Radius Of Torus Research Articles

Modal stability analysis of toroidal pipe flow approaching zero curvature

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature $\delta$ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for $10^{-7} \leq \delta \leq ~10^{-2}$ is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below $1.5 \times 10^{-6}$ , the critical Reynolds number ${{Re}}_{cr}$ is proportional to $\delta ^{-1/2}$ . Hence, the critical Dean number is constant, ${{De}}_{cr} = 2\,{{Re}}_{cr}\,\sqrt {\delta } \approx 113$ . This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.

Application of Shadow Visualization to Study the Drag of Toroidal Bubbles

In this work, the ascent of a toroidal bubble created by the injection of a pulsed air jet into water vertically upwards is studied by the shadow method. When using the shadow method, the bubble border on the image is darkened, and the contrast in comparison with the background is increased, which makes it possible to use software processing algorithms to determine the parameters of the ring on each frame. There are experimental results indicating that, in addition to the buoyancy force, the drag force also acts on the toroidal bubble. In this paper, the experimental data on the change in the torus radius as a function of time are compared with a theoretical model constructed with and without taking into account the drag force. It is shown that taking into account the drag force leads to a much better agreement between theory and experiment. The drag force is concluded to act on toroidal bubbles, but its influence decreases with time, i.e., as the bubble rises. The drag coefficient used in the calculations is determined empirically and assumed to be constant.

Towards quantum turbulence theory: A simple model with interaction of vortex loops

This work investigates a quantized system of thin vortex rings with a flow in the core. Examples show that the set of permissible circulation values has a fractal structure. Circulation values in the domain of a round tube with a radius R${}_{0}$ in the shape of a torus of radius R${}_{1}\ensuremath{\gg}{R}_{0}$ are given by the accompanying expression where quantities ${\ensuremath{\zeta}}_{k}^{(l)}$ stand for zeros of the Bessel function J${}_{l}$(\ensuremath{\rho}). A general expression for the partition function of a turbulent flow is also suggested.

Size Effect of Hemi-Toroidal Quantum Dot on the Electronic Properties in the Presence of an Off-Center Hydrogenic Shallow Donor Impurity

We have studied the electronic properties in presence of an off-center hydrogenic shallow donor impurity confined in GaAs semiconductor quantum dot with toroidal geometry by considering the infinite confinement potential. This study has been performed within the parabolic band and the effective mass approximations in the presence of an off-center donor impurity. Three-dimensional Schrödinger equations are discretized using the finite difference method on a mesh containing Nr*Nθ*Nφ nodes. The numerical results of the analytical calculations demonstrate that the variation of the geometrical and torus radii (Rg and Rc) has a remarkable effect on the donor energy and the average electron-impurity distance, which is quite remarkable in small hemi-Toroidal quantum dot. On the other hand, we've demonstrated that the donor atom's position has a considerable impact on their energy. Furthermore, our numerical results show that the geometrical radius and donor atom's position significantly affect the electron impurity binding energy.

Modeling the Infrared Reverberation Response of the Circumnuclear Dusty Torus in AGNs: An Investigation of Torus Response Functions

Abstract The size and structure of the dusty circumnuclear torus in active galactic nuclei (AGNs) can be investigated by analyzing the temporal response of the torus’s infrared (IR) dust emission to variations in the AGN ultraviolet/optical luminosity. This method, reverberation mapping, is applicable over a wide redshift range, but the IR response is sensitive to several poorly constrained variables relating to the dust distribution and its illumination, complicating the interpretation of measured reverberation lags. We have used an enhanced version of our torus reverberation mapping code (TORMAC) to conduct a comprehensive exploration of the torus response functions at selected wavelengths, for the standard interstellar medium grain composition. The shapes of the response functions vary widely over the parameter range covered by our models, with the largest variations occurring at shorter wavelengths (≤4.5 μm). The reverberation lag, quantified as the response-weighted delay (RWD), is most affected by the radial depth of the torus, the steepness of the radial cloud distribution, the degree of anisotropy of the AGN radiation field, and the volume filling factor. Nevertheless, we find that the RWD provides a reasonably robust estimate, to within a factor of ∼3, of the luminosity-weighted torus radius, confirming the basic assumption underlying reverberation mapping. However, overall, the models predict radii at 2.2 μm that are typically a factor of ∼2 larger than those derived from K-band reverberation mapping. This is likely an indication that the innermost region of the torus is populated by clouds dominated by large graphite grains.

