Common Axis Of Symmetry Research Articles

Leapfrogging vortex rings for the three‐dimensional incompressible Euler equations

AbstractA classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three‐dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3D Euler equations exhibiting this motion pattern.

Design research with the use of visual and symmetry analysis in indigenous woven textiles.

Since ca the 1990s, there have been developments in the weaving process for making tin chok fabrics in the Long district, Phrae province, northern Thailand, with the aim of reducing the production time and increasing the production capacity with respect to poverty alleviation and cultural revitalization. In this article, the symmetry patterns in tin chok fabrics in a collection of 17 vintage (traditional) skirts held in the Komol Antique Textile Museum, in the Long district, are examined to determine how to distinguish tin chok fabrics woven by the integrated method from those woven by the traditional method. The research includes visual and symmetry analysis, literature review, fieldwork, and the creation of a pattern booklet. This research reveals that the hem, which is one of the four parts (supplementary part one, the main part, supplementary part two and the hem) of the detachable tin chok, is key to distinguishing the fabrics made by the two methods. The four parts of tin chok made by the integrated method will always have a common vertical axis of symmetry, whereas the vertical axes of symmetry of the hem of a fabric made by the traditional method may not be aligned with the motifs in the other three parts. The frieze groups of the individual parts of the 17 vintage (traditional) skirts are decoded and possible corresponding patterns for weaving by the integrated method are generated.

Periodic oscillations in the restricted hip-hop $ 2N+1 $-body problem

This manuscript investigates a dynamical system in which $ 2N $ primary particles of equal masses move in space under Newton's law of gravitation forming the vertices of antiprisms while a particle of negligible mass moves along the common axis of symmetry of the antiprisms. This $ n $-body problem that we call the restricted hip-hop $ (2N + 1) $-body problem is an extension of the generalized Sitnikov problem studied in [17] for which the primaries remain in a plane. This work also relies on an early study [14] where certain families of periodic hip-hop solutions to a $ 2N $–body problem were constructed. We prove the existence of a continuous symmetric family of solutions of the restricted hip-hop $ (2N+1) $-body problem for each family of symmetric and periodic hip-hop solutions of the primaries studied in [14]. The main tools for proving our results are the implicit function theorem and a compactness argument. In addition, we present some numerical periodic solutions to the restricted $ 7 $-body problem.

DETERMINATION OF KINEMATIC CHARACTERISTICS OF METAL PLASTIC FLOW UNDER CONDITIONS OF AXIAL SYMMETRY OF THE DEFORMATION PROCESS

With axisymmetric deformation, the deformable body and the load have a common axis of symmetry. Such deformation occurs in numerous technological operations. About 70% of parts obtained by cold extrusion are deformed under conditions of axial symmetry. The plane problem of the theory of plasticity is reduced to a solution in the framework of a two-dimensional statement, when the motion of points in the section of a workpiece is analyzed. Each point can move only in the section plane, and its velocity can be decomposed into two mutually perpendicular directions along the coordinate axes. For plane deformation, it is assumed that the velocity in the direction of the third coordinate axis is equal to zero. The presence of axial symmetry allows us to confine ourselves to studying the behavior of points located on the plane of the meridional section of the workpiece. In this case, each point can move only in the section plane and its velocity can be decomposed into two orthogonal directions: along the axis and along the radius. The component of the velocity vector in the circumferential direction is equal to zero, so only four of the six independent components of the strain rate tensor remain. In this regard, axisymmetric problems of the theory of plasticity are of considerable interest from the point of view of solving applied problems. The article discusses the possibility of using mixed Euler and Lagrange coordinates to determine the components of the strain rate tensor in plastic deformation processes characterized by the axisymmetric nature of metal plastic flow. The vector field of displacements at each point in space reflects the transformation of the initial (non-deformed) configuration into the current one, and therefore determines the configuration of the deformed body in space in a certain frame of reference. The deformation process is first considered in a single Cartesian coordinate system, and then in a cylindrical coordinate system, the use of which is more appropriate for axisymmetric deformation. It is assumed that the functions of the Euler coordinates from the Lagrange coordinates are obtained by approximating the experimental data.

Quartic Integral in Rigid Body-Gyrostat Dynamics

In this work, we investigate the problem of constructing new integrable problems in the dynamics of the rigid body rotating about its fixed point as results of the effect of a combination of potential and gyroscopic forces possessing a common symmetry axis. We introduce two new integrable problems in a rigid body dynamics that generalize some integrable problems in this field, known by names of Chaplygin and Yehia–Elmandouh.

