Prolate Spheroid Geometry Research Articles

The role of axial angular momentum in exact non-linear solutions of multipolar spherical and cylindrical vortices

The time-dependent acceleration of fluid particles in an arbitrary superposition of multipolar spherical and cylindrical vortices in presence of a background rotational rigid flow is the gradient of the difference between their kinetic energy and the product of their total axial angular momentum times the angular velocity of the background flow. If the potential energy is defined as the time-dependent field whose minus gradient equals the acceleration of the flow, then the total energy, defined as the sum of kinetic and potential energies, equals the total axial angular momentum times the angular velocity of the background flow. The total energy of a fluid particle has therefore a linear relationship with the azimuthal velocity field. The role of the axial angular momentum in these oscillating flows is explained also in the classical Lagrangian and Hamiltonian descriptions of motion. The linear relation between total energy and angular momentum makes possible that the free parameters of these flows may be obtained, in a way similar to that developed in the quantum mechanics description of motion, as the eigenvalues of linear operators acting on some components of the spherical modes. Beltrami flow solutions in prolate spheroidal geometry for axisymmetric flows are also discussed.

Design of a Compact 1-MV-Class Radial Closing Switch for Tesla Type Generator

The paper presents the development of a megavolt-class SF6-insulated radial switch. The switch is used as a crowbar in a MV Tesla-type generator to produce pulsed electric fields in very large volumes, for proof-of-concept experimentation of a novel non-invasive food processing technology. The main features of the switch include: (1) Using SF6 gas as the insulation gas; (2) a breakdown channel along the radial direction, rather than axial; (3) a compact configuration with its volume limited to 3.2 L. In order to achieve a ruggedized high voltage insulation as well as an enhanced operation safety, the following design techniques were applied: (1) Structures of the switch were well designed to minimize the local electric filed in the cathode triple junctions; (2) the grooves of surface of the insulators that enclose the switch were finely optimized to keep the surface flashover under control; (3) a prolate spheroid geometry of the high voltage electrode was adopted to achieve a better control of the gas breakdown. This paper describes in detail the design and the preliminary test of this switch.

Mathematical Modeling of the Evolution of the Exterior Boundary in Spheroidal Tumour Growth

The present paper concerns the formulation and the evolution of the non symmetrical growth of an avascular cancerous cell colony in an analytical mathematical fashion. Although most of the existing research considers spherical tumours, here we work in the frame of a more general case of the prolate spheroidal geometry. The tumour lies inside a host spheroidal shell which provides vital nutrients, receives the debris of the dead cells and also transmittes to the tumour the pressure imposed by the surrounding on its exterior boundary. Under the aim of studying the evolution of the exterior tumour boundary, we focus on the exterior conditions under which such a geometrical structure can be sustained. To that purpose, the corresponding nutrient concentration, the inhibitor concentration and the pressure field are calculated analytically providing the necessary data for the evolution equation to be solvable. It turns out that an avascular tumour can exhibit a prolate spheroidal growth only if the external conditions for the nutrient supply and the transversally isotropic pressure field have a specific form, which is consistent with the tumour evolution. Additionally, our model exhibits a geometrical reduction to special cases and, mainly, to the spherical geometry in order to recover the existing results for the sphere.

Modeling and experimentation of continuous and intermittent drying of rough rice grains

Rice (Oryza sativa L.) is one of the most produced and consumed cereals in the world, being characterized as the main food of more than half of the world population, since it is source of energy, protein, vitamins and minerals. Freshly harvested rice from the field generally has a high moisture content to be stored safely and, therefore, a suitable drying process needs to be conducted to decrease the physical-chemical activity of the product and inhibit the associated microbial activities. In this work, experiments were carried out on the intermittent drying of rice grains, in order to evaluate the effects of different drying air temperatures (40, 50, 60 and 70 °C) and tempering times (30, 60, 120, 180 and 240 min). Intermittent drying was simulated by means of a new liquid diffusion model based on a prolate spheroid geometry. To validate the model, the solution was fitted to the experimental data for the drying air temperatures of 40 and 70 °C and tempering periods ranging from 0 to 180 min. The results show that, for all experiments of intermittent drying of rough rice, the effective operating time decreased compared to the continuous drying. In addition, intermittent drying produces lower temperature on the grain surface, which minimizes the thermal damages caused to the product during the process. Numerical simulations provide information on moisture distribution inside the rice grain during the periods of interruption in hot air application. After 15 min of drying at temperatures of 40 and 70 °C, followed by a 60-min pause at room temperature, it is possible to respectively reduce about 72 and 68% of the moisture content gradients inside the rice grain. With tempering of 180 min, it is possible to eliminate almost all moisture content gradients for the drying at 70 °C.