Rapid luminosity decline and subsequent reformation of the innermost dust distribution in the changing-look AGN Mrk 590

ABSTRACT We examine the long-term optical/near-infrared (NIR) flux variability of a ‘changing-look’ active galactic nucleus (AGN) Mrk 590 between 1998 and 2007. Multiband multi-epoch optical/NIR photometry data from the SDSS Stripe 82 data base and the Multicolor Active Galactic Nuclei Monitoring (MAGNUM) project reveal that Mrk 590 experienced a sudden luminosity decrease during the period from 2000 to 2001. Detection of dust reverberation lag signals between V- and K-band light curves obtained by the MAGNUM project during the faint state in 2003–2007 suggests that the dust torus innermost radius Rdust of Mrk 590 had become very small [Rdust ≃ 32 light-days (lt-days)] by the year 2004 according to the aforementioned significant decrease in AGN luminosity. The Rdust in the faint state is comparable to the H β broad-line region (BLR) radius of RH β, BLR ≃ 26 lt-days measured by previous reverberation mapping observations during the bright state of Mrk 590 in 1990–1996. These observations indicate that the innermost radius of the dust torus in Mrk 590 decreased rapidly after the AGN ultraviolet-optical luminosity drop, and that the replenishment time-scale of the innermost dust distribution is less than 4 yr, which is much shorter than the free fall time-scale of BLR gas or dust clouds. We suggest that rapid replenishment of the innermost dust distribution can be accomplished either by new dust formation in radiatively cooled BLR gas clouds or by new dust formation in the disc atmosphere and subsequent vertical wind from the dusty disc as a result of radiation pressure.

Axial and radial dynamics of compressible vortex rings

Vortex rings are typically generated by a piston within a tube for the incompressible case and a bursting membrane in a shock tube for the compressible case. Despite these different origins, they show a common general behaviour which allows us to compare both with a single numerical setup. The two main control parameters are the non-dimensional mass supply which can be expressed as the ratio of length and diameter of the tube ( L ∕ D ) and the pressure ratio from the reservoir to ambient ( p 0 ∕ p ∞ ) . They basically control the blowing duration and the compressibility. We performed Direct Numerical Simulations and studied the influence of variations in ( L ∕ D ) as well as compressibility by keeping the discharge-time constant and appropriate combination of both parameters on four aspects: torus radius R , core radius r , axial position and the axial velocity of the vortex rings. Comparison with a large number of available data from the literature shows good agreement and we were able to reconcile an apparent contradiction in the community about the exponent of the power law describing the axial dynamics of the vortex ring being 1 ∕ 2 or 3 ∕ 2 . In particular, an increase in L ∕ D leads to faster and longer growth of the vortex ring, while the maximum is reached earlier. The transition to the ( t ∗ ) 1 ∕ 2 power law commences also earlier and the axial velocity is higher for larger values of L ∕ D . Keeping the discharge time constant and increasing the Mach number, we found increased torus radii R , the maximum core radius r is reached later, and lower propagation velocities.

Analyzing temporal variations of AGN emission line profiles in the context of (dusty) cloud structure formation in the broad line region

The formation processes and the exact appearance of the dust torus and broad line region (BLR) of active galactic nuclei (AGN) are under debate. Theoretical studies show a possible connection between the dust torus and BLR through a common origin in the accretion disk. However observationally the dust torus and BLR are typically studied separately. NGC 4151 is possibly one of the best suited Seyfert 1 galaxies for simultaneous examinations because of its high number of both photometric and spectroscopic observations in the past. Here we compare changes of the dust radius to shape variations of broad emission lines (BEL). While the radius of the dust torus decreased by almost a factor of two from 2004 to 2006 shape variations can be seen in the red wing of BELs of NGC 4151. These simultaneous changes are discussed in a dust and BEL formation scheme. We also use the BEL shape variations to assess possible cloud distributions, especially in azimuthal direction, which could be responsible for the observed variations. Our findings can best be explained in the framework of a dust inflated accretion disk. The changes in the BELs suggest that this dusty cloud formation does not happen continuously, and over the whole accretion disk, but on the contrary in spatially confined areas over rather short amount of times. We derive limits to the azimuthal extension of the observed localized BEL flux enhancement event.

