Infinite Tube Research Articles

Numa Manson on velocity deficits and detonation stability

This memorial paper pays tribute to Professor Numa Manson’s contributions to the understanding of detonation velocity deficits and wave stability. Manson and his colleague Guenoche postulated that a velocity deficit exists in a tube because the chemical reactions are inhibited in a thin layer adjacent to the tube walls. The hydrodynamic theory of detonation was modified to account for this, and it was shown that the deficit varies inversely with the tube diameter. Manson and his students measured detonation velocities in tubes of various diameter. An estimate of the detonation velocity for an infinite tube diameter was obtained by plotting the velocity against the reciprocal of the tube diameter, \({\phi^{-1}}\) , and extrapolating the line through the data to \({\phi^{-1}=0}\) . The relative contributions of tube geometry and surface roughness to the deficits were systematically studied. Manson was also one of the early investigators to shed light on the cellular structure of detonation by reporting “vibratory phenomena” seen as striations in streak schlieren photographs. An attempt was made to relate this phenomenon to “dispersions” in the propagation velocity and hence the wave stability. The author has extended Manson’s work by investigating detonations in tubes with yielding walls. Whereas boundary layers were responsible for the gasdynamic expansion and deficits in Manson’s rigid tubes, it was the moving boundaries that caused similar effects in the author’s investigations. The author has repeated the “nozzle” analysis of Fay and Dabora using the detonation cell length as the relevant chemical kinetic length scale, and found reasonable agreement between his experimental results and the model. When the Poitiers data are reinterpreted in light of the modified model, the trends are described quite well. More recent studies have shown that the measured deficits for mixtures characterized by irregular cellular structures do not agree with the Fay–Dabora model. Possible reasons for the discrepancy are discussed.

Post-bifurcation analysis of a thin-walled hyperelastic tube under inflation

We consider the problem of bulging, or necking, of an infinite thin-walled hyperelastic tube that is inflated by an internal pressure, with the axial stretch at infinity maintained at unity. We present a simple procedure that can be used to derive the bifurcation condition and to determine the near-critical behaviour analytically. It is shown that there is a bifurcation with zero mode number and that the associated axial variation of near-critical bifurcated configurations is governed by a first-order differential equation that admits a locally bulging or necking solution. This result suggests that the corresponding bifurcation pressure can be identified with the so-called initiation pressure which featured in recent experimental studies. This is supported by good agreement between our theoretical predictions and one set of experimental data. It is also shown that the Gent material model can support both bulging and necking solutions whereas the Varga and Ogden material models can only support bulging solutions. Relevance of the present method to the study of non-linear wave propagation in a fluid-filled distensible tube is also discussed.

A Hardy Inequality in Twisted Waveguides

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.

PROPAGATION OF WAVES IN A BIFURCATED CYLINDRICAL WAVEGUIDE WITH WALL IMPEDANCE DISCONTINUITY

In the present work the radiation of sound from a bifurcated circular waveguide formed by a semi-infinite rigid duct inserted axially into a larger infinite tube with discontinuous wall impedance is reconsidered through an alternative approach which consists of using the mode matching technique in conjunction with the Wiener-Hopf method. By expressing the total field in the appropriate waveguide region in terms of normal modes and using the Fourier transform technique elsewhere, we end up with a single modified Wiener-Hopf equation whose solution involves an infinite system of algebraic equations. This system is solved numerically and the influence of some parameters on the radiation phenomenon is shown graphically. The equivalence of the direct method described in (1) and the present mixed method are shown numerically.

Crystal structure of the tritylated product of 3-hydroxymethylbicyclo[3.3.1]nonan-2-on-7-ol ethylene acetal cyclization

A tritylated product of 3-hydroxymethylbicyclo[3.3.1]nonan-2-on-7-ol ethylene acetal cyclization, 4-hydroxymethyl-3-[2-(trityloxy)-ethoxy]-2-oxatricyclo[4.3.1.03,8]decane, was found to have a crystal structure comprising close packed molecular layers intersected by infinite tubes filled with solvent molecules.

Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator.

Micromagnetic simulations of hysteresis in an array of cobalt nanotubes

Here we perform modeling of hysteresis measurements, described earlier by Crowley et al., of an array of cobalt nanotubes using a single infinite tube simulation. The modeling has been performed under the assumptions that (i) the long axis of the tube is the hard axis of the anisotropy and (ii) in the experiment we deal with a distribution of different tubes, each having a different anisotropy constant. The best fit to the experimental result is achieved using a log-normal anisotropy distribution, with a peak value close to −0.2 MJ/m 3. The distribution parameters are nearly the same for both experimental temperatures, T exp=1.8 and 300 K. These parameters are only weakly dependent on the tube-wall width and the material parameters.

Zigzag carbon nanotubes—Molecular/structural mechanics and the finite element method

This paper discusses in details the relation between the bond bending stiffness used in molecular mechanics and the bending stiffness used in structural mechanics for zigzag carbon nanotubes (CNTs). Recent publications assumed the structural bending stiffness EI/ a to be a constant and set it equal to the molecular bond bending stiffness C. By developing a closed form expression for the deformation of zigzag CNTs under simple tension, we suggest that the relation between EI/ a and C is more complex. It actually depends on the bond bending stiffness C, the torsional angle φ and the lattice translational index n. In the limit of an infinite tube radius, which represents a graphene sheet, EI/ a tends to C/2. Numerical simulations are also presented that validate the results.

