Finite Cylindrical Geometry Research Articles

Damping of dark envelope soliton in a viscous bounded dusty plasma

The propagation of dust-acoustic waves in a dusty plasma bounded in finite symmetrical cylindrical geometry has been theoretically investigated. By using the reductive perturbation method, we obtain a quasi-nonlinear Schrodinger equation. It is shown that there is a dark envelope soliton under certain conditions, while its amplitude will decrease with the time. The dependence of both the dispersion relation and group velocity of envelope solitary waves on the radius of cylinder has been given. Moreover, we define the duration time of dark envelope solitons and give the theoretical ones. It is found that the duration time increases as the radius of cylinder increases, while it decreases as the viscosity coefficient of dusty plasmas increases.

Computing the Karhunen–Loève dimension of an extensively chaotic flow field given a finite amount of data

The use of Karhunen–Loève decomposition (KLD) to explore the complex fluid flows that are common in engineering applications is increasing and has yielded new physical insights. However, for most engineering systems the dimension of the dynamics is expected to be very large yet the flow field data is available only for a finite time. In this context, it is important to establish the amount of data required to compute the asymptotic value of the Karhunen–Loève dimension given a finite amount of data. Using direct numerical simulations of Rayleigh–Bénard convection in a finite cylindrical geometry we compute the asymptotic value of the Karhunen–Loève dimension. The amount of time required for the Karhunen–Loève dimension to reach a steady value is very slow in comparison with the time scale of the convection rolls. We show that the asymptotic value of the Karhunen–Loève dimension can be determined using much less data if one uses the azimuthal symmetry of the governing equations prior to performing a KLD. The Karhunen–Loève dimension is found to be extensive as the system size is increased and for a dimension measurement that captures 90% of the variance in the data the Karhunen–Loève dimension is approximately 20 times larger than the Lyapunov dimension.

A diffusion based model for intermittent drying of rough rice

In this study, intermittent drying behavior of single layer rough rice with a moisture content of between 22 and 24% on the dry basis was simulated by means of a liquid diffusion model based on a prolate spheroid geometry. For this purpose, solution of the liquid diffusion equation was fitted to the experimental data for the drying air temperature 40°C, drying velocity 1.5 ms−1 and tempering periods ranging from 0 to 1 h. In order to make a comparison, solution of the liquid diffusion equation for a finite cylindrical geometry was also fitted to the experimental data. The results show that the liquid diffusion model based on a prolate spheroid geometry explains the drying behavior of rough rice more accurately. The results also show that greater variations occur in diffusion coefficient with increasing tempering time for prolate spheroid geometry which is more realistic geometry for a rough rice grain.

Eigenfunctions of the curl in annular cylindrical and rectangular geometry

A method for determining the eigenfunctions of the equation ∇⃗×B⃗=λB⃗ for finite cylindrical geometry with normal boundary condition B⃗∙n̂=0 and nonaxisymmetric modes ∼eimθ,m≠0 was given by Morse [J. Math. Phys. 46, 113511 (2005)] This method uses series solution of a Chandrasekhar-Kendall [Astrophys. J. 126, 157 (1957)] function. Here this method is generalized to include annular (coaxial) cylindrical geometry of finite length. The method is also applied to the case of rectangular duct geometry with periodic z-dependence. For slender aspect ratios in the annular cylinder, the eigenfunctions and eigenfields correspond to the rectangular duct. The eigenfunctions for finite-length rectangular boxes can also be obtained from the rectangular duct case. The robust convergence properties (an∼1∕n4) and simple eigenvalue equations from Morse are retained in these other geometries.

Simulation of Dynamic Levitation Force Taking Flux Creep Into Account

This work compares dynamic levitation force measurements, at different approaching speeds, with levitation force simulations that take flux creep into account. This was done using a power-law electric field current-voltage relationship that can be derived from the Anderson-Kim model. The algorithm for the simulation starts from the analytical expressions of the classical electromagnetic theory to write the integral equation of the time derivative of the current density inside the superconductor, depending on the geometry of the system and the configurations of the applied field. Then, using the Method of Moments, the analytical integral equation was written in its matrix formulation. The current density in each time step is obtained by a simple integral rule (method of Euler). From the current density profile in the superconductor, the levitation force between a permanent magnet and a superconductor, with finite cylindrical geometry, is calculated. The current density profile depends on the approaching speed. The results of the simulations were compared favorably with the experimental measurements.