Long-Lived Quantum Vortex Knots

Dynamics of simplest quantum vortex knots of torus type in a superfluid at zero temperature has been simulated within a regularized Biot-Savart law (the torus radii $R_0$ and $r_0$ for initial vortex configuration were large in comparison with a vortex core width $\xi$). Computations of evolution times of knots until their significant deformation were carried out with a small step on parameter $B_0=r_0/R_0$ for different values of parameter $\Lambda=\log(R_0/\xi)$. It has been found that at $\Lambda\gtrsim 3$, bands of quasi-stability appear in a region of $B_0\lesssim 0.2$, which correspond to long knot lifetimes and to very large traveling distances --- up to several hundreds of $R_0$. This result is new and quite unexpected, because previously it was believed that maximal lifetime of torus knots until reconnection does not exceed several typical periods. The opening of quasi-stable 'windows' at increasing $\Lambda$ is due to narrowing main parametric resonances of the dynamical system on parameter $B_0$.

Determination of the size of the dust torus in H0507+164 through optical and infrared monitoring

The time delay between flux variations in different wavelength bands can be used to probe the inner regions of active galactic nuclei (AGN). Here, we present the first measurements of the time delay between optical and near-infrared (NIR) flux variations in H0507+164, a nearby Seyfert 1.5 galaxy at z = 0.018. The observations in the optical V -band and NIR J, H and Ks bands carried over 35 epochs during the period October 2016 to April 2017 were used to estimate the inner radius of the dusty torus. From a careful reduction and analysis of the data using cross-correlation techniques, we found delayed responses of the J, H and Ks light curves to the V -band light curve. In the rest frame of the source, the lags between optical and NIR bands are found to be $27.1^{+13.5}_{-12.0}$ days (V vs. J), $30.4^{+13.9}_{-12.0}$ days (V vs. H) and $34.6^{+12.1}_{-9.6}$ days (V vs. K ). The lags between the optical and different NIR bands are thus consistent with each other. The measured lags indicate that the inner edge of dust torus is located at a distance of 0.029 pc from the central UV/optical AGN continuum. This is larger than the radius of the broad line region of this object determined from spectroscopic monitoring observations thereby supporting the unification model of AGN. The location of H0507+164 in the {\tau} - MV plane indicates that our results are in excellent agreement with the now known lag-luminosity scaling relationship for dust in AGN.

Mid-infrared Flare of TDE Candidate PS16dtm: Dust Echo and Implications for the Spectral Evolution

Abstract PS16dtm was classified as a candidate tidal disruption event in a dwarf Seyfert 1 galaxy with a low-mass black hole ( ) and has presented various intriguing photometric and spectra characteristics. Using the archival Wide-field Infrared Survey Explorer and the newly released NEOWISE data, we found that PS16dtm is experiencing a mid-infrared (MIR) flare that started ∼11 days before the first optical detection. Interpreting the MIR flare as a dust echo requires close pre-existing dust with a high covering factor and suggests that the optical flare may have brightened slowly for some time before it became bright detectable from the ground. More evidence is given at the later epochs. At the peak of the optical light curve, the new inner radius of the dust torus has grown to a much larger size (i.e., a factor of seven of the initial radius) due to the strong radiation field. At ∼150 days after the first optical detection, the dust temperature has dropped well below the sublimation temperature. Other peculiar spectral features shown by PS16dtm are the transient, prominent Fe ii emission lines and outflows indicated by broad absorption lines detected during the optical flare. Our model explains the enhanced Fe ii emission from iron that is newly released from the evaporated dust. The observed broad absorption line outflow could be explained by accelerated gas in the dust torus due to the radiation pressure.

Evolution of Molecular Clouds in the Superwind Galaxy NGC 1808 Probed by ALMA Observations

Abstract ALMA imaging of the cold molecular medium in the nearby starburst galaxy NGC 1808 is presented. The observations reveal the distribution of molecular gas, traced by 12CO (1–0) and 12CO (3–2), and continuum (93 and 350 GHz) across the central 1 kpc starburst region at a high resolution of . A molecular gas torus (radius ∼ 30 pc) is discovered in the circumnuclear disk (CND; central 100 pc), with a high CO (3–2)/CO (1–0) ratio of ∼1, surrounded by massive (106– ) clouds with high star formation efficiency ( yr−1), molecular spiral arms, and a 500 pc pseudo-ring. The CND harbors a continuum core and molecular gas exhibiting peculiar motion. The new data confirm the line splitting along the minor galactic axis, interpreted as a nuclear gas outflow with average velocity ∼180 km s−1, and show evidence of a velocity gradient of km s−1 pc−1 along the axis. In addition, supershells expanding from the 500 pc ring with maximum velocities of ∼75 km s−1 are revealed. The distribution and CO luminosities of molecular clouds in the central 1 kpc starburst region indicate an evolutionary sequence, from gas accretion onto the 500 pc ring from the large-scale bar to enhanced star formation in the ring, and outflow as feedback.