Supramolecular crystal structures of per(3,6-anhydro)-α-cyclodextrin grown from KCl or NaI solutions

Crystals of per(3,6-anhydro)-α-cyclodextrin{hexakis(3,6- anhydro)cyclomaltohexaose} ( 1) grown in presence of KCl or NaI evidence the similar distorted conformation of macrocycle and provide the same supramolecular scaffolding of cyclodextrins based on CH⋯O hydrogen bonds. Infinite tubes parallel to symmetry axis 6 are filled by water molecules or by iodine atoms in 1/KCl and 1/NaI crystals, respectively. In 1/KCl crystals an inclusion complex of stoichiometry 1:2 has been found. Coordination polyhedra of the dimeric K units fuse with the coordination polyhedra of one pentacoordinated chloride atom to form an infinite helix along the c direction crossing over a helical organic envelope of cyclodextrin molecules.

Carbon chains and the (5,5) single-walled nanotube: Structure and energetics versus length

Reliable thermochemistry is computed for infinite stretches of pure-carbon materials including acetylenic and cumulenic carbon chains, graphene sheet, and single-walled carbon nanotubes (SWCNTs) by connection to the properties of finite size molecules that grow into the infinitely long systems. Using ab initio G3 theory, the infinite cumulenic chain (:C[double bond]C[double bond]C[double bond]C:) is found to be 1.9+/-0.4 kcal/mol per carbon less stable in free energy at room temperature than the acetylenic chain (.C[triple bond]C-C[triple bond]C.) which is 24.0 kcal/mol less stable than graphite. The difference between carbon-carbon triple, double, and single bond lengths (1.257, 1.279, and 1.333 A, respectively) in infinite chains is evident but much less than with small hydrocarbon molecules. These results are used to evaluate the efficacy of similar calculations with the less rigorous PM3 semiempirical method on the (5,5) SWCNT, which is too large to be studied with high-level ab initio methods. The equilibrium electronic energy change for C(g)-->C[infinite (5,5) SWCNT] is -166.7 kcal/mol, while the corresponding free energy change at room temperature is -153.3 kcal/mol (6.7 kcal/mol less stable than graphite). A threefold alternation (6.866, 6.866, and 6.823 A) in the ring diameter of the equilibrium structure of infinitely long (5,5) SWCNT is apparent, although the stability of this structure over the constant diameter structure is small compared to the zero point energy of the nanotube. In general, different (n,m) SWCNTs have different infinite tube energetics, as well as very different energetic trends that vary significantly with length, diameter, and capping.

Density Functional Study of the13C NMR Chemical Shifts in Small-to-Medium-Diameter Infinite Single-Walled Carbon Nanotubes

NMR chemical shifts were calculated for semiconducting (n,0) single-walled carbon nanotubes (SWNTs) with n ranging from 7 to 17. Infinite isolated SWNTs were calculated using a gauge-including projector-augmented plane-wave (GIPAW) approach with periodic boundary conditions and density functional theory (DFT). In order to minimize intertube interactions in the GIPAW computations, an intertube distance of 8 A was chosen. For the infinite tubes, we found a chemical shift range of over 20 ppm for the systems considered here. The SWNT family with lambda = mod(n, 3) = 0 has much smaller chemical shifts compared to the other two families with lambda = 1 and lambda = 2. For all three families, the chemical shifts decrease roughly inversely proportional to the tube's diameter. The results were compared to calculations of finite capped SWNT fragments using a gauge-including atomic orbital (GIAO) basis. Direct comparison of the two types of calculations could be made if benzene was used as the internal (computational) reference. The NMR chemical shifts of finite SWNTs were found to converge very slowly, if at all, to the infinite limit, indicating that capping has a strong effect (at least for the (9,0) tubes) on the calculated properties. Our results suggest that (13)C NMR has the potential for becoming a useful tool in characterizing SWNT samples.

Distorted Born Diffraction Tomography Applied to Inverting Ultrasonic Field Scattered by Noncircular Infinite Elastic Tube

This study focuses on the application of ultrasonic diffraction tomography to noncircular 2D-cylindrical objects immersed in an infinite fluid. The distorted Born iterative method used to solve the inverse scattering problem belongs to the class of algebraic reconstruction algorithms. This method was developed to increase the order of application of the Born approximation (in the case of weakly-contrasted media) to higher orders. This yields quantitative information about the scatterer, such as the speed of sound and the attenuation. Quantitative ultrasonic imaging techniques of this kind are of great potential value in fields such as medicine, underwater acoustics and nondestructive testing.

The Structure of Onoratoite, Sb8O11X2 (X: Cl, Br) Revisited.

