Generalized heat-transport equations: parabolic and hyperbolic models

We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman–Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.     

Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

Fast precise integration method for hyperbolic heat conduction problems

A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suit- able for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples.     

The hyperbolic heat conduction equation in an anisotropic material

In this work, the phase-lag concept in the wave theory of heat conduction is extended to describe the thermal behavior of anisotropic material. This is achieved by assuming that there are phase-lags of different magnitudes between each component of the heat flux vector and the summation of temperature gradients in all directions of the orthogonal coordinate system. Also, expressions are provided to specify the locations of the principal coordinate axes, the principal thermal conductivities and the principal thermal relaxation times. Received on 26 March 1999     

A generalized thermal boundary condition for the hyperbolic heat conduction model

 A generalized thermal boundary condition is derived for the hyperbolic heat conduction equation to include all thermal effects of a thin layer, whether solid-skin or fluid film, moving or stationary, in perfect or imperfect thermal contact with an adjacent domain. The thin layer thermal effects include, among others, thermal capacity of the layer, thermal diffusion, enthalpy flow, viscous dissipation within the layer and convective losses from the layer. Six different kinds of thermal boundary conditions can be obtained as special cases of the generalized boundary condition. The importance of the generalized boundary condition is demonstrated comprehensively in an example. The effects of different geometrical and thermophysical properties on the validity of the generalized thermal boundary condition are investigated. Received on 23 May 2001 / Published online: 29 November 2001     

Effect of thermal losses on the microscopic hyperbolic heat conduction model

The effects of radiative losses on the thermal behavior of thin metal films, as described by the microscopic two-step hyperbolic heat conduction model, are investigated. Different criteria, which determine the ranges within which thermal radiative losses are significant, are derived. It is found that radiative losses from the electron gas are significant in thin films having [(C_R e_e^4/₃ T_￥ ⁴ )/(k_e^1/₃ L^2/₃ G)] 3 4.6 ×10⁷{{C_R \epsilon _e^{{4 \over 3}} T_\infty ^4 } \over {k_e^{{1 \over 3}} L^{{2 \over 3}} G}}\geq 4.6 \times 10^7 for /_o > 4 and F_F < 1 and [(C_R e_e^3/₂ T_￥ ^9/₂)/(k_e^1/₂ L^1/₂ G)] 3 7.4 ×10¹⁰{{C_R \epsilon _e^{{3 \over 2}} T_\infty ^{{9 \over 2}}} \over {k_e^{{1 \over 2}} L^{{1 \over 2}} G}}\geq 7.4 \times 10^{10} for /_o < 4 and F_F > 1.     

Transient hyperbolic heat conduction in thick-walled FGM cylinders and spheres with exponentially-varying properties

This paper focuses on non-Fourier hyperbolic heat conduction analysis for heterogeneous hollow cylinders and spheres made of functionally graded material (FGM). All the material properties vary exponentially across the thickness, except for the thermal relaxation parameter which is taken to be constant. The cylinder and sphere are considered to be cylindrically and spherically symmetric, respectively, leading to one-dimensional heat conduction problems. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin’s numerical inversion method. The transient responses of temperature and heat flux are investigated for different inhomogeneity parameters and relative temperature change values. The comparisons of temperature distribution and heat flux between various time and material properties are presented in the form of graphs.     

A Precise time stepping scheme to solve hyperbolic and parabolic heat transfer problems with radiative boundary condition

This paper develops a precise discretized algorithm in the time domain solving hyperbolic and parabolic heat conduction problems with radiative boundary condition. By expanding variables at a discretized time interval, FEM based recurrent formulae are derived, by virtue of which, a self-adaptive computing procedure, without requirement of iteration for the non-linear solutions, can be carried out for different sizes of time steps. Numerical validation gives satisfactory results.     

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

Homogenized equations for steady heat conduction in composite materials with dilutely-distributed impurities

In this paper.by using the two-space method,homogenizedequations for steady heat conduction in the composite ma-terial cylinders with dilutely-distributed elliptic cylin-ders of impurities are derived.and the ezplicit ezpres-sions for the corresponding effective heat conductivity ofthose which are concerned are obtained.It is also shownthat the macroscopic heat conduction is anisotropic whenthe cross-sections of the impurity cylinders are unidirec-tionally oriented and isotropic when the angular distribu-tion of the cross-sections is uniform.     