Determining the Spheroid Geometry of Individual Metallic Nanoparticles by Two-Dimensional Single-Particle Dynamic Light Scattering

Single-particle dynamic light scattering (SP-DLS) is a recently developed technique that uses dark-field illumination, active real-time three-dimensional single-particle tracking, and measurements of scattered photon polarizations to nonperturbatively evaluate the shapes of single, freely diffusing particles under the assumption of the particle having either prolate or oblate spheroid geometry. As originally developed, however, SP-DLS is incapable of unambiguously assigning either of these geometries to a single particle. In this contribution, we resolve this ambiguity by introducing a second experimental observable—the scattering spectrum—so that both the scattering polarization and spectrum are simultaneously recorded and analyzed. We used numerical simulations of SP-DLS to characterize the performance of this new approach as well as the effects of key experimental parameters. We anticipate that the analyses presented here will not only form a straightforward guide for researchers seeking to optimize th...

Image Theory for Neumann Functions in the Prolate Spheroidal Geometry

Interior and exterior Neumann functions for the Laplace operator are derived in terms of prolate spheroidal harmonics with the homogeneous, constant, and nonconstant inhomogeneous boundary conditions. For the interior Neumann functions, an image system is developed to consist of a point image, a line image extending from the point image to infinity along the radial coordinate curve, and a symmetric surface image on the confocal prolate spheroid that passes through the point image. On the other hand, for the exterior Neumann functions, an image system is developed to consist of a point image, a focal line image of uniform density, another line image extending from the point image to the focal line along the radial coordinate curve, and also a symmetric surface image on the confocal prolate spheroid that passes through the point image.

Current from a nano-gap hyperbolic diode using shape-factors: Theory

Quantum tunneling by field emission from nanoscale features or sharp field emission structures for which the anode-cathode gap is nanometers in scale (“nano diodes”) experience strong deviations from the planar image charge lowered tunneling barrier used in the Murphy and Good formulation of the Fowler-Nordheim equation. These deviations alter the prediction of total current from a curved surface. Modifications to the emission barrier are modeled using a hyperbolic (prolate spheroidal) geometry to determine the trajectories along which the Gamow factor in a WKB-like treatment is undertaken; a quadratic equivalent potential is determined, and a method of shape factors is used to evaluate the corrected total current from a protrusion or wedge geometry.

Radiation damping of, and scattering from, an arbitrarily shaped bubble.

The work by Strasberg on small volume oscillations of bubbles with arbitrary shape [J. Acoust. Soc. Am. 25, 536-537 (1953)] is here extended to include the effects of radiation damping. An expression for the far-field scattering amplitude from an arbitrarily shaped bubble is given in terms of a quantity that is mathematically equivalent to the electrostatic capacitance of the bubble shape. This general approach is then applied to prolate and oblate spheroidal geometries, for which simple analytic expressions are available for the electrostatic capacitance, and the resulting far-field scattering amplitudes are compared to previous work in the literature.

Green’s function and image system for the Laplace operator in the prolate spheroidal geometry

In the present paper, electrostatic image theory is studied for Green’s function for the Laplace operator in the case where the fundamental domain is either the exterior or the interior of a prolate spheroid. In either case, an image system is developed to consist of a point image inside the complement of the fundamental domain and an additional symmetric continuous surface image over a confocal prolate spheroid outside the fundamental domain, although the process of calculating such an image system is easier for the exterior than for the interior Green’s function. The total charge of the surface image is zero and its centroid is at the origin of the prolate spheroid. In addition, if the source is on the focal axis outside the prolate spheroid, then the image system of the exterior Green’s function consists of a point image on the focal axis and a line image on the line segment between the two focal points.

2D/3D image charge for modeling field emission

Analytic image charge approximations exist for planar and spherical metal surfaces but approximations for more complex geometries, such as the conical and wirelike structures characteristic of field emitters, are lacking. Such models are the basis for the evaluation of Schottky lowering factors in equations for current density. The development of a multidimensional image charge approximation, useful for a general thermal-field emission equation used in space charge studies, is given and based on an analytical model using a prolate spheroidal geometry. A description of how the model may be adapted to be used with a line charge model appropriate for carbon nanotube and carbon fiber field emitters is discussed.

