A class of translation-invariant Gibbs measures for models with a continuum of spin values on a Cayley tree was examined. The results show that the problem of describing such measures reduces to studying positive solutions of a nonlinear integral equation of Hammerstein type. A connection between the solutions of the integral equation and the properties of a corresponding polynomial was established. A theorem on the existence and uniqueness of a positive solution of the Hammerstein equation for an arbitrary tree order 𝑘 ≥ 2 was proved. Explicit formulas for the solutions were obtained via the roots of a polynomial of degree 𝑘 + 1, constructively describing the set of translation-invariant Gibbs measures.
The influence of pore space characteristics of terrigenous reservoir rocks on two-phase flows was investigated. Computational experiments were carried out using the lattice Boltzmann method on a series of digital models of natural sandstones with various filtration and capacity properties. It was established that pore space heterogeneity, which is described by the standard deviation of pore size distribution, emerges as a key factor determining displacement efficiency. The negative impact of increasing heterogeneity on displacement efficiency and front stability was shown. A map of the influence of the capillary number on displacement efficiency, depending on standard deviation and permeability, was created. The results are of fundamental importance for the mechanics of multiphase flows in porous media and can be used in predicting hydrocarbon field development measures.
This article describes different modeling options for mixing the hot lubricant of the upstream pad of a fluid film bearing and the fresh lubricant of the oil-feeding inter-pad groove. The outcomes of the mixing process affect the lubricant temperature distribution near the leading edge of the next pad located downstream. Ultimately, this distribution determines the thermal state of the lubricant film, the output characteristics, and the power losses of the bearing. The fixed pad thrust bearing of a centrifugal or screw compressor was investigated. The mixing simulation was reduced to different boundary conditions for the truncated Navier–Stokes equation of motion, which are also part of the general periodic thermoelastohydrodynamic (PTEHD) mathematical model of bearing operation. The results of the numerical calculations of the thrust bearing characteristics under different flow boundary conditions were compared with those obtained using other programs. For these purposes, the numerically implemented Sm2px3Txt calculation program was employed. Based on the results, conclusions were drawn about the need for a sufficient flow of the fresh lubricant in the oil supply groove to compensate for lateral leaks of the pad and when the thrust bearing operation mode becomes more complicated, for example, if the gap decreases, the rotational speed increases, or the thrust collar moves.
The stability of the Hodgkin–Huxley model was examined with respect to the following three ionic parameters: the equilibrium potentials of potassium, sodium, and leak ions. A two-parameter bifurcation analysis of the Hodgkin–Huxley model was performed in the plane of the equilibrium potentials of potassium and sodium. Regions of stable and unstable steady states were identified. The Hopf bifurcation boundary was located. Using numerical continuation, the influence of the leak potential on shifts in the detected bifurcation values was investigated, and the two-dimensional stability maps were extended into a three-dimensional parameter space. The results clarify the role of the three equilibrium potentials in the modulation of the excitability of the Hodgkin–Huxley system and contribute to a better understanding of the dynamical consequences of ionic disturbances such as hyperkalemia, hyponatremia, and leak-channel dysfunction.
The spectra of grid operators in some finite element method (FEM) schemes for solving twodimensional problems of elasticity theory were investigated. It was revealed that, for a number of classical FEM schemes, a mutual influence between the spectra of volumetric and shear deformations may occur, which can lead to undesirable effects such as volumetric and shear locking. The analysis of the spectral errors in the FEM schemes demonstrated the effectiveness of the reduced integration technique and underscored the importance of fulfilling the group properties of grid differential operators that ensure consistent approximation of partial derivatives. A number of numerical schemes with improved spectral properties were described. The results were illustrated by the solutions of model problems.