Measurements and modeling of ion divergence from a gridded inertial electrostatic confinement device using laser induced fluorescence

Inertial electrostatic confinement (IEC) is a method of confining and heating a plasma at benchtop scales to sufficient energies for nuclear fusion to occur. Ion velocity and flow direction were measured in an IEC discharge using laser induced fluorescence (LIF) on argon ions. A cathode of two parallel rings, with a common axis of symmetry, resulted in predominant discharge beams, otherwise known as microchannels, along this axis. The device was operated in the abnormal glow discharge regime where both current and voltage increase monotonically, replicating a conventional high voltage IEC device. It was found that argon ions accelerated and flowed outward from the midpoint between the rings along the axis; we have labeled this ion motion as being divergent. The predominant direction of ion flow in the discharge is opposite to the conventional ion focus model, where the discharge at the cathode center is assumed to be the result of ion flow toward it from outside of the cathode. An ion sheath model is shown to produce a virtual anode at the axial midpoint between the rings. The model also shows that ions within the virtual anode are accelerated outward with a spatial velocity profile that replicates those measured using LIF.

Minimal surfaces on mirror-symmetric frames: a fluid dynamics analogy

Chaplygin’s hodograph method of classical fluid mechanics is applied to explicitly solve the Plateau problem of finding minimal surfaces. The minimal surfaces are formed between two mirror-symmetric polygonal frames having a common axis of symmetry. Two classes of minimal surfaces are found: the class of regular surfaces continuously connecting the supporting frames forming a tube with complex shape; and the class of singular surfaces which have a partitioning film closing the tube in between. As an illustration of the general solution, minimal surfaces supported by triangular frames are fully described. The theory is experimentally validated using soap films. The general solution is compared with the known particular solutions obtained by the Weierstrass inverse method.

On the movement of briquetted mass in extruder. Exact solutions

The increased interest in cold agglomeration in recent years has been largely due to the successful experience in operating briquette lines based on stiff vacuum extrusion (SVE). The high performance of SVE extruders and the satisfactory metallurgical properties of extrusion briquettes obtained this way (brex) make it possible to consider this cold agglomeration technology as promising. SVE extruders allow efficient briquetting of the materials with moisture contents values in the range of 12 – 16 % and compacting pressure of 3.5 – 4.5 MPa, leading to the possibility of achieving high values of mechanical strength of raw briquettes and eliminates the need for drying briquetted charge and heat treatment of the green briquettes. The growing scale of practical use of extruders in the steel industry necessitated the development of simple and effective methods for determining their operating parameters. The briquetted mass is a moistened plastic mass, driven by the blades of a rotating auger and squeezed out further through the holes in the extruder die in the form of elongated briquettes, repeating in cross section the shape of the hole. In application to the optimization problems of extrusion briquette technology, the exact solution of the Navier–Stokes equations for a viscous incompressible medium shifted between coaxial cylinders along the common axis of symmetry and twisted around it by longitudinal displacement and axial rotation of the inner cylinder is given, respectively, under sticking conditions and given longitudinal pressure drop. In particular, it was found that the speed of transportation of the mixed mass cannot exceed the speed delivered by the supplied pressure, and the formula of the latter is transferred to the compressible medium as a special viscosity integral for a viscoplastic medium, where it serves as a generalization of known exact solutions. A similar solution for a compressible medium is being sought. The obtained analytical dependencies can be used to calculate the parameters of industrial briquette extruders operating in both the stiff extrusion mode and the semi-stiff and soft extrusion mode, differing in the moisture values of the briquetted mass and the values of the applied pressure.

On the Movement of Briquetted Mass in Extruder. Exact Solutions. 1

The increased interest in cold agglomeration in recent years has been largely due to the successful experience in operating briquette lines based on stiff vacuum extrusion (SVE). The high performance of SVE extruders and the satisfactory metallurgical properties of extrusion briquettes obtained in this way (brex) make it possible to consider this cold agglomeration technology as promising. SVE Extruders allow efficient briquetting of the materials with moisture contents values in the range of 12–16% and compacting pressure of 3.5–4.5 MPa, leading to the possibility of achieving high values of mechanical strength of raw briquettes and eliminates the need for drying briquetted charge and heat treatment of the green briquettes. The growing scale of practical use of extruders in the steel industry necessitated the development of simple and effective methods for determining their operating parameters. The briquetted mass is a moistened plastic mass, driven by the blades of a rotating auger and squeezed out further through the holes in the extruder die in the form of elongated briquettes, repeating in cross section the shape of the hole. In application to the optimization problems of extrusion briquette technology, the exact solution of the Navier–Stokes equations for a viscous incompressible medium shifted between coaxial cylinders along the common axis of symmetry and twisted around it by longitudinal displacement and axial rotation of the inner cylinder is given, respectively, under sticking conditions and given longitudinal pressure drop. In particular, it was found that the speed of transportation of the mixed mass cannot exceed the speed delivered by the supplied pressure, and the formula of the latter is transferred to the compressible medium as a special viscosity integral for a viscoplastic medium, where it serves as a generalization of known exact solutions. A similar solution for a compressible medium is being sought. The obtained analytical dependencies can be used to calculate the parameters of industrial briquette extruders operating in both the stiff extrusion mode and the semi-stiff and soft extrusion mode, differing in the moisture values of the briquetted mass and the values of the applied pressure.

