Equal Radii Research Articles

Dot-pattern design of a light guide in an edge-lit backlight using a regional partition approach

This paper employs a regional partition approach to establish the optimal distribution of the diffuser dots in the light guide of an edge-lit backlight. The proposed approach commences by dividing the light guide into a number of partitions and assigning an equal radius to every dot. An iterative optimization procedure is then performed in which the illuminance in each partition is calculated and the dot radius in each partition progressively adjusted until uniform luminance is obtained over the entire light-guide surface. To simplify the geometry construction process and improve the efficiency of the optimization procedure, a simple optical model is proposed for the diffuser dots and the bare surface of the light guide, and an analytical procedure is implemented to take account of the diffuser-film and brightness-enhancement-film effects. The simulation results confirm the ability of the proposed approach to accurately reflect the fine regulation of the dot radius required to obtain uniform luminance for typical light-emitting diode configurations used in practical backlight modules.

Calculation of the electric field in arrangements of intersecting spheres

The application of the method of images to the calculation of electric field in arrangements of intersecting conducting spheres is presented. The multipole re-expansions are utilized to determine the multipole images and the potential coefficients. The method is applicable for cases of electrically floating, grounded, or energized conducting spheres that intersect each other. Examples are given for two grounded intersecting spheres of equal radii and those of different radii, respectively, under an external uniform field. The results for the spheres of equal radii agree well with the analytical solutions for the case of completely overlapped spheres and touching spheres, respectively. For the spheres of different radii, the numerical results show that the method may also be used when the degree of intersection is not too high.

A functional limit theorem for the position of a particle in the Lorentz model

We consider a particle moving through a medium under a constant external field. The medium consists of immobile spherical obstacles of equal radii randomly distributed in ℝ3. When the particle collides with an obstacle, it reflects inelastically, with restitution coefficient α ∈, (0, 1). We study the asymptotics of X(t), the position of the particle at time t, as t → ∞. The main result is a functional limit theorem for X(t). Its proof is based on the functional CLT for Markov chains. Bibliography: 10 titles.

Three-Dimensional Simulations of Reactive Gas Uptake in Single Airway Bifurcations

The pattern of lung injury induced by the inhalation of ozone (O(3)) depends on the dose delivered to different tissues in the airways. This study examined the distribution of O(3) uptake in a single, symmetrically branched airway bifurcation. Reaction in the epithelial lining fluid was assumed to be so rapid that O(3) concentration was negligible along the entire surface of the bifurcation wall. Three-dimensional numerical solutions of the continuity, Navier-Stokes and convection-diffusion equations were obtained for steady inspiratory and expiratory flows at Reynolds numbers ranging from 100 to 500. The total rate of O(3) uptake was found to increase with increasing flow rate during both inspiration and expiration. Hot spots of O(3) flux appeared at the carina of the bifurcation for virtually all inspiratory and expiratory Reynolds numbers considered in the simulations. At the lowest expiratory Reynolds number, however, the location of the maximum flux was shifted to the outer wall of the daughter branch. For expiratory flow, additional hot spots of flux were found on the parent branch wall just downstream of the branching region. In all cases, O(3) uptake in the single bifurcation was larger than that in a straight tube of equal inlet radius and wall surface area. This study provides insight into the effect of flow conditions on O(3) uptake and dose distribution in individual bifurcations.

Hydrodynamic assist and the dynamic contact angle in the coalescence of liquid drops

The coalescence of two viscous liquid drops in an inviscid gas or in a vacuum is studied using the interface formation model. In the very early stages of coalescence during the formation of the 'liquid-bridge' connecting the two drops, this model predicts a moving contact line and a dynamic contact angle. This paper examines the dynamic evolution of this contact angle, and for small Reynolds number and small Capillary number, relevant particularly in micro-fluidics, a non-linear differential equation is derived for the contact angle and solved computationally. It is found that the contact angle evolution can only be evaluated by determining information about the flow away from the contact line. This is a manifestation of so-called hydrodynamic assist, studied experimentally in the context of curtain coating by Blake et al. (1999 Experimental evidence of non-local hydrodynamic influence on the dynamic contact angle. Phys. Fluids, 11, 1995-2007). For small Capillary number and small Reynolds number, the free-surface evolution is determined for the coalescence of two cylinders of equal radius. Finally, some comments are made on experiments in coalescence, as well as on issues arising in a computational solution of the full model described here.

