Exact solutions to one-dimensional acoustic fields with temperature gradient and mean flow

An exact solution for one-dimensional acoustic fields in ducts in the presence of an axial mean temperature gradient and mean flow is presented in this paper. The analysis is valid for mean Mach numbers such that the square of the mean Mach number is much less than one. The one-dimensional wave equation for ducts with axial mean temperature gradient and mean flow is derived. By appropriate transformations, the wave equation is reduced to an analytically solvable hypergeometric differential equation for the case of a linear mean temperature profile. The developed solution is applied to investigate the dependence of sound propagation in a duct on factors such as temperature gradient and mean flow. The results obtained using the analytical solution compare very well with the numerical results. The developed solution is also compared with an existing analytical solution.     

立方非线性Schr?dinger方程的Jacobi椭圆函数周期解

本文利用F-展开法,求出了立方非线性Schrodinger方程的由Jacobi椭圆函数表示的行波解;并且在极限情况下,得到了方程的孤波解.     

Mathematical Models for the Propagation of Stress Waves in Elastic Rods: Exact Solutions and Numerical Simulation

In this work, the Bishop and Love models for longitudinal vibrations are adopted to study the dynamics of isotropic rods with conical and exponential cross-sections. Exact solutions of both models are derived, using appropriate transformations. The analytical solutions of these two models are obtained in terms of generalised hypergeometric functions and Legendre spherical functions respectively. The exact solution of Love model for a rod with exponential cross-section is expressed as a sum of Gauss hypergeometric functions. The models are solved numerically by using the method of lines to reduce the original PDE to a system of ODEs. The accuracy of the numerical approximations is studied in the case of special solutions.     

Electronic phenomena in semi-conductors with a temperature gradient

Phenomena connected with the transfer of current carrier concentrations as a result of a temperature gradient in semi-conductors with two kinds of current carriers are theoretically studied. The paper first deals with the general equations which are valid at a certain point of the semi-conductor with a temperature gradient for the concentrations of current carriers and density of electric and thermal current. From this the distribution of the concentrations of current carriers is found for a typical case, the phenomenon of thermal emission of minority current carriers is discussed, relations are derived for the dependence of the thermal emf on the magnitude of the temperature gradient, the possibility is discussed of thermal rectification and equations are given for the thermal conductivity of a semi-conductor with non-equilibrium concentrations of current carriers.     

DIFFERENTIALLY MOVING MEDIA WITH MANY SPECTRAL LINES: STOCHASTIC APPROACH

Based upon the analytical solution of the radiative transfer equation for a given source function and a new approach to account for very many spectral lines contributing to the extinction, the connection between line properties and the emergent intensity is derived under the assumption that the wavelengths of the line centers follow a Poisson point process, whereas the other line parameters may have arbitrary distribution functions.A comparison with the widely used list of Kurucz shows that the Poisson distribution well describes deterministic “real” lines. The presentation by a Poisson point process requires only a modest number of parameters and is very flexible. It allows most operations to be carried out analytically and hence is very suitable to study the intricate influence of many lines on radiation fields in differentially moving media.We consider a simplified case of the solution of the radiative transfer equation in order to demonstrate the basic effects of the velocity field upon the emerging radiation field. Expressions for the expectation value of the intensity are derived, and examples are given for Lorentz line profiles and infinitely sharp lines, in particular as functions of the velocity gradient and the mean line density.     

Acoustic streaming in closed thermoacoustic devices.

A derivation of acoustic streaming in a steady-state thermoacoustic device is presented in the case of zero second-order time-averaged mass flux across the resonator section (nonlooped device). This yields analytical expressions for the time-independent second-order velocity, pressure gradient, and time-averaged mass flux in a fluid supporting a temperature gradient and confined between widely to closely separated solid boundaries, both in the parallel plate and in the cylindrical tube geometries (two-dimensional problem). From this, streaming can be evaluated in a thermoacoustic stack, regenerator, pulse tube, main resonator of a thermoacoustic device, or in any closed tube that supports a mean temperature gradient, providing only that the acoustic pressure, the longitudinal derivative of the pressure, and the mean temperature variation are known.     

Instabilities in buoyant flows under localized heating

We study, from the numerical point of view, instabilities developed in a fluid layer with a free surface in a cylindrical container which is nonhomogeneously heated from below. In particular, we consider the case in which the applied heat is localized around the origin. An axisymmetric basic state appears as soon as a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. We find different kinds of instabilities: extended patterns growing on the whole domain, which include those known as targets, and spiral waves. Spirals are present even for infinite Prandtl number. Localized structures both at the origin and at the outer part of the cylinder may appear either as Hopf or stationary bifurcations. An overview of the developed instabilities as functions of the dimensionless parameters is presented in this article.     

