Localized heat perturbation in harmonic 1D crystals: Solutions for the equation of anomalous heat conduction

In this paper exact analytical solutions for the equation that describes anomalous heat propagation in a harmonic 1D lattices are obtained. Rectangular, triangular and sawtooth initial perturbations of the temperature field are considered. The solution for an initially rectangular temperature profile is investigated in detail. It is shown that the decay of the solution near the wavefront is proportional to \(1/\sqrt t \). In the center of the perturbation zone the decay is proportional to 1/t. Thus, the solution decays slower near the wavefront, leaving clearly visible peaks that can be detected experimentally.     

Line-integral solution for the stress and displacement fields of an arbitrary dislocation segment in isotropic bi-materials in 3D space

The solutions for the stress and displacement fields due to an arbitrary dislocation segment in an isotropic bi-material medium consisting of joined three-dimensional (3D) half spaces are derived and expressed in terms of line integrals, integrands of which are given in an exact analytical form that, in turn, can also be integrated to yield analytical expressions for the stress–displacement field. The solution is constructed by employing a general solution derived by Walpole [Int. J. Eng. Sci. 34 (1996) p.629] for any elastic singularity in joined isotropic half space, and combining it with Mura's integral formula for the displacement gradient of an arbitrary dislocation segment in homogeneous medium. The resulting new solution provides a framework for deriving analytical expressions for stress and displacement fields of dislocation curves of arbitrary shapes and orientations. The benefit of the method developed, as compared with other methods found in the literature, is that the new solution presented is naturally divided into two components, a homogenous component representing the field of a dislocation in an infinitely homogenous medium, and an image component. This makes it easy and straightforward to modify existing dislocation dynamics codes that already include the homogenous part. To illustrate the accuracy of the method, the stress field expressions of an edge dislocation with Burgers vector perpendicular to the bi-material interface are derived as a degenerate case of the general result. It is shown that our solution is identical to that found in the literature for this case.     

Quadratic solitons in nonconservative media

We develop a perturbation theory for the evolution of solitary waves in quadratic nonconservative nonlinear media. The expressions derived for the solitons' amplitude and width permit us to estimate straightforwardly the damping-amplification rate of an arbitrary soliton of the unperturbed system. The analytical results obtained agree well with numerical experiments.     

Analytical solution to the problem of a temperature jump in a metal

An analytical solution to the problem of a temperature jump in a metal is obtained. The jump is caused by a heat flow to the surface. An exact expression for the heat-induced electric field is derived. Based on the exact expressions, the electric field profiles are plotted for different ratios of the electron free path to the Debye length. It is shown that the field profile near the surface may differ substantially from the Debye profile.     

Perturbation Theory for Nonlinear Klein-Gordon Kinks

A direct approach is used to investigate perturbations about a kink solution for nonlinear Klein-Gordon models. Effects of perturbations on a kink are derived in this scheme. After linearizing the initial equations by making use of derivative expansion method from classical perturbation theory, the solution for the first-order correction is derived by means of the method of Laplace transformation. The parameter variations are naturally given by the secular conditions.     

平板在任意周期表面热扰动作用下的非Fourier热传导的求解与分析

分析了平板前表面遭受任意周期热扰动这类非Fourier传热情形下的温度响应. 采用双曲型热传导方程描述平板表面温度急速变化时的热传导问题. 为求解此类方程, 首先 利用分离变量法和Duhamel积分原理, 得到了平板前表面遭受突变热流和简谐热流两种情况下的解析解.然后, 在此基础上应用Fourier级数展开法和叠加原理, 获得了平板前表面热流任意周期变化时非Fourier热传导下温度场的解析表达式. 利用得到的解析表达式进行数值模拟, 分析了不同热松弛时间、 不同时刻和不同位置对温度响应的影响, 讨论了非Fourier热传导模型所给出的温度响应与Fourier热传导模型的差别. 这种方法能够处理许多在生产实际中具有周期边界条件的非Fourier热传导问题. 关键词： 非Fourier 热传导 周期变化 温度响应 平板     

