Heat conduction with radiating boundary conditions

Existence of positive solutions is established for a class of nonlinear boundary value problems that include steady-state heat conduction problems with radiation at the boundary according to the fourth power radiation law. Existence is established using topological methods and a priori bounds.     

Heat conduction with nonlinear boundary condition

The paper is concerned with determination of the solution of the one-dimensional heat equation in a semi-infinite region subject to a nonlinear boundary condition at the surface. The exact solution is obtained and is expressed in infinite series. It is shown that the series is absolutely and uniformly convergent. Two special cases of Newton's cooling and of Stefan-Boltzmann's radiation at the boundary are discussed.

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Zusammenfassung Diese Arbeit befaßt sich mit der Lösung der Gleichung der eindimensionalen Wärmeleitung in einem Halbraum mit nicht-linearen Randbedingungen an der Oberfläche. Es wird eine genaue Lösung gefunden, die die Form einer unendlichen Reihe besitzt. Es wird gezeigt, daß die Reihe absolut und gleichmäßig konvergiert. Als zwei Spezialfälle werden Newton-Kühlung und Stefan-Boltzmann-Strahlung behandelt.

     

Heat conduction in transversally graded laminates with non-uniform microstructure

Two-component laminates made of conductors distributed non-periodically as laminas along one direction are considered in this note. The macroscopic properties of these laminates are described by continuous slowly-varying functions across laminas. In order to analyse heat conduction, the approach, called the tolerance modelling, is used. The aim of this note is to use averaged equations of the tolerance model for transversally graded laminates to analyse stationary heat conduction across laminas. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat conduction in laminates with a weak transversal inhomogeneity

The purpose of this contribution is to formulate a tolerance model, [1], for periodically laminated media with a weak transversal inhomogeneity. Laminates in which layers are of the same material but one of them is reinforced by thin fibers in parallel to the interfaces are this kind of media. Taking into account the weak transversal inhomogeneity makes possible to separate model equations for the averaged temperature and temperature fluctuations. To analyse and solve these equations we apply asymptotic expansions. Using the proposed model selected initial-boundary value problems were solved. Results were verified by direct numerical solutions obtained in the framework of the Fourier model. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

In this paper, we study the heat transfer in a one‐dimensional fully developed flow of granular materials down a heated inclined plane. For the heat flux vector, we use a recently derived constitutive equation that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient, and the velocity gradient in an appropriate frame invariant formulation. We use two different boundary conditions at the inclined surface: a constant temperature boundary condition and an adiabatic condition. A parametric study is performed to examine the effects of the material dimensionless parameters. The derived governing equations are coupled nonlinear second‐order ordinary differential equations, which are solved numerically, and the results are shown for the temperature, volume fraction, and velocity profiles. Copyright © 2013 John Wiley & Sons, Ltd.     

Heat conduction on the ring: Interface problems with periodic boundary conditions

The classical problem of heat conduction in one dimension on a composite ring is examined. The problem is formulated using the heat equation with periodic boundary conditions. We provide an explicit solution of this problem using the Method of Fokas. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. Instead, the physical assumption of continuity at the interface is imposed.     

Heat conduction of shells reinforced by fibers with constant and variable cross sections

We propose a model for heat conduction of a spatially reinforced medium and present its generalization to the case of a polyreinforced layer. We consider the heat-conduction equations for fibrous shells and construct a procedure for reduction of a three-dimensional problem of heat conduction to a two-dimensional one. Analytic solutions of a stationary problem of heat conduction are found for thin conic shells of revolution for various structures of reinforcement, and a graphical comparison of the corresponding results is performed. We study one of the approaches to rational reinforcement of thin shells, according to which the thermal “transparency” of a shell in the transverse direction is taken as a criterion of rational design. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 132–150, April–June, 1998.     

Heat conduction through a rarefied gas between two rotating cylinders at small temperature difference

Summary Numerical calculations of heat transfer between two coaxial rotating cylinders at a small temperature difference are carried out over wide ranges of the Knudsen number and the angular velocity. The calculations have been performed based on the S-model of the Boltzmann equation by the discrete velocity method. It has been confirmed that in a rotating gas a radial temperature gradient causes both radial and tangential heat fluxes. Also, it has been found that the radial heat flux is affected by the rotation.On temporary leave from Department of Physics, Urals State University, 620083 Ekaterinburg, Russia.     

