Spheres Of Equal Radii Research Articles

ПОРОМАСШТАБНОЕ МОДЕЛИРОВАНИЕ ПОТОКА ЖИДКОСТИ В ПРОНИЦАЕМЫХ СФЕРАХ С ИСПОЛЬЗОВАНИЕМ ПРОЕКЦИОННОГО МЕТОДА ДЛЯ НЕСЖИМАЕМЫХ УРАВНЕНИЙ НАВЬЕ-СТОКСА

The direct numerical simulation (DNS) is an effective and useful tool in the two-phase fluid flow studying. The projection method on the staggered grid was applied in this paper to solve the incompressible Navier-Stokes equations in irregular domains at the pore-scale level (irregular boundary is presented by its level-set function). The permeability of porous medium which was constructed by the random positioning of penetrable spheres of equal radii were numerically calculated and validated by comparing with theoretical estimations of permeability based on the numerical solution of the lattice-Boltzmann equation in irregular domains in previous works. All numerical calculations were performed using PARIS simulator.

Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asymptotic expansion for small distances between the two spheres is determined and analytical expressions for the coefficients are given.

Proximity effects in radiative heat transfer

Though the dependence of near-field radiative transfer on the gap between two planar objects is well understood, that between curved objects is still unclear. We show, based on the analysis of the surface polariton mediated radiative transfer between two spheres of equal radii $R$ and minimum gap $d$, that the near--field radiative transfer scales as $R/d$ as $d/R \rightarrow 0$ and as $\ln(R/d)$ for larger values of $d/R$ up to the far--field limit. We propose a modified form of the proximity approximation to predict near--field radiative transfer between curved objects from simulations of radiative transfer between planar surfaces.

The Capacitance of Rosette Ice Crystals

The capacitances of seven bullet rosette ice crystals are computed based on the classical electrostatic analogy theory of diffusional growth. The rosettes simulated have 2, 3, 4, 6, 8, 12, and 16 lobes using mathematical formulas published previously. The Laplace equation for the water vapor density distribution around a stationary rosette is solved explicitly by the finite element method. The total flux of vapor toward the rosette surface and the vapor density on the surface determine the capacitance. The capacitances of these rosettes are smaller than that of spheres of equal radii but greater than columnar ice crystals of the same maximum dimensions. They can be greater or smaller than that of circular plates, depending on the number of lobes. Since many previous estimates of rosette growth rates were based on the assumption that their capacitances are the same as spheres of equal radii, the present finding implies that some of the previous rosette growth rates may be overestimated. The overestimation becomes less important if the rosettes have more lobes. Empirical power equations are given to fit the relations between the capacitance and the number of lobes, surface area, and volume of rosettes. Possible implications of rosette capacitance on the atmospheric heating by cirrus clouds are also discussed.

Polynomial-time construction of codes .II. spherical codes and the kissing number of spheres

A spherical code is a finite set X of points lying on the unit sphere of R/sup n/. For such a set, we define /spl rho/(X) as the minimum of the squared distances /spl par/x-y/spl par//sup 2/, when x, y/spl isin/X and x/spl ne/y. Define R(/spl rho/)=lim sup n/spl rarr//spl infin/, /spl rho/(X)=p log/sub 2/CardX/n. Chabauty in 1953 and Shannon in 1959 have given a lower bound for R(/spl rho/), namely, R(/spl rho/)>R/sub CS/(/spl rho/)=1-1/3log/sub 2//spl rho/(4-p). The complexity of construction of the spherical codes used in order to get this bound is doubly exponential. The polynomially constructible spherical bound R/sub pol/(/spl rho/) is defined as above with the additional restriction that only families of codes with polynomial complexity of construction are considered. We prove R/sub pol/(/spl rho/)/spl ges/R/sub CS/(/spl rho/)/2, if /spl rho//spl les/1.535. Denote by /spl tau//sub X/(n) the number of spheres of equal radius that touch one sphere in the n-dimensional space given by some explicit family X, that is, a family of arrangements of spheres). The asymptotic polynomially constructible kissing number is /spl theta//sub pol/=lim sup(log/sub 2//spl tau//sub X/(n))/n, when X ranges over all polynomially constructible families. We prove /spl theta//sub pol//spl ges/2/15=0.133/spl middot//spl middot//spl middot/. >

