reference grating and analyzer grating for the released stress can be evaluated.A validation test has been conducted on the weld joint of a stainless steel tube and the obtained results demonstrate the ability of the method in measuring the residual strain of curved surface.

Judging or setting weight steady-state of rational Bézier curves and surfaces

Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.

Studying Scalar Curvature of Two Dimensional Kinematic Surfaces Obtained by Using Similarity Kinematic of a Deltoid

We consider a similarity kinematic of a deltoid by studying locally the scalar curvature for the corresponding two dimensional kinematic surfaces in the Euclidean space . We prove that there is no two dimensional kinematic surfaces with scalar curvature K is non-zero constant. We describe the equations that govern such the surfaces.

ANOTHER TYPE OF GENERALIZED BALL CURVES AND SURFACES

In 2000, Wu presented two new types of generalized Ball curves, one of which is called an NB1 curve located between the Wang-Ball curve and the Said-Ball curve. In this article, the authors aim to discuss properties of NB1 curves and surfaces, including the recursive algorithms, conversion algorithms between NB1 and Bezier curves and surfaces, etc. In addition the authors compare the computation efficiency of recursive algorithms for the NB1 and above mentioned two generalized Ball curves and surfaces.