A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models

Recent advances in computational speed have resulted in the ability to model composite materials using larger representative volume elements (RVEs) with greater numbers of inclusions than have been previously studied. Imposing periodic boundary conditions on very large RVEs can mean enforcing thousands of constraint equations. In addition, a periodic mesh is essential for enforcing the constraints. The present study investigates a method that uses a local implementation of the constraints that does not adversely affect the computational speed. The present study demonstrates the method for two-dimensional triangular and square RVEs of periodically-spaced regular hexagonal and square arrays of composite material containing fibers of equal radii. To impose the boundary conditions along the edges, this study utilizes a cubic interpolant to model the displacement field along the matrix edges and a linear interpolant to model the field along the fiber edges. It is shown that the method eliminates the need for the conventional node-coupling scheme for imposing periodic boundary conditions, consequently reducing the number of unknowns to the interior degrees of freedom of the RVE along with a small number of global parameters. The method is demonstrated for periodic and non-periodic mesh designs.

The stability of two connected drops suspended from the edges of circular holes

The stability of an equilibrium system of two drops suspended from circular holes in a horizontal plate is examined. The drop surfaces are the disconnected axisymmetric surfaces pinned to the edges of the holes. The holes lie in the same horizontal plane and the two drops are connected by a liquid layer that lies above the plate. The total liquid volume is constant. For identical pendant drops pinned to holes of equal radii, axisymmetric perturbations are always the most dangerous. The stability region for two identical drops differs considerably from that for a solitary pendant drop. A bifurcation analysis shows that the loss of stability leads to a continuous transition from a critical system of identical drops to a stable system of axisymmetric non-identical drops. With increasing total protruded liquid volume this system of non-identical drops reaches its own collective stability limit (to axisymmetric perturbations) which gives rise to dripping or streaming from the holes. Critical volumes and heights for non-identical drops have been calculated as functions of the dimensionless hole radius (associated with the Bond number). For unequal hole radii, there are three intervals of the larger dimensionless hole radius, R 0 1 , with qualitatively different bifurcation patterns which in turn can depend on the smaller dimensionless hole radius, R 0 2 . Loss of stability may occur when the drop suspended from the larger hole reaches its stability limit (to non-axisymmetric perturbations) as a solitary drop or when the system reaches the collective stability limit (to axisymmetric perturbations). Typical situations are illustrated for selected values of R 0 1 , and then the basic characteristics of the stability for a dense set of R 0 1 are presented.

Examination of the Validity of the Ionic Additivity Principle in the Determination of Single Ion Thermodynamic Properties

Single ion thermodynamic values of hydration or solvation (or solution) cannot be experimentally determined. The values are determinable from the thermodynamics of solvation ΔX0solv or soln(MX) [X=H,GorS] based on ionic additivity and using extrathermodynamic assumptions.However, the principle of ionic additivity is not valid in the determination of single ion thermodynamic values [i.e.ΔG0solv(M+or X−) or ΔG0soln(M+or X−), ΔH0solv(M+or X−) or ΔH0soln(M+or X−)] but additivity principle appears to be valid for the determination of ΔS0solv or soln(M+or X−).Moreover, appropriation of ΔG0solvor ΔH0solv(MX) equally on the basis of equality of radii of cations and anions is also not tenable due to differences in the effective charges of the cations and anions of equal radii.

Finite deformation model of simple shear of fault with microrotations: apparent strain localisation and en-echelon fracture pattern

Strain localisation is a widespread phenomenon often observed in shear and compressive loading of geomaterials, for example, the fault gouge. It is believed that the main mechanisms of strain localisation are strain softening and mismatch between dilatancy and pressure sensitivity. Observations show that gouge deformation is accompanied by considerable rotations of grains. In our previous work as a model for gouge material, we proposed a continuum description for an assembly of particles of equal radius in which the particle rotation is treated as an independent degree of freedom. We showed that there exist critical values of the model parameters for which the displacement gradient exhibits a pronounced localisation at the mid-surface layers of the fault, even in the absence of inelasticity. Here, we generalise the model to the case of finite deformations characteristic for the gouge deformation. We derive objective constitutive relationships relating the Jaumann rates of stress and moment stress to the relative strain and curvature rates, respectively. The model suggests that the pattern of localisation remains the same as in the linear case. However, the presence of the Jaumann terms leads to the emergence of non-zero normal stresses acting along and perpendicular to the shear layer (with zero hydrostatic pressure), and localised along the mid-line of the gouge; these stress components are absent in the linear model of simple shear. These additional normal stresses, albeit small, cause a change in the direction in which the maximal normal stresses act and in which en-echelon fracturing is formed.