FREE VIBRATIONS OF FUNCTIONALLY GRADED PIEZOCERAMIC HOLLOW SPHERES WITH RADIAL POLARIZATION

By introducing displacement functions as well as stress functions, two independent state equations with variable coefficients are established from the three-dimensional equations of a radially inhomogeneous spherically isotropic piezoelastic medium. By virtue of the laminated approximation method, the state equations are then transformed into the ones with constant variables in each layer, and the state variable solutions are presented. Based on the solutions, linear algebraic equations about the state variables only at the inner and outer spherical surfaces are derived by utilizing the continuity conditions at each interface. Frequency equations corresponding to two independent classes of vibrations are finally obtained from the free surface conditions. Numerical calculations are presented and the effect of the material gradient index on natural frequencies is discussed.     

An analytical superposition approach to wall heat conduction under arbitrary temperature perturbations for air-conditioning applications

An analytical solution is derived for the transient wall heat conduction problem under arbitrary outdoor air temperature perturbations. This is obtained by expanding the arbitrary perturbation in a Fourier series, consisting of a step function and an infinite sum of cosine and sine functions. The analytical solutions in the cases of step, cosine and sine temperature perturbations are derived separately from each other and are then superimposed to produce the resultant solution. Simple linear expressions are derived, which express the maximum indoor heat flow provoked by the arbitrary temperature perturbation in terms of parameters characterizing the perturbation. Such simple expressions are very useful in air-conditioning calculations for predicting peak loads and sizing heating or cooling equipment.     

The influence of temperature gradients on the kinetic boundary layer problem for a condensing droplet

We extend an earlier method for solving kinetic boundary layer problems to the case of particles moving in aspatially inhomogeneous background. The method is developed for a gas mixture containing a supersaturated vapor and a light carrier gas from which a small droplet condenses. The release of heat of condensation causes a temperature difference between droplet and gas in the quasistationary state; the kinetic equation describing the vapor is the stationary Klein-Kramers equation for Brownian particles diffusing in a temperature gradient. By means of an expansion in Burnett functions, this equation is transformed into a set of coupled algebrodifferential equations. By numerical integration we construct fundamental solutions of this equation that are subsequently combined linearly to fulfill appropriate mesoscopic boundary conditions for particles leaving the droplet surface. In view of the intrinsic numerical instability of the system of equations, a novel procedure is developed to remove the admixture of fast growing solutions to the solutions of interest. The procedure is tested for a few model problems and then applied to a slightly simplified condensation problem with parameters corresponding to the condensation of mercury in a background of neon. The effects of thermal gradients and thermodiffusion on the growth rate of the droplet are small (of the order of 1%), but well outside of the margin of error of the method.     

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrödinger equation. Using the particular integral of the inhomogeneous Schrödinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.     

Fluctuations in a fluid under a stationary heat flux II. Slow part of the correlation matrix

The general formulas, derived in a previous paper, are used to calculate the correlation functions of the hydrodynamic variables in the Rayleigh-Bénard system. The behavior of the correlation functions on a time scale slow compared to that of sound propagation is determined, using systematically nonequilibrium hydrodynamic eigenmodes. These (slow) eigenmodes of the linearized Boussinesq equations in the presence of gravity and a temperature gradient are the viscous and the visco-heat modes. They are determined for ideal heat-conducting plates with stick boundary conditions. The visco-heat modes are found to behave qualitatively different from those obtained with slip boundary conditions. Using these eigenmodes, the slow part of the correlation functions can be determined explicitly. On a small length scale, as probed by light scattering, we recover the same expression for the Rayleigh line as quoted in the literature. On larger length scales, as probed by microwaves, the coupling of gravity to the temperature gradient gives rise to a convective instability (heating form below) or to propagating visco-heat modes (heating from above). The corresponding correlation functions and the Rayleigh line are calculated and discussed.     