A liquid diffusion model for thin-layer drying of rough rice

In this study, the drying behavior of single-layer rough rice with a moisture content of between 22 and 24% on the dry basis was simulated by means of a liquid diffusion model, based on a prolate spheroid geometry. For this purpose, the solution of liquid diffusion equation was fitted to the experimental moisture ratios for drying air temperatures between 40 and 60 °C and velocity 1.5 m s−1. In order to make a comparison, the predictions of liquid diffusion equations for a spherical and finite cylindrical geometry were also fitted to the experimental results. Modeling was performed by selecting the diffusion coefficients in diffusion equations in such a manner as to minimize the sum of the squared differences between the experimental results and the theoretical predictions. It was found that the liquid diffusion model, based on a prolate spheroid geometry, explains single-layer drying behavior of rough rice well. It was also found that the model, based on a prolate spheroid geometry, has better agreement with the experimental results than the other geometries.

Eigenfunctions of the curl in cylindrical geometry

Eigenfunctions of the equation ∇⃗×B⃗=λB⃗ are found for finite cylindrical geometry with normal boundary condition B⃗∙n̂=0 and nonaxisymmetric modes ∼eimθ,m≠0. The vector field B⃗ can be represented by a scalar generating function of the Chandrasekhar-Kendall type with radial Bessel functions for the nondegenerate cases. A general set of solutions can also be generated by transformation of variables. A series solution in terms of radial Bessel functions is found which has excellent convergence properties (an∼1∕n4) and a robust method of locating eigenvalues is described.

Computer simulation of food sterilization using an alternating direction implicit finite difference method

A computer simulation model for food sterilization was developed based on the solution of Fourier’s heat conduction equation for finite cylindrical geometry. The solution is based on the alternating direction finite difference method. Experimental verification of the model was performed using two sets of published data, which shows good agreement with the simulation results. Furthermore, the model performance was tested by subjecting a constant retort temperature profile to an upset condition, the model responded quite well indicting its potential for investigating the influence of constant retort temperature upset conditions on lethality and the effects of variable retort temperature profiles on quality retention and process optimization. In addition, the model can also be used for process control.

A realistic model of spin-transport in dilute [formula omitted] in [formula omitted] in a finite geometry

We have investigated the spin-motion of spin-polarised solutions of Fermi-liquid 3 He in 4 He below 100 mK . The linearised version of the equation of motion is solved for a finite cylindrical geometry with fully spin-reflecting boundary conditions, and is used to model an actual NMR spin-echo experiment. We include non-linear gradients as well as the applied, linear gradient and we have also incorporated the effect of finite-amplitude NMR excitation pulses. We find that the presence of boundaries causes marked deviation from the analytical expressions previously used for data analysis. The non-uniform gradient causes spin-echoes to be delayed and it is noted that the finite amplitude pulses also have a significant effect on the appearance and timing of spin-echoes. Previous values for the ‘anisotropy temperature’ in the spin-polarised Fermi-liquid, derived experimentally from spin-echoes, are shown to be overestimates.

Spherical harmonics — Finite element treatment of neutron transport in cylindrical geometry

A variational spherical harmonics (DP N ) method is used in conjunction with the finite element method to solve the transport of neutron beams. The method is developed for finite axisymmetric cylindrical geometry. In this model, the angular distribution along the path of neutron travel is separated into two sub-ranges of the forward and the backward moving particles. The system of the model equations is cast in terms of an even and an odd vector analogue to the even-parity transport technique. The finite element method is applied in space to solve the resulting coupled system of equations. A source-free cylinder with specified incident flux on the z = 0 surface is considered in this study. The model is based on the DP 1 and DP 3 orders of expansion. The study demonstrates the advantages of the DP N method in accounting for the discontinuity of the angular flux in the direction of particles with no approximation. Results for the scalar flux are presented in isodose curve (contour) form for homogeneous and non-homogeneous media with different absorbing and purely scattering media. Numerical results compared favorably with those obtained by the exact solutions.