Efficient algorithm for placing a given number of base stations to cover a convex region

In the context of mobile communication, an efficient algorithm for the base-station placement problem is developed in this paper. The objective is to place a given number of base-stations in a given convex region, and to assign range to each of them such that every point in the region is covered by at least one base-station, and the maximum range assigned is minimized. It is basically covering a region by a given number of equal radius circles where the objective is to minimize the radius. We develop an efficient algorithm for this problem using Voronoi diagram which works for covering a convex region of arbitrary shape. Existing results for this problem are available when the region is a square [K.J. Nurmela, P.R.J. Ostergard, Covering a square with up to 30 equal circles, Research Report HUT-TCS-A62, Laboratory for Theoretical Computer Science, Helsinky University of Technology, 2000] and an equilateral triangle [K.J. Nurmela, Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, Exp. Math. 9 (2) (2000)]. The minimum radius obtained by our method favorably compares with the results presented in [K.J. Nurmela, P.R.J. Ostergard, Covering a square with up to 30 equal circles, Research Report HUT-TCS-A62, Laboratory for Theoretical Computer Science, Helsinky University of Technology, 2000; K.J. Nurmela, Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, Exp. Math. 9 (2) (2000)]. But the execution time of our algorithm is a fraction of a second, whereas the existing methods may even take about two weeks’ time for a reasonable value of the number of circles ( ⩾ 27 ) as reported in [K.J. Nurmela, P.R.J. Ostergard, Covering a square with up to 30 equal circles, Research Report HUT-TCS-A62, Laboratory for Theoretical Computer Science, Helsinky University of Technology, 2000; K.J. Nurmela, Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, Exp. Math. 9 (2) (2000)].

Dust-acoustic solitary waves in an inhomogeneous magnetized hot dusty plasma with dust charge fluctuations

Using a standard reductive perturbation theory, a Zakharov-Kuznnetsov (ZK) equation is derived in an inhomogeneous dusty plasma comprised of negatively charged dust grains of equal radii, Boltzmann distributed electrons and positive ions. The effects of dust thermal pressure, dust charge variation, dust-neutral collision, and the external magnetic field are taken into account. Either compressive or rarefective solitons are shown to exist depending on the critical value of the phase velocity which in turn, depends on the dust temperature and pressure. An analytic soliton solution is obtained and discussed. The behaviors of the soliton amplitude, width, and the Mach number are investigated numerically for different plasma parameters.

A frequency-domain BEM for 3D non-synchronous crack interaction analysis in elastic solids

A frequency-domain boundary element method (BEM) is presented for non-synchronous crack interaction analysis in three-dimensional (3D), infinite, isotropic and linear elastic solids with multiple coplanar cracks. The cracks are subjected to non-synchronous time-harmonic crack-surface loading with contrast frequencies. Hypersingular frequency-domain traction boundary integral equations (BIEs) are applied to solve the boundary value problem. A collocation method is adopted for solving the BIEs numerically. The local square-root behavior of the crack-opening-displacements at the crack-front is taken into account in the present method. For two coplanar penny-shaped cracks of equal radius subjected to non-synchronous time-harmonic crack-surface loading, numerical results for the dynamic stress intensity factors are presented and discussed.

Resonance effect of a magnetic quantum ring on the conductance of a two-dimensional sample

The resonant tunneling of electrons through a magnetic quantum ring in a two-dimensional sample and the effect of this resonant tunneling on the conductivity of the sample are studied in this work. The ballistic conductance of the sample is calculated for several values of the ratio (inner radius)/(outer radius) of the ring. The limit of a conductance plateau is recovered for approximately equal radii and previous results for the conductance of a magnetic quantum dot are recovered in the case of an approximately zero inner radius.

On the Dynamics of Belt Drives

AbstractIn modelling the dynamics of belt drives, one has to distinguish between taut and slack belts. While the former can be described by a linear model, the latter is inherently nonlinear and needs a more sophisticated analysis. The paper at hand presents an analytical approach to modelling slack belts in plane motion. To simplify the setting, the pulleys are assumed to have equal radii, such that only one quarter of the total system needs to be considered. The extension to a more general geometry is straightforward. The belt is assumed to have non‐zero bending stiffness. The slackness of the belt should be high enough to allow .exural vibrations even if the belt is inextensible. The static and the stationary solutions are formulated analytically in terms of elliptic functions. As a preliminary study, the special case of a travelling string is considered yielding a .rst insight into the complexity of belt dynamics. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Magnetic dynamics of simple collective modes in a two-sphere plasma model

A plasma blob is modeled as consisting of two homogeneous spheres of equal radius and equal but opposite charge densities that can move relative to each other. Relative translational and rotational motion are considered separately. Magnetic effects from the current density caused by the relative motion are included. Magnetic interaction is seen to cause an inductive inertia. In the relative translation case the Coulomb attraction, approximately a linear force for small amplitudes, causes an oscillation. For a large number of particles, the corresponding oscillation frequency will not be the Langmuir plasma frequency, because of the large inductive inertia. For rotation an external magnetic field is included and the energy and diamagnetism of the plasma in the model is calculated. Finally, it is noted how the neglect of resistivity is motivated by the results.