Nonlinear Oscillations of a Pair of Charged Rings

Two electrically charged rings of different sizes are assembled along their common vertical symmetry axis through their centers. The bottom ring is secured on a horizontal support while the top one is loose. For a set of practical values characterizing the charged rings we envision a scenario where the mutual electric repulsion between the rings and the weight of the top ring results in stable nonlinear oscillations. To quantify the characteristics of the oscillations, we utilize a Computer Algebra System specifically Mathematica [1]. We accompany the analysis with a simulation for a comprehensive visual understanding.

Experimental investigation of the collective stimulated Brillouin and Raman scattering of multiple laser beams in inertial confinement fusion experiments

The direct and indirect drive schemes for inertial confinement fusion (ICF) make use of a large number of laser beams arranged in a symmetric angular distribution. The preferential decay geometry of the three waves resonant couplings, mainly responsible for backscattered light in single beam experiments, may then be deeply modified in the region of crossing beams where collective laser plasma instabilities could develop. Such instabilities can occur for laser beams having a common symmetry axis along which they drive a common daughter wave. The collective coupling results in an increase of the growth gain with the increase of the number of interacting beams and produce energy losses in new backward directions. We have taken advantage of the multiple beams of the Omega laser facility and of its large battery of diagnostics to study the physics related to this multiple beams interaction in the regimes of high temperature plasmas relevant of the direct and indirect drive schemes to ICF. Experiments performed in a planar open geometry have evidenced the large amplification of stimulated Raman scattering (SRS) electromagnetic waves almost transverse to the density gradient as theoretically predicted 40 years ago. This was achieved in long scale-length high-temperature plasmas in which two beams couple to the same scattered electromagnetic wave further demonstrating this multiple-beams collective SRS interaction. The collective nature of the coupling and the amplification at large angles from the density gradient increase the global SRS losses and produce light scattered in novel directions out of the planes of incidence of the beams. Indirect drive experiments were performed in rugby ball shaped Hohlraum irradiated by 40 beams. Large instantaneous (peak reflectivity >30%) Brillouin sidescattering was evidenced to originate from the collective Brillouin amplification of a shared ion acoustic wave driven along the Hohlraum axis by a cone of 10 beams. In this paper, the scattering geometry is detailed for the two types of collective instabilities showing that they produce light scattered in novel very precise directions located far from the original aperture of the beams where the diagnostics are usually set-up. This scattered light could be measured on Omega thanks to the flexibility of the facility. Key features of the light scattered by collective instabilities are identified that would allow to recognize their signatures in more complex, less diagnosed experiments.

Harmonic functions which vanish on coaxial cylinders

It was recently established that a function which is harmonic on an infinite cylinder and vanishes on the boundary necessarily extends to an entire harmonic function. This paper considers harmonic functions on an annular cylinder which vanish on both the inner and outer cylindrical boundary components. Such functions are shown to extend harmonically to the whole of space apart from the common axis of symmetry. One of the ingredients in the proof is a new estimate for the zeros of cross product Bessel functions.

Photon position eigenvectors, Wigner’s little group, and Berry’s phase

We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x^, reflects the E(2) symmetry of the photon little group. The eigenvectors of x^ form a basis of localized states that have definite angular momentum, J^, parallel to their common axis of symmetry. This basis is well suited to the description of “twisted light” that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that x^1,x^2,J^3 is a realization of the two dimensional Euclidean e2 algebra that effects genuine infinitesimal displacements in configuration space.

Reynolds Number Effects in the Flow of a Vočadlo Electrorheological Fluid in a Curved Gap

Many electrorheological fluids (ERFs) as fluids with micro-structure demonstrate a non-Newtonian behaviour. Rheometric measurements indicate that some flows of these fluids may by modelled as the flows of a Vočadlo ER fluid. In this paper, the flow of a Vočadlo fluid – with a fractional index of non-linearity – in a narrow gap between two fixed surfaces of revolution with a common axis of symmetry is considered. The flow is externally pressurized and it is considered with inertia effect. In order to solve this problem the boundary layer equations are used. The Reynolds number effects (the effects of inertia forces) on the pressure distribution are examined by using the method of averaged inertia terms of the momentum equation. Numerical examples of externally pressurized flows in the gap between parallel disks and concentric spherical surfaces are presented.

