Spherical Rings Research Articles

Existence and isoperimetric characterization of steady spherical vortex rings in a uniform flow in RN

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

Axisymmetric acoustic scattering by vortices

The scattering of acoustic waves by compact three-dimensional axisymmetric vortices is studied using direct numerical simulation in the case where the incoming wave is aligned with the symmetry axis and the direction of propagation of the vortices. The cases of scattering by Hill’s spherical vortex and Gaussian vortex rings are examined, and results are compared with predictions obtained by matched asymptotic expansions and the Born approximation. Good agreement is obtained for long waves, with the Born approximation usually giving better predictions, especially as the difference in scale between vortex and incoming waves decreases and as the Mach number of the flow increases. An improved version of the Born approximation which takes into account higher-order effects in Mach number gives the best agreement.

Singular solutions, homogeneous norms, and quasiconformal mappings in Carnot groups

In any Carnot (nilpotent stratified Lie) group G of homogeneous dimension Q, Green's function u for the Q-Laplace equation exists and is unique. We prove that there exists a constant \(\gamma=\gamma(G)\) so that \(N=e^{-\gamma u}\) is a homogeneous norm in G. Then the extremal lengths of spherical ring domains (measured with respect to N) can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the Carnot-Caratheodory metric. Applications include regularity properties of quasiconformal mappings and a geometric characterization of bi-Lipschitz mappings.

Model etch profiles for ion energy distribution functions in an inductively coupled plasma reactor

Rectangular trench profiles are modeled with analytic etch rates determined from measured ion distribution functions. The pattern transfer step for this plasma etch is for trilayer lithography. Argon and chlorine angular ion energy distribution functions measured by a spherical collector ring analyzer are fit to a sum of drifting Maxwellian velocity distribution functions with anisotropic temperatures. The fit of the model ion distribution functions by a simulated annealing optimization procedure converges adequately for only two drifting Maxwellians. The etch rates are proportional to analytic expressions for the ion energy flux. Numerical computation of the etch profiles by integration of the characteristic equations for profile points and connection of the profiles points is efficient.

Synthesis, structure, and gas sensitivity properties of SnO2–CuO mixture phase obtained by pyrolysis of an aerosol

SnO2–CuO mixture phase has been prepared by pyrolysis of an aerosol produced by ultrahigh frequency of different precursor solutions. As-received samples were annealed in different conditions to study the influence of the temperature on the microstructure. Scanning electron microscopy showed that the samples consisted of hollow spherical particles and rings, depending on the precursor solution utilized. The evolution of conductance under both pure and polluted air is discussed.

3-D scanning-particle-image-velocimetry: Technique and application to a spherical cap wake flow

A 3-D time-resolving and whole-volume Digital-Particle-Image-Velocimetry (DPIV) technique based on the concept of a scanning light-sheet is presented and applied here to the 3-D transient wake phenomena in the spherical cap wake flow. The technique uses a scanning light-sheet for rapid sampling of the flow in depth and a two-camera recording system for stereoscopic 3-D DPIV. Application of a correlation technique in combination with a calibration yields, aside from the correct in-plane displacement, also the out of plane component and thus the total velocity vectors within the planes of the scanning light-sheet. With a high scanning rate in comparison to the characteristic time scales the method provides the 3-D velocity field in space and time. Through the use of conventional video-techniques the temporal evolution of the complete velocity and vorticity field can be obtained quantitatively from experiments. This is demonstrated for the 3-D starting flow around a spherical cap at Re=300. During the starting process, the flow in the wake evolves into a spherical vortex ring where the velocity distribution is very close to the theoretical solution of the Hill-type vortex. Later on, the Hill-type vortex ring deforms and the flow changes from a rotational symmetric stage to a planar symmetric flow with a double-threaded vortical structure which consists of two counter-rotating streamwise vortices similar to the ones observed in sphere wake flow.

Analytical results for the electronic shell structure in mesoscopic systems: Balian‐Bloch type theory for spheres, discs, rings and anti‐dot lattices

AbstractThe electronic shell structure resulting from the interference of closed orbital paths is determined for mesoscopic systems like spherical clusters, discs and rings by extending the semiclassical theory of Balian and Bloch. Analytical results for the shell structure in the density of states are obtained. Thus, the dependence of the shell structure on dimension, size and geometry and potential of the mesoscopic system and on an external magnetic field can be studied systematically. Comparison of the semiclassical results and those of quantum mechanical calculations permits analysis of typical quantum mechanical effects and shows the validity of the semiclassical theory. Our results should stimulate new experiments, can be used to calculate oscillations in the binding‐energy, ionization‐potential, and can be applied to analyze oscillations in the electronic density of states of quantum dot systems like anti‐dot lattices.

