On the constitutive relations for second sound in elastic solids

We give a derivation of the constitutive relations of an elastic heat conductor for which the heat flux and the temperature obey a frame-invariant form of Cattaneo's equation.     

Uniform asymptotics for the linearized Boltzmann equation describing sound wave propagation

The multiple-scale expansionmethod is used for constructing a uniformly applicable asymptotic approximation of the solution of the linearized Boltzmann equation for small Knudsen numbers. The asymptotic expansion is constructed for the particular example of a sound wave generated by a plane oscillation source and dissipating in a half-space. The simplicity of the problem makes it possible clearly to demonstrate the appearance of secular terms in the expansion and the introduction of multiple scales opens the way to eliminating them.     

Modeling and simulation of first and second sound in solids

We investigate the propagation of mechanical and thermal waves in solids at cryogenic temperatures. The latter are known as the second sound phenomenon. It occurs, e.g., in dielectric solids and differs greatly from the classical case in which the heat transport proceeds by diffusion. Since Fourier’s law of heat conduction fails for modeling second sound, we apply a non-classical one. During the last two decades, the non-classical thermoelastodynamic theory of Green and Naghdi enjoys steadily growing research activities.     

On hyperbolic heat conduction in solids: minimum mean free path of energy carriers and a method of estimating thermal diffusivity and heat propagation characteristics

The paper gives the analytical solution to the one dimensional hyperbolic heat conduction equation in an insulated slab-shaped sample that is heated uniformly on the front face with δ or laser impulse. The solution results in a formula that enables to estimate the minimum mean free path of energy carriers in the sample to detect the second sound (i.e. the thermal wave) at the sample rear face. A method of experimental data evaluation at the second sound effect is proposed, which gives the thermal diffusivity of the sample and the parameters of heat propagation.     

Chaos in a non-autonomous nonlinear system describing asymmetric water wheels

We derive a water wheel model from first principles under the assumption of an asymmetric water wheel for which the water inflow rate is in general unsteady (modeled by an arbitrary function of time). Our model allows one to recover the asymmetric water wheel with steady flow rate, as well as the symmetric water wheel, as special cases. Under physically reasonable assumptions, we then reduce the underlying model into a non-autonomous nonlinear system. In order to determine parameter regimes giving chaotic dynamics in this non-autonomous nonlinear system, we consider an application of competitive modes analysis. In order to apply this method to a non-autonomous system, we are required to generalize the competitive modes analysis so that it is applicable to non-autonomous systems. The non-autonomous nonlinear water wheel model is shown to satisfy competitive modes conditions for chaos in certain parameter regimes, and we employ the obtained parameter regimes to construct the chaotic attractors. As anticipated, the asymmetric unsteady water wheel exhibits more disorder than does the asymmetric steady water wheel, which in turn is less regular than the symmetric steady state water wheel. Our results suggest that chaos should be fairly ubiquitous in the asymmetric water wheel model with unsteady inflow of water.     

Solution and analysis of hyperbolic heat propagation in hollow spherical objects

The hyperbolic heat conduction process in a hollow sphere with its two boundary surfaces subject to sudden temperature changes is solved analytically by means of integration transformation. An algebraic analytical expression of the temperature profile is obtained. Accordingly, the non-Fourier hyperbolic heat propagation in hollow spherical medium is analyzed and possible hyperbolic anomalies are discussed.     

Stability of heterogeneous equilibrium in a system with liquid phases

Gibbs's method is used to study the equilibrium and stability in a heterogeneous system consisting of single-component liquid phases. The coefficient of surface tension is assumed to be a constant independent of the state of the phases. An expression is obtained for the second variation of the corresponding functional, and this expression can be used to analyze the stability of nucleating centers. It is shown that in the absence of external force fields and surface tension forces the equilibrium state of a closed thermodynamic system consisting of single-component liquid phases satisfying the classical Gibbs inequalities is always stable.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 88–94, March–April, 1982.     