Effects of heat conduction and sample size on ultrasonic absorption measurements

The absorption coefficient of a material can be determined by measuring the heating which occurs as a result of ultrasonic irradiation. When narrow focused beams are used to heat a sample, or when the available volume of a material is restricted to small dimensions, then the effect of heat conduction to surrounding unheated regions becomes significant, complicating the relation between measured temperatures and acoustic parameters. In this paper new analytical expressions, which account for radial and axial heat flow in a medium, are derived for the case of Gaussian-shaped ultrasonic beam patterns in thin or semi-infinite absorbing materials. Solutions are given for temperature histories resulting from an ultrasonic impulse (pulse decay method) or a step input (rate of heating method). The use of these equations in absorption measurements is discussed, and experimental results are given. These expressions provide flexibility in choice of laboratory ultrasonic parameters, and the results are especially useful for many biomedical measurements where the volume of tissue available is restricted.     

Soliton dynamics of symmetry-endowed two-soliton solutions of the nonlinear Schrodinger equation

A comprehensive analysis is presented of the propagation of symmetry-endowed two-soliton solutions under the influence of various perturbations important in nonlinear optics. Thus, we begin by introducing the analytical expressions of these two-soliton solutions. Then, by considering perturbations which preserve the initial symmetry of the two-soliton solutions, the dependence of the soliton parameters on the propagation distance is determined by using an adiabatic perturbation method. As perturbations of this kind, important for soliton-based communication systems, we consider the bandwidth-limited amplification, nonlinear amplification, and amplitude and phase modulation. Moreover, the results obtained by the adiabatic perturbation method are compared with those obtained by direct numerical simulations of the corresponding governing differential equations. (c) 2000 American Institute of Physics.     

One-dimensional ising model with next-nearest-neighbour interaction in magnetic field

The exact solution for the one-dimensional Ising model with nearest-neighbour and next-nearest-neighbour interactions in an external magnetic field is 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, magnetic field, and the signs and values of the interaction parameters.     

Perturbation technique for unsteady MHD mixed convection periodic flow, heat and mass transfer in micropolar fluid with chemical reaction in the presence of thermal radiation

An analytical study is presented for the problem of unsteady hydromagnetic heat and mass transfer for a micropolar fluid bounded by semi-infinite vertical permeable plate in the presence of first-order chemical reaction, thermal radiation and heat absorption. A uniform magnetic field acts perpendicularly to the porous surface which absorbs the micropolar fluid with a time-dependent suction velocity. The basic partial differential equations are reduced to a system of nonlinear ordinary differential equations which are solved analytically using perturbation technique. Numerical calculations for the analytical expressions are carried out and the results are shown graphically. The effects of the various dimensionless parameters related to the problem on the velocity, angular velocity, temperature and concentration fields are discussed in detail.     

Higher harmonics in the voltage on a superconducting wire carrying AC electrical current

The problem of determining the harmonic content in the voltage that appears on a superconducting wire carrying cosine-like AC current was resolved theoretically, using two approaches. First, the Fourier components of the voltage spectrum were found by numerical integration. Importance of individual terms was established, leading to two conclusions: a) it is the cosine component of the 3rd harmonic that represents the bulk of harmonic distortion, b) for the practical purposes it is sufficient to consider higher harmonics with n ≤ 7. Then, the analytical formulas were derived. While for the sine components a general expression containing an infinite series was found, closed-form formulas were derived for the cosine components of the harmonics 1, 3, 5, 7. Consequences of the results to the experimental technique used to study the AC transport properties of superconductors are discussed.     

Perturbations of the seismic reflectivity of a fluid-saturated depth-dependent poroelastic medium

Analytical formulas are derived to compute the first-order effects produced by plane inhomogeneities on the point source seismic response of a fluid-filled stratified porous medium. The derivation is achieved by a perturbation analysis of the poroelastic wave equations in the plane-wave domain using the Born approximation. This approach yields the Frechet derivatives of the P-SV- and SH-wave responses in terms of the Green's functions of the unperturbed medium. The accuracy and stability of the derived operators are checked by comparing, in the time-distance domain, differential seismograms computed from these analytical expressions with complete solutions obtained by introducing discrete perturbations into the model properties. For vertical and horizontal point forces, it is found that the Frechet derivative approach is remarkably accurate for small and localized perturbations of the medium properties which are consistent with the Born approximation requirements. Furthermore, the first-order formulation appears to be stable at all source-receiver offsets. The porosity, consolidation parameter, solid density, and mineral shear modulus emerge as the most sensitive parameters in forward and inverse modeling problems. Finally, the amplitude-versus-angle response of a thin layer shows strong coupling effects between several model parameters.     