On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

In this paper we analyze the stress field of a solid torus made of an incompressible isotropic solid with a toroidal inclusion that is concentric with the solid torus and has a uniform distribution of pure dilatational finite eigenstrains. We use a perturbation analysis and calculate the residual stresses to the first order in the thinness ratio (the ratio of the radius of the generating circle and the overall radius of the solid torus). In particular, we show that the stress field inside the inclusion is not uniform. This is in contrast with the corresponding results for infinitely-long and finite circular cylindrical bars and spherical balls with cylindrical and spherical inclusions, respectively. We also show that for a solid torus of any size made of an incompressible linear elastic solid with an inclusion with uniform (infinitesimal) pure dilatational eigenstrains the stress inside the inclusion is not uniform.

Modal instability of the flow in a toroidal pipe

The modal instability encountered by the incompressible flow inside a toroidal pipe is studied, for the first time, by means of linear stability analysis and direct numerical simulation (DNS). In addition to the unquestionable aesthetic appeal, the torus represents the smallest departure from the canonical straight pipe flow, at least for low curvatures. The flow is governed by only two parameters: the Reynolds number $\mathit{Re}$ and the curvature of the torus ${\it\delta}$, i.e. the ratio between pipe radius and torus radius. The absence of additional features, such as torsion in the case of a helical pipe, allows us to isolate the effect that the curvature has on the onset of the instability. Results show that the flow is linearly unstable for all curvatures investigated between 0.002 and unity, and undergoes a Hopf bifurcation at $\mathit{Re}$ of about 4000. The bifurcation is followed by the onset of a periodic regime, characterised by travelling waves with wavelength $\mathit{O}(1)$ pipe diameters. The neutral curve associated with the instability is traced in parameter space by means of a novel continuation algorithm. Tracking the bifurcation provides a complete description of the modal onset of instability as a function of the two governing parameters, and allows a precise calculation of the critical values of $\mathit{Re}$ and ${\it\delta}$. Several different modes are found, with differing properties and eigenfunction shapes. Some eigenmodes are observed to belong to groups with a set of common characteristics, deemed ‘families’, while others appear as ‘isolated’. Comparison with nonlinear DNS shows excellent agreement, confirming every aspect of the linear analysis, its accuracy, and proving its significance for the nonlinear flow. Experimental data from the literature are also shown to be in considerable agreement with the present results.

Symbolic derivation of bicycle kinematics with toroidal wheels

Bicycle kinematics with toroidal wheels is presented in this paper. Using symbolic mathematic tool Maple, we obtain two holonomic and four nonholonomic constraint equations due to front and rear wheels of a bicycle. We show that the two holonomic constraints cannot be expressed in quartic form for bicycle rear body pitch angle unless the minor (crown) radius of the torus are the same for both the front and rear wheels. In addition, we show that all the constraints can be written in differential form, from which a constraint matrix is constructed, according to standard procedure for developing dynamics in robotics.

Computational simulation analysis for torus radius of edge contact in hip prostheses.

Stripe wear occurs when the components of hip prostheses move a sufficient distance laterally to contact the edge of the acetabular cup, causing abnormally high contact stresses. In this research, edge loading contact of prosthetic hip is analyzed in the most commonly used material pairs. The contact dimensions and maximal contact pressure are investigated in mutative normal edge loading with 3 different inclination angles (15°, 20°, 25°) and in alterable edge torus radius with edge loading of 2500 N and inclination of 20°. A computational case was conducted for a 14 mm radius alumina ceramic bearing with a radial clearance of 0.1 mm using a normal edge loading ranged from 0 N to 3000 N. Additionally, the Hertzian theory successfully captures principal curvature trends of the edge torus on the influence of maximal contact pressure and obtains the appropriate edge radius range for lower maximal contact pressure. This work elucidates the methods of applying classical contact theory to design the edge radius of hip bearings to better resist severe edge loading contact stresses and reduce the stripe wear.