Extended planar defects and the rapid incorporation of Ti4+ into olivine

The formation of extended planar defects in minerals such as olivine is related to high point defect concentration and can be driven by large gradients in chemical potential, where the energy of the system is lowered by the ordering of defects along specific planes in the crystal. The presence of extended defects has the potential to create the (apparently) anomalous ionic diffusion in olivine as reported recently (Spandler and O’Neill in Contrib Mineral Petrol 159(6):791–818, 2010). High-resolution transmission electron microscopy and energy-filtered imaging were done using experimental samples designed to examine the impact of a TiO2 and f O2 on the potential to form such defects in ferromagnesian olivine. Doped basalt (5 wt% TiO2)–olivine reaction couple experiments were run at 1 atm and 1,310 and 1,410 °C for 50 h at various f O2, ranging from 102 below to 102 above the quartz–fayalite–magnetite buffer. Our results show that extended planar defects in olivine, parallel to {101}ol and occurring in ordered “clusters” with a prolate spheroid geometry ~5–25 nm across and extending up to 150 nm into the olivine, are present near the olivine–glass interfaces in all of our experimental high-TiO2 basalt–olivine samples. Increased Ti content in the olivine is associated with the defects; ordering of Ti4+ and octahedral site vacancies leads to a two- or three-layer superstructure in the olivine. Defect nucleation and growth is driven by the large TiO2 chemical potential gradient across the phase boundary at the start of the experiments, which provides access to microstructures not otherwise present.

On the R‐semiseparation of the Stokes bi‐stream operator in inverted prolate spheroidal geometry

An analytical method, for deriving the complete general solution of the fourth order partial differential equation E4ψ=0, in the inverted prolate spheroidal coordinates is presented. This equation governs the axisymmetric Stokes flow and is employed in many applications where particle‐fluid systems are encountered. The complete solution is obtained by employing the method of the R‐separation of variables and the notion of semiseparability. The general Stokes stream function ψ is represented as a sum of two different kinds of functions where the first one belongs to the 0‐eigenspace of the operator E′2, and it is given in R‐separable form with R equals to the Euclidean distance r, whereas the second one belongs to the generalized 0‐eigenspace of the Stokes operator E′2 and it is given in R‐semiseparable form, with R equals now to r3. This result reveals that the equation E′4ψ = 0, in the inverted prolate spheroidal system, allows R‐semiseparation. In order to illustrate the method, we derive explicitly a particular eigenfunction of the generalized 0‐eigenspace of E′2. Copyright © 2013 John Wiley & Sons, Ltd.

The Dielectric Behavior of Nonspherical Biological Cell Suspensions: An Analytic Approach

The influence of the cell shape on the dielectric and conductometric properties of biological cell suspensions has been investigated from a theoretical point of view presenting an analytical solution of the electrostatic problem in the case of prolate and oblate spheroidal geometries. The model, which extends to spheroidal geometries the approach developed by other researchers in the case of a spherical geometry, takes explicitly into account the charge distributions at the cell membrane interfaces. The presence of these charge distributions, which govern the trans-membrane potential Δ V, produces composite dielectric spectra with two contiguous relaxation processes, known as the α-dispersion and the β-dispersion. By using this approach, we present a series of dielectric spectra for different values of the different electrical parameters (the permittivity ɛ and the electrical conductivity σ, together with the surface conductivity γ due to the surface charge distribution) that define the whole behavior of the system. In particular, we analyze the interplay between the parameters governing the α-dispersion and those influencing the β-dispersion. Even if these relaxation processes generally occur in well-separated frequency ranges, it is worth noting that, for certain values of the membrane conductivity, the high-frequency dispersion attributed to the Maxwell-Wagner effect is influenced not only by the bulk electrical parameters of the different adjacent media, but also by the surface conductivity at the two membrane interfaces.

SUNYAEV-ZEL'DOVICH EFFECT OBSERVATIONS OF THE BULLET CLUSTER (1E 0657–56) WITH APEX-SZ

We present observations of the Sunyaev-Zel'dovich effect (SZE) in the Bullet cluster (1E 0657–56) using the APEX-SZ instrument at 150 GHz with a resolution of 1'. The main results are maps of the SZE in this massive, merging galaxy cluster. The cluster is detected with 23σ significance within the central 1' radius of the source position. The SZE map has a broadly similar morphology to that in existing X-ray maps of this system, and we find no evidence for significant contamination of the SZE emission by radio or IR sources. In order to make simple quantitative comparisons with cluster gas models derived from X-ray observations, we fit our data to an isothermal elliptical β model, despite the inadequacy of such a model for this complex merging system. With an X-ray-derived prior on the power-law index, β = 1.04+0.16 –0.10, we find a core radius r c = 142'' ± 18'', an axial ratio of 0.889 ± 0.072, and a central temperature decrement of –771 ± 71 μKCMB, including a ±5.5% flux calibration uncertainty. Combining the APEX-SZ map with a map of projected electron surface density from Chandra X-ray observations, we determine the mass-weighted temperature of the cluster gas to be T mg = 10.8 ± 0.9 keV, significantly lower than some previously reported X-ray spectroscopic temperatures. Under the assumption of an isothermal cluster gas distribution in hydrostatic equilibrium, we compute the gas mass fraction for prolate and oblate spheroidal geometries and find it to be consistent with previous results from X-ray and weak-lensing observations. This work is the first result from the APEX-SZ experiment, and represents the first reported scientific result from observations with a large array of multiplexed superconducting transition-edge sensor bolometers.

