Conserved Quantities for Equations with Infinite Group

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model) and makes it possible to calculate on non-structured non uniform grids in velocity and physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

Article demonstrates the applicability of modeling non-stationary non-isothermal gas flow along a linear section of a gas transmission system by means of using various numerically simulated models and sophisticated numerical techniques. There are described several models of non-stationary non-isothermal regimes of gas flow along the pipeline section. They are included in the considered general model and their comparative analysis is carried out by the virtue of numerical simulation. The finite difference algorithm is used to solve the simultaneous equations of the numerically simulated model for the pipeline section. The results of calculating the gas flow parameters using various models are presented: both with and without taking into account kinetic energy, as well as both with and without taking into account the Joule–Thompson effect. The matter of choosing the appropriate model is discussed. The obtained results can be used at the stage of transfer pipeline system operation in order to develop scientifically well-founded recommendations for improving the safety and efficiency of the pipeline transportation system.

Infinite discrete symmetry group for the Yang-Baxter equations: Spin models

The influence of nonisothermality and hydrodynamic nonstationarity on turbulence generation and development is analyzed. A physical model of a hydrodynamically nonstationary turbulent flow is proposed, the model is created on the basis of previous experimental investigations on the structure of a nonstationary turbulent gas flow in the channel.

One-dimensional flows in the quantum Hall system

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.

Thermo-fluid dynamics of the unsteady channel flow

The time-dependent flow of a neon–argon mixture in a microchannel interrupted by a row of oscillating barriers is numerically studied using the Direct Simulation Monte Carlo method in a range of Knudsen numbers from 0.1 to 10 and in a wide range of oscillation frequencies. The emphasis of the study is on the effect of mixture separation. It is demonstrated that in addition to a mid-frequency (“resonance”) regime, as discovered in the author's previous works [Kosyanchuk et al., “Numerical simulation of novel gas separation effect in microchannel with a series of oscillating barriers,” Microfluid. Nanofluid. 21, 116 (2017) and Kosyanchuk and Pozhalostin, “Non-stationary rarefied gas flow in a plane channel with a series of oscillating barriers,” Eur. J. Mech.-B/Fluids 92, 90–99 (2022)], two other enhanced separation regimes at very low and at very high oscillation frequencies are present. It is also demonstrated that the effect in the mid-frequency regime degrades with decreasing Knudsen number and is almost absent for Kn values around 0.1. The effect in the high-frequency regime is shown to be dictated both by the high frequency of barrier oscillations and by the high speed of barrier motion, and it is shown that with decreasing Knudsen number, the impact of barriers speed becomes dominant. The effect in the low-frequency regime is present for all Knudsen numbers and significantly depends on the phases of barrier motion, which is not observed in other regimes. The separation factor in the low-frequency regime also increases with the number of barriers but only up to the level of molecular diffusion. It was also shown that in the low-frequency regime, there is a trade-off between the separation factor and the gas flow rate.

The heat exchange process at the non-stationary separated supersound, trans- and subsonic gas current developing on a surface of the cylinder set across the flow is studied. The cylinder moving together with the evenly unperturbed flow at the time moment t = 0 stop instantaneously and further becomes still. The problem consist in gas current simulation at t >0 with regard to heat exchange. The solution of the non-stationary two-dimensional Navier-Stokes equations was performed by the Arbitrary Lagrangian-Eulerian computing method. Proposed in the present paper the non-linear, non-reflecting conditions at the outer boundary of the calculation region in comparison with the application of the ordinary “mild” conditions allow one to considerably reduce the dimensions of the calculation region and to make the calculations on the sufficiently detailed calculation nets actual. The results are presented by fields of velocity vectors, pressure, density, temperature and integral characteristics as well. A good agreement with the experiment is obtained.

A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we have found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for non-integrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappearand the Poisson distribution be recovered on (non Yang--Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion we have tested the generic calculability of the model on these algebraic varieties.

Celestial holography proposes a duality between gravitational scattering in asymptotically flat space-time and a conformal field theory living on the celestial sphere. Its dictionary relates the infinite dimensional space-time symmetry group to Ward identities of the CFT. The spontaneous breaking of these asymptotic symmetries governs the dynamics of the soft sector in the CFT. Here we show that this sector encodes non-trivial backreaction effects that exhibit characteristics of maximal quantum chaos. A key element in the derivation is the identification of the Hilbert space of celestial CFT, defined through radial quantization, with that of a constantly accelerating Rindler observer. From the point of view of the bulk, Rindler particles exhibit Lyapunov behavior due to shockwave interactions that shift the observer horizon. From the point of view of the boundary, the superrotation Goldstone modes affect the relevant representations of the celestial Virasoro symmetry in a manner that induces Lyapunov behavior of out-of-time-ordered celestial correlators.

Conservation laws of rheo-linear dynamical systems with one-and two-degrees-of-freedom

A finite element method (FEM) for solution of the conservative full potential flow equation is presented as a means for improving the computation of transonic flows about complex geometries. The method does not require an orthogonal mesh, thus removing a major constraint in grid generation. A standard Galerkin finite element formulation is used in conjunction with the artificial compressibility method which stabilizes the algorithm in transonic flow and allows the capture of embedded shock waves. Solution comparisons, including one test case which stresses the'capabilities of state-of-the-art finite difference methods, verify the FEM accuracy and geometric capability. I. Introduction W HILE the computation of inviscid two- and threedimensional flowfields by finite difference methods has now become commonplace in industry, one of the most important technological deficiencies is in the area of geometric modeling, particularly for transonic flows. The finite element method (FEM) is presented as a way of alleviating some problems of geometric modeling which limit the effectiveness of current finite difference methods. Certain specific advantages of the FEM, if fully exploited, may improve greatly the computational state-of-the-art. Since the governing equations of inviscid transonic potential flow are nonlinear, the method of superposition (i.e., panel methods) is insufficient to yield a proper solution, even though solutions have been obtained by a method combining surface singularities and field differences.1 Also, because of the nonlinearity, similarity laws do not exist which allow subsonic flows to be related to transonic flows. Field methods must be used, typically finite difference or finite element methods, requiring a grid of points to be defined, both on the body surface arid in the flowfield, on which solutions to the governing equations are approximated. Finite difference method (FDM) computer codes generally

The dynamics of particles around time conformal AdS-Schwarzschild black hole

In a medical study we are often interested in graphically displaying the relationship between continuous variables and clinical events indicating disease progression. Often, it is reasonable to make the minimal assumption that the risk of progression is an arbitrary monotone function of the continuous variable. Sometimes the continuous variable is a disease marker which is recorded longitudinally, and so the goal is to provide a graphical display of the relationship between the hazard for progression and the most recent measurement of the longitudinal marker. For example, we know that for a patient with HIV infection, declining CD4 count is associated with an increased risk of opportunistic infection. The goal of this paper is to extend isotonic regression techniques to failure time data with a continuous covariate, to obtain a non-parametric estimate for the hazard of disease progression as a monotonic function of the continuous variable. We propose two methods for modelling the relationship of the hazard and covariate: the first assumes that the hazard is constant over time, and the second allows the hazard to be an arbitrary function of time. These methods will be applied to graphically display the risk for an AIDS patient of an opportunistic infection as a function of CD4 count.