An asymptotic theory of the sound dispersion in a binary mixture of gases

The sound propagation in a binary mixture of multiatomic non-relaxing gases is considered using Burnett's equations (for the case of rapid exchanges of the internal and translational energies of the molecules). Asymptotic expressions (for small Knudsen numbers) are obtained for the absorption and dispersion coefficients, which are expressed in terms of the Navier–Stokes and the Navier–Stokes and the Burnett transport coefficients respectively. “Working” expressions of different levels of accuracy for these coefficients are known for the case of a binary mixture of monatomic gases. For this case, the results obtained are compared with known ones and the drawbacks and errors of previous papers are pointed out.     

具有两个驰豫时间的热弹性立方晶体材料中平面波的传播

研究具有两个驰豫时间的、两个不同弹性和热性质的、广义传热立方晶体固体半空间的有缺陷结合面上,热弹性平面波的反射和折射问题.具有两个驰豫时间的广义热弹性理论,是1972年由Green和Lindsay提出并应用于问题的研究.对有缺陷边界,给出了反射系数和折射系数(即反射波和折射波振幅与入射波振幅之比)的表达式,并推演了法向刚性边界、横向刚性边界、接触传热边界、滑动边界和结合面边界时的表达式.给出了在不同边界条件及出射角时,不同的反射波和折射波的振幅比,在不同的入射波时的变化图.发现反射波和折射波振幅比受到介质刚性和热性质的影响.     

The hydrodynamic equations for chemically equilibrium flows of a multielement plasma with exact transport coefficients

Explicit expressions for all of the effective transport coefficients are derived for thermochemically equilibrium flows using the exact mass and heat transfer equations, which are resolved with respect to the “forces” (the gradients of the hydrodynamic variables) via the flukes. It is shown that, in a mixture where the components have different diffusion properties, separation (diffusion) of the chemical elements occurs which leads to a state of affairs where the equilibrium concentrations, and together with them, the effective transport coefficients will be functions not only of the pressure and temperature but will also depend on the concentrations of the elements, determined when solving the problem (self-consistent concentrations of the elements). It is shown that the existence of an electric current and lack of quasineutrality (flow around electrically conducting walls—electrodes) does not change the structure of the expressions for the effective transport coefficients and does not add anything new. The approximate and incomplete treatment of thermochemically equilibrium flows of multicomponent gas mixtures and a plasma in previously published papers are especially noted. Numerical estimates of the effective transport coefficients are presented for an air plasma and the domains in the pressure-temperature plane with the required number of approximations in order to obtain results with an error of no worse than 5% are indicated.     

The arithmetic of cuts in models of arithmetic

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic.     

On the linear problem of swelling porous elastic soils

In this paper we consider the linear theory of swelling porous elastic soils. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas. It proposes some new mathematical difficulties. Logarithmic convexity and/or the Lagrange identity method is used in the case of fluid and gas saturation. Continuous results dependent on initial conditions and supply terms are obtained in the general case. Spatial decay estimates are also obtained by means of comparison arguments. This last result is only valid when suitable conditions on the viscosity coefficients are satisfied.     

Dual mean field annealing scheme for binary optimization under linear constraints

It is shown that certain general classes of constrained binary optimization tasks can be solved with increasing accuracy by a first order mean field approximation of the Boltzmann distribution of the associated Lagrangian as the instance size grows. The formalism is thoroughly analyzed for the quadratic and multidimensional knapsack models. In these cases analytical expressions for the convergence of the optimality gaps are given, which are experimentally verified.     

On the Evaluation of Polynomial Coefficients

The evaluation of the coefficients of a polynomial from its zeros is considered. We show that when the evaluation is carried out by the standard algorithm in finite precision arithmetic, the accuracy of the computed coefficients depends on the order in which the zeros are introduced. An ordering that enhances the accuracy for many polynomials is presented.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Computational methods for comparing two binary diagnostic tests in the presence of partial verification of the disease

In the presence of partial verification of the disease, confidence interval estimation for the difference of sensitivities and specificities cannot be carried out by applying confidence intervals for the difference of binomial proportions, and the comparison of the accuracy of the two binary tests cannot be carried out by applying McNemar’s test. In this article we propose two methods for comparing the accuracy of two binary tests in the presence of partial verification of the disease. The first method is based on the application of the EM and SEM algorithms, and the second method consists of the calculation of confidence intervals for the difference in sensitivities and specificities applying confidence intervals for the difference in the two binomial proportions from the last table obtained applying the EM algorithm. We carried out simulation experiments in order to study and compare the coverage of several confidence intervals for the difference of the sensitivities and specificities.     