Moisture diffusion modeling of parboiled paddy accelerated tempering process with extended application to multi-pass drying simulation

Parboiled paddy grain tempering process, often employed in multi-pass drying for milling quality improvement, is theoretically modeled considering a multi-component prolate spheroid geometry in prolate spheroidal coordinate system. The finite difference formulation analyzed the moisture diffusion during tempering and established the effect of vacuum in tempering acceleration. Experimental procedure reported already is essentially a double-pass drying (90 and 75 °C) of parboiled paddy with tempering stage at a critical moisture content (20.48% d.b.), where the moisture equilibration is accelerated by the application of vacuum (0–700 mm of Hg vacuum gauge). Boundary conditions of previously developed drying model were appropriately modified to model the tempering process. A supplemental fixed boundary condition with the regular derivative boundary condition and incorporation of tempering diffusivity factor modeled the tempering process and explained the effect of vacuum in tempering acceleration. Analysis of moisture history of nodes indicated that starch component moisture moved towards husk through bran component and the moisture profiles clearly demonstrated the effect of vacuum in temperature acceleration. An exponential relationship ( R 2 = 0.9813) adequately modeled the variation of diffusivity factor with the applied vacuum in accelerated tempering. The developed tempering model with drying model can simulate any multi-pass drying processes as well as help perform sensitivity analysis on factors, design equipment, and optimize operations.

Large spectral tuning of liquid microdroplets standing on a superhydrophobic surface using optical scattering force

We demonstrate large spectral tuning of glycerol/water microdroplets standing on a superhydrophobic surface using the optical scattering force exerted by a 1064nm Nd3+:YVO4 solid-state laser. Spectral tuning up to 30nm is presented in the whispering gallery modes as a result of the deformation of the microdroplets toward a truncated prolate spheroid geometry. Observed large spectral tuning is also reported to be highly reversible. This demonstration can inspire novel, largely tunable optical switches or filters based on liquid microdroplets kept in a sealed chamber.

Modeling heat and mass transfer during drying of green coffee beans using prolate spheroidal geometry

A heat and mass transfer model in prolate spheroidal coordinates system was proposed to describe the drying of green coffee beans. The model describes the 3D moisture and temperature profiles inside the bean. The results were integrated over volume in order to obtain a drying kinetic equation for prolate spheroidal geometry. The average effective diffusivity of water as function of temperature and moisture was estimated at 45 and 60 °C by slope method from experimental drying kinetic of green coffee beans. The expression obtained applied to drying kinetic equation reproduced approximately the experimental behavior.

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.

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.

A four-dimensional viscoelastic deformation model for Long Valley Caldera, California, between 1995 and 2000

We investigate the effects of viscoelastic (VE) rheologies surrounding a vertically dipping prolate spheroid source during an active period of time-dependent deformation between 1995 and 2000 at Long Valley caldera. We model a rapid magmatic inflation episode and slip across the South Moat fault (SMF) in late 1997. We extend the spherical VE shell model of Newman et al. [Newman, A.V., Dixon, T.H., Ofoegbu, G., Dixon, J.E., 2001. Geodetic and seismic constraints on recent activity at Long Valley caldera, California: Evidence for viscoelastic rheology. J. Volcanol. Geotherm. Res. 105, 183–206.] to include a prolate spheroid geometry more accurately representing the probable source geometry inferred from other studies. This paper presents the first attempt to geodetically constrain the volcanic deformation source volume at Long Valley, a parameter for hazard assessment. Including fault slip along the SMF explains significant deformation observed with several EDM baselines and components of two continuous GPS time series. Additionally, the model explains the spatial extent of deformation observed by InSAR data covering the 1997–98 inflation episode. For the time period studied, the VE model requires modest pressure changes (maximum of 14.3 MPa) that are far lower than the overburden pressure (∼115 MPa), and less than the maximum for a purely elastic model with the same geometry and elastic strength (∼45 MPa). Thus, the inclusion of a realistic VE component significantly lowers the inferred pressures necessary to explain observed surface deformation. Though our model is non-unique, it is consistent with a broader variety of data compared to purely elastic models. Only right-lateral slip, and not dilitation, was necessary to explain offsets in EDM data near and crossing the SMF.