Shape-preserving signal forms in heat conduction

Let us consider the heat conduction problem described by a parabolic equation. We study under which conditions is the time-dependence on the boundary preserved inside the solid. The question is how information entering on the boundary penetrates the solid. E.g. consider a heat conducting solid subject to sinusoidally varying boundary condition. After decay of the transients, the temperature at any inner point varies in time sinusoidally with the same circular frequency, with space dependent amplitude and phase delay. So, sinusoidal signals inserted on the boundary are preserved. Information is also preserved in case of linear signals. Farkas and Mudri [H. Farkas, I. Mudri, Shape-preserving time-dependences in heat conduction, Acta Phys. Hung. 55 (1984) 267–273] have formulated this phenomenon, defined the notion of the boundary following solution and the shape-preserving signal forms, determined necessary and heuristic sufficient conditions for the shape-preserving signal forms.

Their work is extended by rigorous proofs of some sufficient conditions in this paper, and the minimum of the phase delay, expected to be attained on the boundary for physical reasons, is examined.     

Energy fluctuations in simple conduction models

We introduce a class of stochastic weakly coupled map lattices, as models for studying heat conduction in solids. Each particle on the lattice evolves according to an internal dynamics that depends on its energy, and exchanges energy with its neighbors at a rate that depends on its internal state. We study energy fluctuations at equilibrium in a diffusive scaling. In some cases, we derive the hydrodynamic limit of the fluctuation field.     

End effects in three-phase-lag heat conduction

In this article we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under null initial data, a Phragmen–Lindelof alternative is obtained. An upper bound for the amplitude term in terms of the boundary data is also established. For the case of decay solutions, an improvement is obtained. We prove that the decay can be controlled by the exponential of a second-degree polynomial in the distance from the finite end of the cylinder. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T ₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

The nonlinear optimization problem in heat conduction

In this paper we study the existence and geometric properties of an optimal configuration to a nonlinear optimization problem in heat conduction. The quantity to be minimized is , where D is a fixed domain. A nonconstant temperature distribution is prescribed on and a volume constraint on the set where the temperature is positive is imposed. Among other regularity properties of an optimal configuration, we prove analyticity of the free boundary.Received: 6 October 2004, Accepted: 19 October 2004, Published online: 22 December 2004     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems. Received: July 14, 2003     

The Stefan problem in nonlinear heat conduction

Summary An exact parametric solution for the planar solidification of a liquid metal occupying the infinite half-space is presented. The metal is assumed to exhibit nonlinear thermal characteristics of the Storm type. Both the idealized one phase and the full two phase problems are considered. For both problems an approximate analysis of the underlying coupled transcendental equations is presented which provides initial estimates for use in a numerical scheme. Typical numerical results are given which illustrate the monotonic nature of the solution.

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Zusammenfassung Eine genaue Lösung für die ebene Solidifizierung eines flüssigen Metalls, das in einem unendlichen Halbraum liegt, wird dargestellt. Es wird angenommen, daß das Metall nonlineare thermale Elgenschaften der Stormschen Art aufweist. Das idealisierte Einphasen-Problem wie auch das voile Zweiphasen-Problem werden betrachtet. Für beide Probleme wird eine ungefähre Analyse der zugrundeliegenden gekuppelten transzendentalen Gleichungen gegeben, die Anfangswerte zur Verwendung in einer numerischen Darstellung liefert. Es werden typische Zahlenwerte gegeben, die die Monotonie der Lösung illustrieren.

     

Steady-state heat conduction in a circular cone

Zusammenfassung Diese Arbeit befasst sich mit der Bestimmung des stationären Temperaturfeldes, das in einem Kreiskegel durch eine unstetige Verteilung der vorgegebenen Oberflächentemperatur hervorgerufen wird. Die Temperatur der Mantelfläche wird oberhalb und unterhalb einer festen Entfernung von der Kegelspitze konstant angenommen. Dieses Problem wird mit Hilfe der Mellin-Transformation streng gelöst. Die so erhaltene Lösung wird in einer später erscheinenden Arbeit auf die Bestimmung der zugehörigen Temperaturspannungen angewendet.

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The results communicated in this paper were obtained in the course of an investigation conducted under Contracts Nonr 562(20) and Nonr 562(25) of Brown University with the Office of Naval Research in Washington, D.C.     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems.     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.