Study of the effects of three types of time-periodic vertical oscillations on the linear and nonlinear realms of Rayleigh-Bénard convection in hybrid nanoliquids

In the paper, the effect of time-periodic gravity modulation with trigonometric sine, triangular, and square waves-forms on Rayleigh Bénard convection in water-alumina nanoliquids and water-alumina-copper hybrid nanoliqiuds is studied by using a single-phase model. Using a perturbation method, linear stability analysis is performed for all the three wave-forms. A generalized Lorenz model that has the influence of nanoliquids and modulation incorporated in it is derived using a truncated Fourier series representation. The Lorenz model is then transformed into a Ginzburg-Landau model using the method of multiscales, and the solution is used to study heat transport. For trigonometric sine, triangular and square wave-forms of modulations comparison are made on their effect on the onset of convection and the heat transport. The linear stability analysis reveals that the critical Rayleigh number obtained in the case of a triangular wave-form is less compared to the value obtained in the cases of trigonometric sine and square wave-forms. This leads to an enhanced heat transfer situation in the case of triangular wave-form compared to that in the other two wave-forms. It is also found that such an enhancement in heat transport increases with amplitude and decreases with the modulation frequency. Thus, the modulation is found to be a regulating mechanism on heat transport. Further, it is observed that water-alumina-copper facilitates maximum heat transport compared to that by water-alumina and water, leading to the conclusion that hybrid nanoliquids facilitate enhanced heat transport compared to that by mono nanoliquids.     

Tunnelling across a potential barrier

Generalized expressions for the tunnelling probability spectrum and the tunnelling time are derived for a potential barrier of arbitrary shape independent of the use of the perturbation theory. A method to obtain the tunnelling current density by solving the continuity equation in the barrier region is proposed.     

On the theory of electromagnetic radiation absorption during electron intersubband transitions in superlattices

An exact solution to the Schrödinger equation for electrons in superlattices with rectangular potential barriers and a periodic potential was obtained. An exact analytical expression for the electromagnetic radiation absorbance during electron intersubband transitions in such superlattices was derived in the first-order perturbation theory with arbitrary parameters. A number of extreme cases were considered.     

A regular solution model for phases with several components and sublattices,suitable for computer applications

A thermodynamic model, based on the regular solution approximation is presented and a formalism, suitable for phases with an arbitrary number of sublattices, is developed. A new concept, the component array, is introduced in order to simplify the analytical expressions for the integral Gibbs energy. The definition of the component array allows a straightforward procedure for the derivation of expressions describing the Gibbs energy for any kind of phase. Expressions for the partial Gibbs energy are derived. The implementation of the model on a computer is discussed.     

Gravitational perturbations in the expanding universe in a coordinate representation

Gravitational perturbations in the expanding universe are analyzed within the framework of Newtonian cosmology. In contrast to classical work, the perturbations are found not in a Fourier but in a coordinate representation. This makes it possible to obtain general expressions for finding the distributions of perturbations in the density and velocity of matter as a function of the coordinates and time from the known distribution at the initial time. In the simplest cases of plane-symmetric and spherically symmetric distributions, analytical equations are obtained for density perturbations as a function of the coordinates and time. The final conclusion is that it is possible for gravitational perturbations to grow even when the characteristic size of a perturbation is less than the Jeans wavelength. Kazan’ State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 45–52, March, 1997.     

Cooling kinetics of a granular gas of viscoelastic particles

The evolution of a granular gas of viscoelastic particles in the homogeneous cooling state is studied. The velocity distribution function of granular particles and the time dependence of the mean kinetic energy of particles (granular temperature) are found. The noticeable deviation of the distribution function from the Maxwell distribution and its non-monotonous evolution are established. The perturbation theory with respect to the small dispersion parameter is elaborated and the analytical expressions for the asymptotic time dependence of the velocity distribution function and the granular gas temperature are derived.