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Gravitational model of the string

AbstractIt is shown that an infinite gravitational flux tube solution in 5D Kaluza‐Klein gravity with the cross section in the Planck region after 5D ⇒ 4D reduction and isometrical embedding in a Minkowski spacetime can be considered as a moving infinite string‐like object. Such an object carries an electric and a magnetic flux. The 4D gravitational waves on the tube are considered.

The structure of onoratoite, [formula omitted] ( [formula omitted], Br) revisited

The crystal structure of Sb 8O 11Cl 2 (onoratoite) is redetermined in the triclinic space group P-1. The crystal structure was previously described by a partially disordered model in the monoclinic space group C 2 / m containing some partially occupied oxygen atoms. The new structural model features exclusively atomic positions that are fully occupied. The crystal structure was solved from single crystal data and the unit cell parameters are a = 10.317 ( 2 ) Å , b = 11.269 ( 2 ) Å , c = 13.939 ( 3 ) Å , α = 79.29 ( 2 ) ° , β = 76.29 ( 2 ) ° , γ = 73.00 ( 2 ) ° , Z = 2 , R = 0.0637 . It contains antimony oxide in the form of infinite tubes interleaved with halide ions, a main feature very similar to the previous model. However, in the new model the tubes contain both [SbO 4E] (E = 5s 2 lone pair of electrons) trigonal bipyramids and [SbO 3E] tetrahedral, whereas the old model was composed entirely of trigonal bipyramidal [SbO 4E] units. The isostructural compound Sb 8O 11Br 2 was synthesized as well.

Radiation of sound from a semi-infinite rigid duct inserted axially into a larger infinite tube with wall impedance discontinuity

In the present work the radiation of sound from a bifurcated circular waveguide formed by a semi-infinite rigid duct inserted axially into a larger infinite tube with discontinous wall impedance is analyzed. The formulation of the boundary-value problem in terms of Fourier integrals leads to a matrix Wiener-Hopf equation which is uncoupled by the introduction of infinite sum of poles. The exact solution is then obtained in terms of the coefficients of the poles, where these coefficients are shown to satisfy infinite system of linear algebraic equations. This system is solved numerically and the influence of the parameters such as the outer cylinder radius and the discontinuity of the surface impedances on the radiation phenomenon is shown graphically.

Gravitational model of the string

It is shown that an infinite gravitational flux tube solution in 5D Kaluza-Klein gravity with the cross section in the Planck region after $5D \to 4D$ reduction and isometrical embedding in a Minkowski spacetime can be considered as a moving infinite string-like object. Such object has a flux of the electric and magnetic fields. The 4D gravitational waves on the tube are considered.

N-spheres in general relativity: Regular black holes without apparent horizons, static wormholes with event horizons and gravastars with a tube-like core

We consider a way to avoid black hole singularities by gluing a black hole exterior to an interior with a tube-like geometry consisting of a direct product of two-dimensional AdS, dS, or Rindler spacetime with a two-sphere of constant radius. As a result we obtain a spacetime with either “cosmological” or “acceleration” (event) horizons but without an apparent horizon. The inner region is everywhere regular and supported by matter with the vacuum-like equation of state pr+ρ=0 where pr=Trr is the longitudinal pressure, ρ=−T00 is the energy density, Tμν is the stress–energy tensor. When the surface of gluing approaches the horizon, surface stresses vanish, while pr may acquire a finite jump on the boundary. Such composite spacetimes accumulate an infinitely large amount of matter inside the horizon but reveal themselves for an external observer as a sphere of a finite ADM mass and size. If the throat of the inner region is glued to two black hole exteriors, one obtains a wormhole of an arbitrarily large length. Wormholes under discussion are static but not traversable, so the null energy condition is not violated. In particular, they include the case with an infinite proper distance to the throat. We construct also gravastars with an infinite tube as a core and traversable wormholes connected by a finite tube-like region.

Measuring lateral efficiency of optical traps: The effect of tube length

An optical tweezer setup is made based on a custom-designed inverted microscope, which can work both in finite and infinite tube length microscopy modes. It is shown that the spherical aberration due to the mismatch in the refractive indices of the specimen (water) and the immersion oil as well as the wavelength can be partially compensated by introducing another source for the spherical aberration provided it has the opposite sign. Changing the tube length is shown to be a good candidate for this effect: an improvement of up to a factor of four has been observed in the lateral efficiency of the trap.

Mathematical modeling of the residence time distribution in loop reactors

The axial dispersion model was applied to the riser and down-comer, respectively, of an airlift loop reactor and a mathematical model was developed to describe its residence time distribution (RTD). A numerical solution was first obtained for the case where there is no gas–liquid separator. A tracer impulse was considered to flow in an infinite tube comprising alternating riser, gas–liquid separator and down-comer. The tracer concentration at a point in the reactor is the sum of all the tracer concentrations at the corresponding points in the infinite tube. Based on this model, an analytical solution was obtained for the mathematical model. The comparison between the analytical solution and experimental data shows that the model can be used to simulate the RTD in airlift loop reactors. By using the analytical solution in a loop reactor with partial recirculation, the solution with a recycle factor is obtained. A comparison with data in the literature shows that the solution with the recycle factor can be used for the RTD in a loop reactor with partial recirculation.