Relations between matrix elements of the nonlinear Boltzmann collision integral in the axisymmetric case

The properties of the nonlinear Boltzmann collision integral for the axisymmetric velocity distribution are studied. Expansions in spherical Hermitian polynomials orthogonal to the Maxwellian weighting function are employed. It is shown that the nonlinear matrix elements of the collision operator are related to each other by simple relations, which are valid for arbitrary cross sections of particle interaction even if a preferential direction exists. The relations are derived from the invariance of the collision operator under the choice of basis functions or, more precisely, under both temperature and the mean velocity of the Maxwellian weighting function. The recurrent relations found allow one to calculate the matrix elements at large values of indices. This makes it possible to construct exact solutions to complicated kinetic problems.     

Shear Viscosity and Temperature Inequalities for Multi-Species Plasmas

The first coefficients in the orthogonal expansions of the velocity distribution functions with respect to Grad's tensorial (three-dimensional) Hermite polynomials are shown to be proportional to ther-modynamic fluxes, if the weight functions are local Maxwellians centered at the mean mass velocity and widened with a mean temperature. Balance equations for the stress tensors are established and reduced to linear algebraic systems under certain restrictions. Explicit solutions for the traces and the tracefree parts of the stress tensors are given.     

Sound propagation in acoustically lined elliptical ducts

Sound propagation in ducts with elliptical cross-sections can be described in terms of modes characterized by Mathieu functions of orders specified by the boundary conditions. For ducts with locally reacting liners there is coupling between modes because the admissible solutions are linear combinations of Mathieu functions of different orders and the eigenvalues are roots of an infinite determinant. The amount of mode coupling depends on the eccentricity of the duct. For the case of small eccentricity of the duct, approximate general solutions are derived and an example is discussed, where solutions are found.     

Analytical investigation into laser pulse heating and thermal stresses

Laser pulse heating of metallic surfaces results in rapid rise of temperature in the region irradiated by the laser beam. This in turn results in high temperature gradient in this region. The irradiated substrate material expands as a response to the temperature gradient. Consequently, high thermal stress levels are developed in the region of the high temperature gradient. In the present study, closed form solutions for temperature and stress fields due to a laser pulse decaying exponentially in time are presented. A Laplace transformation method is employed in the analysis. The resulting equations are non-dimensionalized with the appropriate parameters. It is found that temperature rises rapidly during the early heating period in the surface region. In this case, internal energy gain dominates the conduction losses from the surface vicinity. The thermal stress levels attain high values in the surface region. The stress wave developed is compressive and it propagates with a wave speed c₁ inside the substrate.     

Lattice vibrations and local order in substitutional solid solutions. I

Expressions are derived for the effective mean square amplitudes of the thermal vibrations of atoms and the Debye characteristic temperatures for the case of binary, substitutional, metallic solid solutions. Both solid solutions with random distribution of atoms and also those with a partially ordered distribution of atoms are investigated. The calculations respect the change in the vibrational state of atoms on solid solution formation. The parameterα, characterizing this change, can be determined from X-ray diffraction experiments or from the concentration dependence of the melting temperature and lattice constant. The results are applied to a solid solution of Ag in Al. A comparison with the experimental values measured by M. Simerská (Czech. J. Phys.B 13 (1963), 737) shows good agreement.     

A mathematical solution for the parameters of three interfering resonances

The multiple-solution problem in determining the parameters of three interfering resonances from a fit to an experimentally measured distribution is considered from a mathematical viewpoint. It is shown that there are four numerical solutions for a fit with three coherent Breit-Wigner functions. Although explicit analytical formulae cannot be derived in this case, we provide some constraint equations between the four solutions. For the cases of nonrelativistic and relativistic Breit-Wigner forms of amplitude functions, a numerical method is provided to derive the other solutions from that already obtained, based on the obtained constraint equations. In real experimental measurements with more complicated amplitude forms similar to Breit-Wigner functions, the same method can be deduced and performed to get numerical solutions. The good agreement between the solutions found using this mathematical method and those directly from the fit verifies the correctness of the constraint equations and mathematical methodology used.     

Modified 6-state and 8-state potts models in magnetic field

The exact solutions for the modified 6-state and 8-state Potts models in an external magnetic field are obtained within the framework of the Kramers-Wannier transfer matrix. The explicit analytical expressions for thermodynamic quantities such as magnetization, magnetic susceptibility and specific heat are derived and analyzed at length as functions of temperature and magnetic field.     

One-dimensional 3-state and 4-state standard Potts models in magnetic field

The exact solutions for 3-state and 4-state standard Potts models in external magnetic field are obtained within the framework of the Kramers-Wannier transfer matrix. The explicit analytical expressions for thermodynamic quantities such as magnetization, magnetic susceptibility and specific heat are derived and analysed at length as functions of temperature and magnetic field.