REVERBERATION MEASUREMENTS OF THE INNER RADIUS OF THE DUST TORUS IN 17 SEYFERT GALAXIES

We present the results of a dust reverberation survey for 17 nearby Seyfert 1 galaxies, which provides the largest homogeneous data collection for the radius of the innermost dust torus. A delayed response of the K-band light curve after the V-band light curve was found for all targets, and 49 measurements of lag times between the flux variation of the dust emission in the K band and that of the optical continuum emission in the V band were obtained. The lag times strongly correlated with the optical luminosity in the luminosity range of M_V=-16 to -22 mag, and the regression analysis was performed to obtain the correlation log $\Delta t$ (days) = -2.11 -0.2 M_V assuming $\Delta t \propto L^{0.5}$, which was theoretically expected. We discuss the possible origins of the intrinsic scatter of the dust lag-luminosity correlation, which was estimated to be about 0.13 dex, and we find that the difference of internal extinction and delayed response of changes in lag times to the flux variations could have partly contributed to intrinsic scatter. However, we could not detect any systematic change of the correlation with the subclass of the Seyfert type or the Eddington ratio. Finally, we compare the dust reverberation radius with the near-infrared interferometric radius of the dust torus and the reverberation radius of broad Balmer emission lines. The interferometric radius in the K band was found to be systematically larger than the dust reverberation radius in the same band by about a factor of two, which could be interpreted by the difference between the flux-weighted radius and the response-weighted radius of the innermost dust torus. The reverberation radius of the broad Balmer emission lines was found to be systematically smaller than the dust reverberation radius by about a factor of 4-5, which strongly supports the unified scheme of the Seyfert type of active galactic nuclei. (Abridged)

VLTI/AMBER observations of the Seyfert nucleus of NGC 3783

Context. The putative tori surrounding the accretion disks of active galactic nuclei (AGNs) play a fundamental role in the unification scheme of AGNs. Infrared long-baseline interferometry allows us to study the inner dust distribution in AGNs with unprecedented spatial resolution over a wide infrared wavelength range. Aims. Near- and mid-infrared interferometry is used to investigate the milli-arcsecond-scale dust distribution in the type 1.5 Seyfert nucleus of NGC 3783. Methods. We observed NGC 3783 with the VLTI/AMBER instrument in the K-band and compared our observations with models. Results. From the K-band observations, we derive a ring-fit torus radius of 0.74 +/- 0.23 mas or 0.16 +/- 0.05 pc. We compare this size with infrared interferometric observations of other AGNs and UV/optical-infrared reverberation measurements. For the interpretation of our observations, we simultaneously model our near- and mid-infrared visibilities and the SED with a temperature/density-gradient model including an additional inner hot 1400 K ring component.

Dimensionally reduced SYM4 at large-N: an intriguing Coulomb approximation

We consider the light-cone (LC) gauge and LC quantization of the dimensional reduction of super Yang Mills theory from four to two dimensions. After integrating out all unphysical degrees of freedom, then on-local LC Hamiltonian exhibits an explicit \( \mathcal{N} = \left( {2,2} \right) \) supersymmetry. A further SUSY-preserving compactification of LC-space on a torus of radius R, allows for a large-N numerical study where the smooth large-R limit of physical quantities can be checked. As a first step, we consider a simple, yet quite rich, “Coulomb approximation” that maintains an \( \mathcal{N} = \left( {1,1} \right) \) subgroup of the original supersymmetry and leads to a non-trivial generalization of ’t Hooft’s model with an arbitrary –but conserved– number of partons. We compute numerically the eigenvalues and eigenvectors both in momentum and in position space. Our results, so far limited to the sectors with 2, 3 and 4 partons, directly and quantitatively confirm a simple physical picture in terms of a string-like interaction with the expected tension among pairs of nearest-neighbours along the single-trace characterizing the large-N limit. Although broken by our approximation, traces of the full \( \mathcal{N} = \left( {2,2} \right) \) supersymmetry are still visible in the low-lying spectrum.

Nano-scale structural features of stratified hydrocarbon films formed at interaction of plasma with surface in T-10 tokamak

Hydrocarbon films and flakes (i.e. free standing films) with high atomic deuterium to carbon ratio formed at deuterium plasma discharges in T-10 tokamak are non-crystalline. They contain structure elements of cluster/nanopore type with typical sizes of about 1nm and interplane distances between X-ray scattering structures of 0.7, 0.24 and 0.12nm. Structural sp2- carbon states were found mainly on the wall side of flakes, and sp3-hybridized carbon states—on their plasma side. In situ monitoring of films growth and hydrogen isotope accumulation in them by means of Fourier-transform infrared (FT-IR) spectroscopy is proposed. The computational modeling of X-ray scattering by random ensembles of carbon nanostructures of various types suggests that a strong peak of X-ray diffraction (XRD) intensity may be caused by the toroids (toroidal carbon nanotubes) with torus radius of about 1nm.