Bias voltage in finite length, cylindrical and coaxial radio‐frequency discharges

We obtain the ratio Va/Vb of the direct‐current glow‐to‐electrode voltages at the powered (a) and grounded (b) electrodes in low pressure capacitive radio‐frequency (rf) discharges having unequal electrode areas Aa and Ab in finite length cylindrical and coaxial geometry. We use a transport model in the glow in which ions are generated by thermal electron ionization and are lost by ambipolar diffusion in the glow, with a constant diffusion coefficient. Three electrode sheath models are considered: collisionless ions, collisional ions with a constant mean free time, and collisional ions with a constant mean free path. Using these and the continuity of the rf current flow, we obtain the voltage ratio as a function of the electrode area ratio, the length‐to‐radius ratio of the cylinder, and, for coaxial systems, the ratio of inner‐to‐outer radii. The theory is in good agreement with experiments in cylindrical and coaxial geometry.

Transfer-matrix spectrum for systems with interfaces.

We consider Ising-type systems in the finite cross-section cylindrical geometry, with an interface imposed by the antiperiodic or + - boundary conditions. Closed forms are derived for the largest, asymptotically degenerate transfer-matrix eigenvalues. For rough interfaces ($0<T<{T}_{c}$ in 2D, ${T}_{R}<T<{T}_{c}$ in 3D), the eigenvalues are split algebraically, and the spectral gaps are governed by the surface stiffness coefficient. For "rigid" interfaces ($0<T<{T}_{R}$ in 3D), the splitting is exponential, and determined by the step-free energy.

Stability and heat transfer of rotating cryogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection

This report presents data describing convection in a rotating cylindrical Bénard cell filled with He I. In particular, convection modes are observed at Rayleigh numbers substantially below those predicted by linear stability analyses for a horizontally infinite layer. Both the Rayleigh numbers associated with the convective onset and the initial-slope measure of heat transport of these modes are found to depend on the rotation rate Ω and the aspect ratio Γ of the cell. A discussion of the relevant literature reveals that these convective modes are probably the same as those observed by Rossby (1969) and are reasonably well characterized by the recent analysis of Buell & Catton (1983) assuming asymmetric modes.

The integral-transform method for solving neutron transport problems with general anisotropic scattering in a cylinder of finite height

In this paper, we present a mathematical technique for solving the integral transport equation for the criticality of a homogeneous cylinder of finite height. The purpose of the present paper is two-fold : firstly, to show that our earlier formalism can be generalized to any order of anisotropy, and secondly to generate the numerical results, which could serve as benchmarks when scattering is linearly anisotropic. We expand the scattering function in spherical harmonics to retain the Lth order of anisotropy. Thereafter, we write the integral transport equations for the Fourier-transformed spherical harmonic moments of the angular flux. In conformity with the integral-transform method for multidimensional geometry, the kernels of these integral equations are represented in their respective factorized form, which consists of a series of products of suitable spherical Bessel functions. The Fourier-transformed spherical harmonic moments are also represented in their separable form by expanding them in a series of products of spherical Bessel functions, commensurate with the symmetry of finite cylindrical geometry. The criticality problem for the cylinder of finite height is then posed as a matrix eigen value problem whose eigen vector is composed of the expansion coefficients mentioned above. The general matrix element is expressed as a product of certain integrals of Bessel functions, which can be evaluated by recursion relations derived in this paper. Finally, a comparison between the present benchmark results and S N results ( twotran) in ( r–z) goemetry is presented when scattering is linearly anisotropic.

Phenomena in unsteady rotating flows for a finite cylindrical geometry

Abstract

Spin-Up and Spin-Down of Rotating Flows in Finite Cylindrical Containers

A simplified theoretical model is used for the successful description of measured impulsively started spin-up and impulsively stopped spin-down times in a finite cylindrical geometry at a Reynolds number R of 1002. The essential physical idea used in the analysis is very rapid establishment of steady-state boundary-layer flows in times that are short compared with spin-up and spin-down times in the interior (core) of the liquid (water).