A phenomenological and kinetic description of diffusion and heat transport in multicomponent gas mixtures and plasma

Different forms of expressing diffusion and heat fluxes in multicomponent mixtures, obtained by methods of non-equilibrium thermodynamics and the kinetic theory of gas mixtures, are analysed and compared. It is shown that an alternative representation of the linear relations of non-equilibrium thermodynamics is possible, which enables them to be written in a form similar to that of the well-known Stefan–Maxwell equations. A relation between the phenomenological coefficients of non-equilibrium thermodynamics and the corresponding transport coefficients obtained in kinetic theory is established, with a confirmation that the Onsager reciprocity relations are satisfied. It is shown that there is an advantage in writing the transport relations on the basis of the “forces in terms of fluxes” representation, compared with the classical “fluxes in terms of forces” representation, used in standard schemes of phenomenological non-equilibrium thermodynamics and the Chapman–Enskog method, traditional for kinetic theory. A generalization of the Stefan-Maxwell equations and the equation for the heat flux is considered, which takes into account the contribution to these equations of the time and space derivatives of the fluxes. The relaxation form of the equations obtained enable one to approach the analysis of the propagation of small heat and concentration perturbations in gas mixtures to be justified, which, within the framework of classical transport relations, propagate with infinitely high velocity. The results presented in this review enable one to determine the areas of effective application of different methods of describing diffusion and heat transfer in multicomponent gas mixtures when solving specific gas-dynamic problems.     

Exact and chapman-enskog solutions for the carleman model

The Chapman-Enskog procedure is applied to the Carleman model of the Boltzmann equation. It has been proved that the Carleman equations possess a solution on the time interval on which a smooth solution of the fluid-like equation exists. The calculations have been performed up to the first order i.e., to the Navier-Stokes-like equation. It has been shown that in this case a difference between an exact solution and the Chapman-Enskog solution is of order ?². Extension of the results to higher orders is also possible. This gives a justification of the Chapman-Enskog procedure as an asymptotic expansion method.     

The level probabilities for a simple loop ordering

Bartholomew's statistics for testing homogeneity of normal means with ordered alternatives have null distributions which are mixtures of chisquared or beta distributions depending on whether the variances are known or not. The mixing coefficients depend on the sample sizes and the order restriction. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a loop ordering is appropriate. Exact expressions for the mixing coefficients for a loop ordering and arbitrary sample sizes are given for five or fewer populations and approximations are developed for more than five populations. Also, the mixing coefficients for a loop ordering with equal sample sizes are computed. These mixing coefficients also arise in testing the ordering as the null hypothesis, in testing order restrictions in exponential families and in testing order restrictions nonparametrically.This research was supported by the National Institutes of Health under Grant 1 R01 GM42584-01A1     

Improved asymptotic predictions for the effective slip over a corrugated topography

Accurate analytical prediction of the effective slip boundary condition in shear-driven Stokes flows directed longitudinally and transversely to a one-dimensional sinusoidal no-slip topography is explored. First, the domain perturbation technique is extended through spectral analysis and symbolic computations to obtain polynomial approximations (Taylor polynomials) of arbitrary specifiable order for the effective slip length. However, when assessed for numerical accuracy against fully-resolved numerical simulations using the finite-element-method, higher order Taylor polynomials give progressively inferior predictions in comparison to lower-order ones, unless the product of amplitude and wave-number is restricted below unity. From Domb–Sykes plots, the reason for the poor accuracy of higher order Taylor polynomials is assessed to be the finite convergence radii, approximately equaling unity, of the asymptotic power series for both longitudinal and transverse flows. For either of the flows, application of Euler transformation to the expansion parameter provide polynomial-form approximations that are accurate for amplitude values exceeding the convergence radius. The slow convergence of the Euler-transformed series can be remedied through Shanks transformation, at the cost of losing the convenience of closed forms. Finally, Padé approximants are shown to provide even more accurate but still closed-form alternatives to polynomials that work accurately at amplitudes much exceeding the above-identified convergence radii.     

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 437–454, June, 2005.     

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic circular cylindrical shells

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic (FGMETE) circular cylindrical shell are carried out in the present work. The Hamilton principle, higher order shear deformation theory, constitutive equation considering coupling effect between mechanical, electric, magnetic, thermal are considered to derive the equations of motion and distribution of electrical potential, magnetic potential along the thickness direction of FGMETE circular cylindrical shell. The influences of various external loads, such as axis force, temperature difference between the bottom and top surface of shell, surface electric voltage and magnetic voltage, on the buckling response of FGMETE circular cylindrical shell are investigated. The natural frequency obtained by present method is compared with results in open literature and a good agreement is obtained.     

Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation

We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions.     

Rational approximation of decay-type functions

Numerically given functions with exponential asymptotic behavior are approximated by rational expressions. The numerators are obtained from the nodes of the functions, and the denominators are powers of polynomials, whose coefficients are obtained by expansion in orthogonal polynomials. Different weights are discussed for uniform absolute or relative errors.An iterative procedure is further described to improve the least-squares approximation to one of Chebyshev-type best fit.The research reported in this paper was sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, Knut and Alice Wallenberg's Foundation, The Swedish Natural Science Research Council, The Swedish Technical Research Council, and in part by the Aeronautical Research Laboratory, OAR, through the European Office, Aerospace Research, United States Air Force.Part of the work on this paper was carried out while the author was at the Physics Department, University of Florida, Gainesville, Florida, U.S.A.     

Numerical prediction of recirculation flows with free convection encountered in gas-agitated reactors

Flows in a gas-agitated reactor have been predicted by a finite difference procedure. The free-convection phenomena in the gas-liquid mixtures have been accounted for by the calculation of a void fraction determined from the gas flow rate. Computations have been performed for two different situations: first, with the allowance of slip between gas and liquid phases, and second, without any slip. Reasonable agreement has been achieved between the measurements.