On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.     

Simulation of structure formation in disperse systems

The main rules governing the redistribution of microparticles in disperse systems were revealed by computer simulation. These rules predetermine the character and properties of the structure being formed, which opens up the possibility to control the properties of disperse materials, including sorbents and catalysts based on high-dispersity solid phases.     

Primary dependence of 7-aryl-2,5-di-t-butyl-1,3,5-cycloheptatriene valence tautomerism on the π-acceptor strength of the 7-aryl substituent

¹³C NMR studies showed that the population of the norcaradiene form of the title systems containing p-CH₃O, H, and p-CF₃ on the 7-aryl group increases in this order. The result is consistent with the prediction from the π-acceptor strength of the aryl group estimated by INDO calculations.     

A new model for the cross spectrum and wavenumber-frequency spectrum of turbulent pressure fluctuations in a boundary layer

A model is developed for the statistical characteristics of the turbulent pressure field in a boundary layer. It is shown that the coherence scales are limited at low frequencies by the finiteness of the boundary layer thickness and at high frequencies by the effect of viscous forces acting in the flow. A relationship between the behavior of the coherence scales and that of the power spectra is demonstrated. The model is characterized by convertibility, which allows changes from cross spectra to wavenumber-frequency spectra and back via Fourier transformations. Calculations performed using the proposed model agree well with experimental results.     

Farthest-point queries with geometric and combinatorial constraints

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log²n) time. Our data structure needs O(nlogn) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other.     

Anisotropic field of background internal waves on a sea shelf and its effect on low-frequency sound propagation

The space-time spectral characteristics of the field of background internal waves (IW) are obtained for two oceanic shelf regions (the Atlantic shelf of the United States and the Kamchatka shelf) and analyzed. Within the framework of a numerical experiment, it is shown that the observed anisotropy of the IW field may considerably affect the low-frequency sound fluctuations in the aforementioned regions and, in particular, may change the interference invariant of the sound field.     

Spectral manifestations of the splitting of the lowest triplet levels in Pd complexes of nonsymmetric porphyrins

Based on the study of the phosphorescence and phosphorescence excitation spectra of Pd(II) tetramethylporphyrin (PdTMP) and Pd(II) tetramethyldiethylporphyrin (PdTMDEP) in solutions in 2-methyltetrahydrofuran and dodecane in the temperature range 77–283 K, the occurrence of the splitting of the lowest degenerate singlet (S ₁) and triplet (T ₁, T ₂) levels of porphyrin molecules is established. In the absorption of molecules of the compounds studied, two components, S ₁ and S ₂, are revealed in the range of allowed long-wavelength Q(0,0) transitions (530–550 nm) and four components, T ₁–T ₄, are observed in the range of spin-forbidden intersystem crossing transitions S ₀ → T _n (560–670 nm), with all the triplet levels T ₁-T ₄ being located below the S ₁ level on the energy scale. It is shown that an increase in the degree of deformation of the porphyrin macrocycle caused by steric interactions between β alkyls and hydrogen meso atoms is accompanied by an increase in the splitting in the system of singlet levels δE(S ₂–S ₁) from 120 cm^?1 for PdTMDEP to 215 cm^?1 for PdTMP, as well as by an increase in the splitting in the system of triplet levels δE(T ₂–T ₁) from 190 cm^?1 for PdTMDEP to 250 cm^?1 for PdTMP.     

Disintegration of a drop in an external electrostatic field

Regular features of the disintegration of both a drop of a perfectly conducting liquid and a drop of a dielectric liquid into two or three parts in an external uniform electric field are studied using the principle of minimizing the potential energy of the final state of a closed system with spontaneous processes.