Velocity Profiles in Steady and Unsteady Rotating Flows for a Finite Cylindrical Geometry

Experimental determinations have been made of tangential velocity profiles in liquid water under conditions of steady and unsteady rotating flows for low (4 to 5) and high (1000) Reynolds numbers. Velocity measurements were performed by using the laser-Doppler velocimeter in various finite cylindrical arrangements with one stationary and one rotating disk. The steady velocity profiles (at appreciable distances from the cylindrical wall) have been found to be in excellent agreement with theoretical predictions based on solutions of the requisite boundary-value problems for cylinders of infinite radius. The observed unsteady velocity profiles (at appreciable distances from the cylindrical wall) for impulsive slow-down and impulsive start-up at small Reynolds numbers (R) are well accounted for by using a linearized theory for an axisymmetric configuration of infinite radial extent. For large values of R, no large differences were observed in the characteristic slow-down times at different radial locations (away from the cylindrical wall). The experimentally observed tangential velocity profiles for impulsive start-up show characteristic development times that are sensitive functions of the finite radial extent used in the geometric arrangement. Speed-up times decrease as the radial distance from the point of observation to the confining cylinder is reduced. In geometric arrangements in which only the cylinder radius was varied, characteristic development times for tangential motion were found to be a function of the dimensionless ratio measuring the radial distance to the wall divided by the cylinder radius.

A Numerical Method for Solving the Neutron Transport Equation in Finite Cylindrical Geometry

A method is presented which provides a numerical solution to the steady state, energy dependent neutron transport equation for finite cylindrical geometry, with anisotropic treatment of elastic scattering and isotropic treatment of inelastic scattering. The main characteristic features of the method are the use of quasi-cartesian coordinates and the application of discrete ordinate numerical integration. A difference form of the Boltzmann equation is derived as the final expression for machine computation. Comparisons are given of the numerical solutions with an analytical solution for a constant source distribution, and with NIOBE calculations and experimental spectra for neutron transport in water, with good agreement obtained between them.

A Numerical Method for Solving the Neutron Transport Equation in Finite Cylindrical Geometry

A Numerical Method for Solving the Neutron Transport Equation in Finite Cylindrical Geometry

Measurement of the thermal neutron diffusion parameters in diphenyl and monoisopropylbiphenyl (MIPBy pulsed neutron methods

The diffusion parameters in diphenyl (C 10H 10) at 77°C and MIPB (C 15H 16) at 30°C were determined using the pulsed neutron source technique. The 1 MeV Van de Graaff accelerator at North Carolina State College, Raleigh, was used as the pulsed source of fast neutrons utilizing the 9Be (d, n) 10B reaction. The decay of the thermal neutron leakage from a finite cylindrical geometry was measured using a 26-channel time analyser. By measuring the thermal neutron decay constant, A, as a function of buckling, B 2, a least squares fit of the data to the equation λ = λ 0 + D 0 B 2 − CB 4 yielded values of D 0, the diffusion constant, and C, the diffusion cooling coefficient. The infinite medium decay constant, λ 0, was calculated from known cross-section data. For diphenyl measurements in the buckling range from 0·0816 cm −2 to 0·2811 cm −2, the diffusion parameter values obtained were D 0 = 5437 ± 0117 × 10 4 cm 2-sec −1 and C = 9·85 ± 5·50 × 10 3 cm 4-sec −1. The results obtained for MIPB in the buckling range from 0·0945 cm −2 to 0·2952 cm −2 were D 0 = 3780 ± 0·033 × 10 4 cm 2-sec −1 and C = 125 ± 1·43 × 10 3 cm 4-sec −1. The values of the transport mean free path, λ tr, and diffusion length, L, derived from these results were found to be less than those values predicted by current theoretical models. A preliminary investigation of λ tr, in MIPB as a function of temperature showed a non-linear behaviour of the transport mean free path with temperature.