The effects of thermal softening and heat conduction on the dynamic growth of voids

This paper seeks to examine the dynamic growth of a single void in an elastic–plastic medium through analytical and numerical approaches. Particular attention is paid to the instability of void growth, and to the effects of inertia, thermal softening and heat conduction. A critical stress is known to exist for the unstable growth of voids. The dependence of this critical stress on material properties is examined, and this critical stress is demonstrated to correspond to the lower limit for the ductile spall strength in many materials. The effects of heat conduction on the dynamic growth of voids strongly depend on the time and length scales in the early stages of the dynamic void growth.     

Non-linear vibrating systems in resonance governed by hyperbolic differential equations

The asymptotic solutions of second order hyperbolic differential equations with weak non-linearities in the case of internal and external resonance are found. The method used is an extension of the Krylov-Bogoliubov-Mitropolskii method. An application is made to the longitudinal vibrations of a rod in which non-linear elastic behaviour and linear viscoelastic damping occur.     

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.     

Hamiltonian description of the heat conduction

It is shown that the linear boundary value problems of the heat conduction in a homogeneous slab can be mapped on the initial value problem for a Hamiltonian motion whose phase-space trajectories are subject to an additional restriction, the “arrival condition”. The physical consequences of this formal analogy for the macroscopic heat conduction are discussed in detail. Received on 27 April 1998     

Some self-similar solutions of two-temperature hydrodynamic equations with allowance for nonlinear heat conduction and exothermic reactions

Processes occurring in plasma produced as a result of the interaction of powerful radiation fluxes with matter can be divided into three stages: absorption of radiation on the matter boundary, subsequent heating and compression of the central part of the target for the purpose of creating the conditions necessary for the initiation of an exothermic reaction and, finally, propagation of an exothermic reaction wave through the ambient matter. The present paper is devoted to an investigation of the last stage, a reaction wave igniting initially cold matter. The main method for the theoretical investigation of the processes described is a numerical solution of the equations of motion of a two-temperature gas with allowance for the physical processes occurring in a completely ionized medium: electron heat conduction, radiative losses, energy transfer between electrons and ions, and others. In view of the complicated nonlinear nature of the system of partial differential equations describing the process, searches for possible self-similar solutions are of interest. These solutions can be used as tests in calculating a complete system of equations; by means of them it is also possible to investigate asymptotic laws of exothermic reaction wave propagation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 145–151, March–April, 1986.     

Boundary integral equations of fluid flow and electrical and heat conduction in a medium composed of blocks

For many applications in the theory of flow through porous media, diffusion, convective transport, and electrical and heat conduction it is important to consider problems involving systems of blocks separated by boundaries (in particular, cracks or conductors), the transport process taking place both through the blocks themselves and along the separating boundaries. Such problems can conveniently be solved by the method of boundary integral equations [1–4]. The object of this study is to propose a form for these equations and methods of solving them specially designed to take into account the particularities of the problems in question. Firstly, the boundary integral equation is given in a form that contains only the net inflow (of fluid, heat or electricity) to a unit area of the boundary and not the individual inflows from each of the blocks separated from that boundary. This almost halves the number of unknowns subject to determination. Secondly, the principal aspects associated with the specifics of the flow along the boundaries and their intersections and discontinuities are discussed. Thirdly, a numerical experiment to realize the proposed form of the equations on a computer using algorithms with different structures leads to quite general conclusions which may be useful for further developing the method of boundary integral equations in relation to the class of applied problems considered. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–78, November–December, 1986. The authors are grateful to their hydrogeologist colleagues at the All-Union Institute of Mining Geomechanics and Surveying (VNIMI) and the Leningrad Mining Institute for taking a stimulating interest in their work over a period of three years and for discussing its various applications.     

Thermal bending associated with heat conduction by holographic interferometry

An experimental technique using real-time holographic interferometry combined with digitized image processing has been developed to measure the thermal diffusivity of polymers. This technique uses a cantilever beam or an annular disk with one side subjected to a pulse of radiant energy from a photographic flash. The resulting thermally induced deflection is measured by holographic interferometry. The observed deflection is due to a resultant thermal moment induced by a temperature gradient through the thickness. As time goes on, the heat conducts from the exposed surface through the thickness, resulting in a decrease of the bending moment and transverse deflection. It is shown that the deflection is proportional to the thermal moment, and the thermal diffusivity can be retrieved by moment analysis without deriving the analytical solution to the thermomechanical problem.