Theory and analysis of adaptive cylindrical array antenna for ultrawideband wireless communications

In this paper, the principle of cylindrical array beamforming based on ultrawideband impulse (UWBI) signals is introduced. A cylindrical array antenna is composed of a number of vertically aligned and concentric circular subarrays of equal radius and equal number of array elements. Theory, analysis, and computer simulation of the cylindrical array antenna are presented based on UWBI signals with the time variation of a generalized Gaussian pulse (GGP). The radiation pattern of the cylindrical array antenna is derived in terms of the inverse Fourier transform of the radiated far-zone GGP signal. The radiation pattern results in different azimuth and elevation beam patterns such as peak amplitude pattern, peak power pattern, and energy pattern. Computer plots of these antenna patterns are generated for different design parameters such as array radius, interelement spacing distance, frequency bandwidth, and steering angle. The phenomenon of beamwidth broadening associated with electronic beamsteering is analyzed, and the resolution angle for the cylindrical array antenna is derived too. Beamforming based on UWBI signals provides a tradeoff between array dimensions, frequency bandwidth, and steering angle for achieving a high angular resolution capability in the azimuth plane as well as in the elevation plane. Such a tradeoff is attractive in practice for UWBI communications, indoor (multimedia) communications, radar, and localization systems.

Quantitative determination of the clustered silicon concentration in substoichiometric silicon oxide layer

We present an analytical methodology, based on electron energy loss spectroscopy (EELS) and energy-filtered transmission electron microscopy, which allows us to quantify the clustered silicon concentration in annealed substoichiometric silicon oxide layers, deposited by plasma-enhanced chemical vapor deposition. The clustered Si volume fraction was deduced from a fit to the experimental EELS spectrum using a theoretical description proposed to calculate the dielectric function of a system of spherical particles of equal radii, located at random in a host material. The methodology allowed us to demonstrate that the clustered Si concentration is only one half of the excess Si concentration dissolved in the layer.

Relativistic effects of our galaxy's motion on circles-in-the-sky in CMB maps

For an observer in the Hubble flow, the last scattering surface (LSS) is well approximated by a 2-sphere. If a non-trivial topology of space is detectable, then this sphere intersects some of its topological images, eventually giving rise, in cosmic microwave background (CMB) radiation maps (which are projections of the LSS onto the observer's sky sphere), to circles-in-the-sky, i.e., pairs of matching circles of equal radii, centred at different points on the sky. We examine the geometric effects due to our galaxy's peculiar motion on circles-in-the-sky in CMB maps, and show that their shape remains circular, as detected by a local observer with arbitrary peculiar velocity. In general, a circle is detected as a circle of different radius, displaced relative to its original position, and centred at a point which does not correspond to its detected centre in the comoving frame. Further, there is an angular displacement of points on the circles. These effects all arise from aberration of cosmic microwave background radiation, exhausting the purely geometric effects due to the peculiar motion of our galaxy, and are independent of the large scale curvature of space, the expansion of the universe and the acceleration of the observer, since aberration is a purely local phenomenon. For a Lorentz-boosted observer with the speed of our entire galaxy, the maximum changes in the angular radius of a circle, its maximum centre displacement as well as the maximum angular distortion are all shown to be of order β = (v/c) ≃ 1.23 × 10−3 radians. In particular, two back-to-back matching circles in a finite universe will have an upper bound of 2|β| in the variation of either their radii, the angular position of their centres, or the angular distribution of points. Although below current WMAP's resolution, these results are within Planck's and other more accurate missions' resolution.

Capabilities of Kinematically Regular Satellite Systems with Symmetry Groups of the Second Type in the Problem of Continuous Single Coverage of the Earth

The problem of a continuous single coverage of the Earth by systems of satellites in circular orbits of equal radii is considered. The results of calculating the characteristics of kinematically regular systems consisting of 6 to 60 satellites and possessing symmetry groups of the second type are presented. It is shown that, when the number of satellites varies from 20 to 38 inclusive, the best systems of this class surpass all currently known systems with respect to their characteristics.

Collapse and merging of cavity regions in a granular material due to viscous flow

Experimental studies are made on the collapse of macroscopic cavity regions placed in an otherwise homogeneous granular material due to viscous flow. First, an initially circular two-dimensional hole is exposed to a uniform flow at infinity and the process of boundary shape deformation due to locally enhanced stress components is clarified. Above a certain critical velocity, particles on the upstream-side boundary lose contact with neighboring ones, and are carried to the other side of the boundary. At the same time the fluidized region develops towards upstream direction in the granular material. A simple scaling law on the reduction of the void area is obtained. Second, attention is paid to the interaction of two circular cavities of equal radii, whose center-to-center distance and angle of attack are varied. In some configuration of cavities, lowering of the critical velocity of collapse is recognized. Successive processes on the collapse of upstream-side and downstream-side cavities are classified depending on the magnitude of velocity and the configuration of cavities. A possible scenario on the growth of global scale patterns in granular material such as water vein formation and landslides is suggested.