Ferromagnetic Flow of Viscous Fluid in a Slot between Fixed Surfaces of Revolution

AbstractIn this paper the steady laminar flow of viscous incompressible ferromagnetic fluid is considered in a slot between fixed surfaces of revolution having a common axis of symmetry. The boundary layer ferromagnetic equations for axial symmetry are expressed in terms of the intrinsic curvilinear orthogonal coordinate system x, θ ,y.The method of perturbation is used to solve the boundary layer equations. As a result, the formulae defining such parameters of the flow as the velocity components vx, vy, and the pressure , were obtained.

On the finite‐time capture of a fast moving target

SummaryIn this work, we propose a feedback control law that enforces capture of a moving target by a slower pursuer in finite time. It is well known that if this problem is cast as a pursuit‐evasion differential game, then the moving target can always avoid capture by taking advantage of its speed superiority, provided that both the target and the pursuer are employing feedback strategies in the sense of Isaacs. Thus, in order to have a well‐posed pursuit problem, additional assumptions are required so that the pursuer can enforce capture of the faster target in finite time provided that it emanates from a set of ‘favorable’ initial positions, which constitute its winning set. In particular, we assume that the target's velocity either is constant and perfectly known to the pursuer (perfect information case) or can be decomposed into a dominant component, which is constant and known to the pursuer, and a second component that is uncertain and unknown to the pursuer (imperfect information case). It turns out that in both cases, the winning sets of the pursuer are pointed convex cones which have a common apex and a common axis of symmetry but different opening angles. We subsequently propose continuous feedback laws that enforce finite‐time capture while the pursuer never exits its winning set before capture takes place, for both cases. Copyright © 2016 John Wiley & Sons, Ltd.

On the symmetry of spatially periodic two-dimensional water waves

We show that a spatially periodic solution to the irrotational two-dimensional gravity water wave problem, with the propertythat the horizontal velocity component at the flat bed is symmetric, while the acceleration at the flat bed is anti-symmetric with respectto a common axis of symmetry, necessarily constitutes a traveling wave. The proof makes use complex variables and structural properties of the governing equations for nonlinear water waves.

Some Relations Between the Spheroidal and Spherical Wave Functions

New relations are found between the spheroidal and spherical wave functions, as well as between the spheroidal functions related to different spheroidal coordinate systems. The systems must have a common origin of coordinates and a common symmetry axis of coordinate surfaces. The applicability ranges of the relations obtained are discussed. Numerical test calculations have demonstrated a high efficiency of the relations, specifically those for wave functions including the radial functions of the first kind. As a particular case, relations are considered between prolate and oblate spheroidal wave functions including radial functions of the first and second kinds. These relations are necessary for solving the light scattering problem for nonconfocal layered spheroidal particles. Bibliography: 15 titles.

Structure and Biophysical Properties of a Triple-Stranded Beta-Helix Comprising the Central Spike of Bacteriophage T4.

Gene product 5 (gp5) of bacteriophage T4 is a spike-shaped protein that functions to disrupt the membrane of the target cell during phage infection. Its C-terminal domain is a long and slender β-helix that is formed by three polypeptide chains wrapped around a common symmetry axis akin to three interdigitated corkscrews. The folding and biophysical properties of such triple-stranded β-helices, which are topologically related to amyloid fibers, represent an unsolved biophysical problem. Here, we report structural and biophysical characterization of T4 gp5 β-helix and its truncated mutants of different lengths. A soluble fragment that forms a dimer of trimers and that could comprise a minimal self-folding unit has been identified. Surprisingly, the hydrophobic core of the β-helix is small. It is located near the C-terminal end of the β-helix and contains a centrally positioned and hydrated magnesium ion. A large part of the β-helix interior comprises a large elongated cavity that binds palmitic, stearic, and oleic acids in an extended conformation suggesting that these molecules might participate in the folding of the complete β-helix.

Strong exchange of stabilities in rapidly rotating parabolic Poiseuille flow

Abstract Poiseuille flow between two coaxial cylindrical pipes, with inner pipe translating at just the rate so that the velocity is parabolic, is rotated around the common axis of symmetry, with a common angular velocity. If the rotation rate is very large and it is assumed that the wave length of the critical disturbances is small, the governing equations take an approximate form that is analogous, remarkably, to Couette flow between coaxial cylinders with a certain difference. It is proved that if the rotation rate meets this criterion, the strong principle of exchange of stabilities (PES) holds; that is, all eigenvalues are real, and there is a first unstable eigenvalue. From a physical viewpoint, the resulting unstable motion initially takes the form of steady spiral vortices.