Spatial distribution of atoms in a magneto-optical trap.

We present a theoretical analysis to explain the origin of different spatial distributions of laser-cooled sodium atoms in a magneto-optical trap. Such distributions, which include spherical clouds, rings, and rings with cores, are attributed to the radiation pressure due to misaligned Gaussian beams acting on each individual atom. The predictions of this theory are confirmed by experimental observations reported earlier. We were able to construct a diagram with several domains corresponding to the stable structures observed

All welded sound isolation flexible connection

ALL WELDED SOUND ISOLATION FLEXIBLE CONNECTION ABSTRACT A flexible pipe joint having an annular housing with a central cavity in which a nipple is partially submerged. The nipple has a flared end on the submerged portion thereof. An elastomeric ring, internally contoured spherically with reference to a common center point spaced from the flared end and located on the longitudinal axis of the point is adhered about its outer surface to the inner surface of the flared end. An elastomeric body having an assembly of embedded concentric spherical rings is adhered to the outer surface of the flared end. The spherical rings are contoured and positioned symmetrically with respect to the common center point. A flange (42 or 144, 146) is welded to the nipple at the end opposite the flared end. The flange is used to bolt the joint to the adjacent pipe sections and can be in two parts (144, 146). A collar welded to the housing encircles the nipple and engages the outer surface of the elastomeric body to maintain the body in position. A support structure secured to the housing has an outward facing spherical surface mating in a sliding relation with the internal surface of the elastomeric ring.

Structure of the central regions of some galaxies with ultraviolet excess. II

Using direct plates obtained at the prime focus of the 2.6-m telescope of the Byurakan Astrophysical Observatory (ZU-21 plates), we have investigated the central regions of ultraviolet galaxies. For seven galaxies, listed in Table 1, double nuclei have been found for the first time. The plates were analyzed using the PDS 1010A--CM4 densitometer complex by means of a program for analyzing extended objects in the framework of the ADA computational system of the Byurakan Astrophysical Observatory. The analysis included scanning the images of the galaxies with a 0:'5  diaphragm, surface photometry of the objects in the B system with construction of contour charts either in units of magnitudes per square second or in densities; estimate of the integrated magnitudes of the galaxies and the identified nuclei; estimate of the geometrical characteristics of the galaxies. The accuracy of the photometry, with transition to the B system, was estimated at +0.m19. We also give a brief description of the central parts of all the investigated galaxies, their isophotes, and geometrical sections through both nuclei. The galaxies for which a complex structure has been discovered for the first time are given in Table 1, which contains the Markaryan numbers of the galaxies, the absolute and apparent integrated magnitudes of the galaxies and their nuclei, the radial velocities (corrected for revolution of the sun, in accordance with [1]), the diameters of the galaxies in seconds of arc and in kiloparsecs, the distances between the components in seconds of arc and kiloparsecs, the apparent and absolute integrated magnitudes of the nuclei, which are designated as in [2], and the morphological types and spectral class according to [4, 5]. We determined the apparent magnitudes of the nuclei visually by the method described in [3]. Figure 1 gives isophote charts of the galaxies. The isophotes were constructed in magnitudes in the B filter per square second. There now follows a description of the galaxies in accordance with the photographs obtained with the 2.6-m telescope. Markaryan 1003. A barred spiral whose arms form a spherical ring. At least two condensations can be identified in the equatorial plane. The fainter condensation is at the geometrical center, and the brighter may be a giant H II region. Markaryan 1126. This is a Seyfert with two nuclei. There is a ring around the bulge. The galaxy has three envelopes. The inner isotopes are spherical, the outer have a cut form. Markaryan 1127. A spiral with faint arms. Within the innermost rectangular isophote at least two condensations are observed. There may be two more which we could not reliably identify but suspected in the photometric section. Markaryan 1131. A spiral galaxy with large bulge in which two nuclei can be identified. The bright nucleus is closer to the center of the galaxy. The second nucleus may be a giant H II region. Being faint, the second component is not clearly revealed in the isophote chart. Markaryan 1140. A barred spiral with two well-defined nuclei; its spiral arms are uniform. The nuclei are well revealed in the photometric section. Markaryan 1156. A somewhat elongated two-nucleus galaxy with weakly developed spiral arms. The nuclei differ strongly in brightness, and neither is at the geometrical center of the galaxy. The outer isophotes are strongly cut. It is possible that the first nucleus is itself divided into two components. Markaryan 1187. This is a Seyfert. In the center there are two nuclei that do not differ strongly in brightness. All the isophotes, the outer and the inner, are very cut. The morphological type is SBc.