Remarks on the density of sphere packings in three dimensions

This paper shows how the density of sphere packings of spheres of equal radius may be studied using the Delaunay decomposition. Using this decomposition, a local notion of density for sphere packings in ℝ3 is defined. Conjecturally this approach should yield a bound of 0.740873... on sphere packings in ℝ3, and a small perturbation of this approach should yield the bound of\({\pi \mathord{\left/ {\vphantom {\pi {\sqrt {18} }}} \right. \kern-\nulldelimiterspace} {\sqrt {18} }}\). The face-centered-cubic and hexagonal-close-packings provide local maxima (in a strong sense defined below) to the function which associates to every saturated sphere packing in ℝ3 its density. The local measure of density coincides with the actual density for the face-centered cubic and hexagonal-close-packings.

Nature of compression strength in concrete

A compression strength theory for concrete based on the capillary porosity of binder paste in concrete and the intrinsic tensional strength of the CSH gel is presented. The capillary pores were modelled as spheres of equal radius and the tensional stress peak on the surface of the voids was calculated using a FEM program. The intrinsic tensional strength of the binder was determined from gas evolution curves of the binder paste by thermogravimetric means. To validitate the theory, eleven different test concretes with compression strength values from 25 to 135 MPa were produced. As binders, five different Portland cement types with and without silica fume were used. The testing ages of the concretes were 28 and 365 days. The porosities of the test concretes were measured by mercury porosimetry, and the microporosity of the hardened binder paste by nitrogen adsorption. The correlation coefficient R-squared between the test results and the results of the theory was 0·941.

Diffusion-controlled reaction kinetics of the binding of carbon monoxide to the heme undecapeptide of cytochrome c (microperoxidase 11) in high viscosity solvents

The kinetics of the binding of carbon monoxide to the covalently linked heme undecapeptide of cytochrome c (ferrous microperoxidase 11) have been studied by flash photolysis in high-viscosity mixtures of glycerol:water, ethylene glycol:water, and polyethylene glycol:water. This was done in order to examine the effects of diffusion-controlled ligand binding to a small heme peptide as a model for hemoglobin and myoglobin. Arrhenius plots of the observed second-order association rate constant are significantly curved consistent with a change from chemical activation control at higher temperatures to diffusion control at the lowest temperatures and highest viscosities. The activation enthalpies in the diffusion-controlled region show a good correspondence with the solvent viscosity activation energies as predicted by theory. Depending upon the solvent, the deviation of the second-order diffusion rate constants are 0.1 to 0.3 of that predicted from simple Smoluchowski theory for reaction between spheres of equal radii. Several models that ascribe these deviations to steric requirements are examined.

Asymptotic solutions of the problem of heat and mass transfer of particles in a fluid

Some questions related to asymptotic analysis (as P → ∞, where P is the Peclet number) of problems involving heat and mass transfer of particles in a fluid are considered. The first part of the paper investigates the stationary convective diffusion of a solute to a particle near its front critical point (incidence point). An explicit expression is obtained for the concentration in the region of the front critical point of a solid or liquid particle around which a Stokes flow occurs. In the second part of the paper, a unified formula is obtained for the concentration distribution behind the particle. Appropriate limits in this formula determine the concentration in the mixing region and the inner and convective boundary-layer regions of the diffusion wake. In the final part of the paper, a study is made of the diffusion to a chain of absorbing solid spheres of equal radius a at distances1, 1 ≪1/a ≪ P1/3, from each other on the axis of an oncoming Stokes flow; an integral equation is obtained for the local diffusion flux when a chemical reaction with arbitrary kinetics takes place on the surfaces of the spheres. A certain heterogeneity of the material in the paper is due to the investigation in it of various questions that arise in the solution of more general problems (see, for example, [1–14]) which have not been considered hitherto.

Statistical Analysis of a Rough-Surface Model

The ‘stereometrical’ characteristics of the rough surface layers of engineering components, which are necessary for the calculation of rough-surface contact parameters, are often identified with the ‘planimetrical’ ones obtained from single profiles. In this paper, a relationship is obtained between these characteristics. Asperities are modelled by segments of spheres of equal radius and arbitrary height.