Decay of a potential vortex and propagation of a heat wave in a second grade fluid

The propagation of a heat wave in an incompressible second grade fluid within the context of a potential vortex is studied. The solutions for the Newtonian fluid can be obtained from those for fluids of second grade as a limiting case.     

Stability of equilibrium of a double-layer system with a deformable interface and a prescribed heat flux on the external boundaries

The stability of mechanical equilibrium of a system of two horizontal immiscible-liquid layers with similar densities is studied. The problem is solved for a prescribed heat flux on the external boundaries. Within the framework of a generalized Boussinesq approximation, which takes the interface deformation correctly into account, the onset of convection caused by heating the system from above or below is considered. Two long-wave instability modes attributable to the presence of the deformable interface and the given heat flux on the external boundaries are detected. The system response to monotonic and oscillatory disturbances with finite wavelengths is investigated. A complete stability map is constructed.     

Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type

A number of problems for the interaction of laser radiation with a heat-conducting half-space and a layer are considered. The obtained solutions are compared with each other and with the solutions of the classic heat equation and the wave equation. A laser impulse is modelled by defining the heat flux at the boundary for the opaque medium, or by defining the distribution of heat sources in the volume for the semitransparent medium. The power of the laser pulse depends on time as the Dirac delta function or as the Heaviside function do. It allows for the simulation of instant and continuous laser exposure on the medium. Temperature distributions are obtained by using Green’s functions for a half-space and a layer with the Dirichlet and Neumann boundary conditions.     

On crack propagation in a coupled thermomechanical system of nonlinear media

In some engineering problems.thermo-mechanical coupling is im-portant and may not be ignored.This paper deals With the crackpropagation problem in a coupled thermo-mechanical system of non-linear media.Various nonlinear media.including nonlinear e-lastic and elastic-plastic cases,have been considered and therelated path-independent integrals are given.To explain thephysical meaning of these integrals,a notched specimen hasbeen considered.and the dynamical crack extension force in acoupled thermo-mechanical system is shown to be equal to thisintegral.Thus.we could consider such integrals as some non-linear criteria for coupled thermo-mechanical fracture dynamics.     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.     

Riccati equations for wave propagation in planarly-stratified solids

Wave propagation and scattering are investigated in three-dimensional, dissipative, anisotropic bodies with one-dimensional inhomogeneity. The constitutive equations are taken to be linear. The incidence is allowed to be oblique. Three different procedures are set up which involve the impedance matrix and two local reflection matrices. All three matrix functions are shown to satisfy appropriate Riccati equations and jump conditions (at discontinuities). The operative aspects are examined to solve reflection–transmission problems for a layer sandwiched between two half spaces. The unknown field within the layer is determined through the impedance or the reflection matrices.     

A hyperbolic mathematical modeling for describing the transition saturated/unsaturated in a rigid porous medium

This work proposes a mathematical model to study the filling up of an unsaturated porous medium by a liquid identifying the transition from unsaturated to saturated flow and allowing a small super saturation. As a consequence the problem remains hyperbolic even when saturation is reached. This important feature enables obtaining numerical solution for any initial value problem and allows employing Glimm’s scheme associated with an operator splitting technique for treating drag and viscous effects. A mixture theory approach is used to build the mechanical model, considering a mixture of three overlapping continuous constituents: a solid (porous medium), a liquid (Newtonian fluid) and a very low-density gas (to account for the mixture compressibility). The constitutive assumption proposed for the pressure gives rise to a continuous function of the fluid fraction. The complete solution of the Riemann problem associated with the system of conservation laws, as well as four examples, considering all the four possible connections, namely, 1-shock/2-shock, 1-rarefaction/2-rarefaction, 1-rarefaction/2-shock and 1-shock/2-rarefaction are presented.     

Periodic solution of a second order, autonomous, nonlinear system

There exist a sufficient condition for the existence of at least one periodic solution for a type of second order autonomous ordinary differential equations. The correctness of the condition has been pointed out by Schauder's fixed point theorem. In order to indicate the validity of the assumptions made, two illustrative examples, showing its application in the nonlinear vibration and relaxation oscillation are presented.