Mean coefficients of quasiconformality of a pair of domains

The notion of the mean coefficients of quasiconformality of a pair of bounded domains in ℝ n is introduced, and a method of their computation is proposed. As an application, the outer β-mean coefficient of a pair of spherical rings is computed, and the extremal mapping corresponding to this coefficient is found.

Electrodynamic loudspeaker having omnidirectional sound emission

An electrodynamic loudspeaker for the high or mid-range sound region emits sound omnidirectionally. The loudspeaker comprises two identical individual loudspeakers arranged on opposite sides of a plane of mirror symmetry. Each loudspeaker has a drive unit with a permanent magnet fixed at a carrier part common to both sides and a moving coil means, including a winding and a bobbin, elastically held by a spider at carrier part. Each moving coil means is rigidly connected with a hemispherical membrane. The carrier part is formed as a ring and is limited by a spherical ring surface having a diameter which is somewhat smaller than twice the diameter of the outer rim of the membrane. The moving coil means has a diameter substantially the same as the outer edge of the membrane.

Radiation field calculations of pulsed ultrasonic transducers: Part 2: spherical disc- and ring-shaped transducers

The exact equation to describe the propagation of ultrasonic pressure waves in a lossless, homogeneous medium was evaluated assuming pulse-excited radiators. Acoustical field patterns were studied using the dimensions and shape of the transducer, the ultrasonic frequency and number of cycles within the pulse as parameters. The results emphasize the influence of the piston shape on interference phenomena within the near — and far-field and on beam-narrowing effects in the intermediate range between near- and far-field. The following transducer shapes are studied: spherical disc and spherical ring.

Vortices and Vortex Rings of Stratified Fluids

Exact solutions for finite-amplitude motion of an inviscid, incompressible and stratified fluid corresponding to rectilinear vortices or vortex rings are found. Some of the rectilinear vortex pairs, in circular-cylindrical form, can propagate with a constant horizontal velocity in a surrounding homogeneous fluid, and some of the spherical vortex rings can be placed in a surrounding quiescent homogeneous fluid without translational propagation. Equations governing finite-amplitude two-dimensional motion and axisymmetric motion with swirl in the presence of a magnetic field are derived for a stratified conducting fluid, and the effects of a magnetic field on vortex motion are ascertained.

Radiative heat transfer in a gas-filled spherical ring

An analysis is made here of the radiative heat transfer in the gas-filled space between two concentric spheres.

Fibrillar crystals of isotactic polystyrene. II. Aggregation in stirred suspensions

Abstract Macroscopic particles of various morphological forms are created by the crystallization of isotactic polystyrene from rapidly stirred solutions. Crystallization from trimethylbenzene over a 150° C temperature range produced spherical particles, globules, fibers, and rings depending upon the growth temperature. Crystallization from cyclohexanol (θ = 83.5°C) produced fibers in the temperature range of 85° to 140° C. Globular structures, similar to those formed by stirrer crystallization in trimethylbenzene developed during the Ziegler-Natta polymerization of styrene in trimethylbenzene between 25° and 75°C. Highly birefringent conical appendages appeared on many of the globules and fibers. The fibers often exhibited a skin-core effect with an average longitudinal and transverse orientation of the chain molecules in the skin and core, respectively. The skin frequently displayed periodic banded extinctions. All of these macrostructures consisted of a highly porous assembly of constituent microfibrill...

Thermal convention and spherical aberration distortion of laser beams in low-loss liquids

Experiments and approximate analysis are given for spherical aberration interference rings and for convection distortion arising in the thermal self-defocusing effects produced by laser beams passing through materials of finite loss. The importance of these thermal effects for moderate-power beams in materials of relatively low loss is stressed.

Localizability of Electrons in Atoms and Molecules—Application to the Study of the Notion of Shell and of the Nature of Chemical Bonds

The importance of the correlations existing between the positions of electrons in atoms and molecules has recently been stressed. A new method for the study of these correlations is presented here. It consists in the decomposition of the space of an electronic system into small parts where there is a very large probability of finding one and only one electron of a given spin. This method allows the association of spherical rings to the shells of atoms and the association of space to the atomic core and to the bonds of molecules.