Separation from the surface of two equal spheres in Stokes flow

In this paper, it is shown that if two spheres of equal radii are placed axisymmetrically in a steady Stokes stream, separation of the flow from the spheres occurs if the distance between their centres is less than approximately 3-67 times the sphere radius. For spheres whose spacing is less than this value, wakes form on both spheres and the fluid within the wakes moves in closed eddy type motion. When the distance between the centres of the spheres is less than approximately 3.22 times the sphere radius, a cylinder of fluid links both spheres, and within this cylinder the fluid rotates in one or more ring vortices, the number of vortices increasing as the distance between the spheres is decreased. When the spheres are in contact, the fluid rotates in an infinite set of nested ring vortices.

Excluded possibilities of geometrodynamical analog to electric charge

in 1956, J. A. Wheeler and C. W. Misner [I] showed in some detail that the existence of charges in the world need not be associated with sources in the electromagnetic field equations, but rather could be described as a consequence of Maxwell's free field equations (i.e. Ffj,,:~,l =0 and F~'";,--0) in a space with a multiply connected topology [2]. The electric lines of force can be ' tra?ped' in the topological structure of space, with the number of trapped lines of force being directly proportional to the charge associated with that field. To make this idea clearer, let us examine a simple example of a multiply connected, or 'v, ormhole' topology trapping electric lines of force. If the interiors of two solid spheres of equal radius are removed from an ordinary three dimensional Euclidean space, and' the appropriate points On the surfaces of the two spheres are identified, we obtain a "wormhole' in our space. If now we imagine that the t**o spheres had been conducting, one having charge q and the other charge q . and that the resultant electric field were fixed during the above reconstruction of our space, we discover that the resultant electric field everywhere obeys the free field equations. However, for an)'one located outside the cutaway portions of the space, there wouk! be no way of telling that the charges were no longer there. For example, integrating the field over any closed surface surrounding one of the wormhole mouths would tell him that the surface still enclosed a charge q. We have created, in Wheeler's words, 'charge without charge'. Due to the symmetry in Maxwell's equations between E and B, magnetic lines of force could similarly be trapped by the topology, creating magnetic charges, ilowever, these have never been observed in nature [3l. Wc theretore make the natural assumption that magnetic lines of force cannot be trapped in a multiply-connected topology. The above concepts can bc put into a rigorous mathematical form [4]. The electromagnetic tensors/-'*~ and ~,,~,~,f'p" are curl free antisymmetric I~ensors, and as such coffespond to closed differential 2+formsfand *f(i.e

Slow motion of an incompressible viscous liquid generated by the rotation of two spheres in contact

1. G. B. Jeffery [1] investigated the axisymmetrical flow of an incompressible viscous fluid caused by two spheres rotating slowly and steadily in the liquid about their line of centres. The numerical results he gave were for spheres of equal radii. The present problem is to investigate what happens when the spheres touch each other either externally or internally and when they are unequal in size. This problem could be approached by taking a limit of Jeffery's solution, but in fact it will be more convenient to use a co-ordinate system different from Jeffery's and, of course, his results would not yield information about unequal spheres.

Capillary condensation of vapors in highly dispersed systems

Exact calculations are made of the volumes of capillary condensate at various relative pressures in systems of two spheres of different radii in contact, and it is shown that a difference of 25% in the dimensions of the spheres in contact results in an appreciable increase in the volume of liquid, as compared with the case of spheres of equal radii.

The Intrinsic Measure Theory of Riemannian and Euclidean Metric Spaces

Euclidean spaces are boundedly compact and have the metric property that any two spheres are similar. In particular, if a closed sphere of radius r can be covered by n open spheres of radius x, then, for any positive a, every closed sphere of radius ar can be covered by n open spheres of radius ax. W\e shall show that this combinatorial similarity property is sufficient in any boundedly compact metric space for the development of ordinarv Lebesgue measure theorya. One result is the validity of the usual formula for the volume of a sphere. That is, apart from a multiplicative constant there is one and only one measure which is a volume in the sense that spheres of equal radii have equal measures, and there is an a such that the volume of a sphere of radius r is ra. The existence proof will be presented in a general enough form to include the development of the intrinsic measure theories of Riemannian metric spaces and of metric spaces like the Cantor sets for which the dimension a is non-integral.