Steady-state temperature distribution in a finite spherical cone

A Naylor transform is employed to determine the axisymmetric steady-state temperature distributions in finite spherical cones where the boundary conditions include a variety of conditions at the spherical surface (insulation or heat loss in accordance with Newton's law) and an assigned temperature variation on the conical surface (in particular,f(r)=r ^m ,m0). Solutions are obtained in terms of inversion integrals and in Fourier-Legendre expansions. Computations based on the inversion integrals are carried out for cones of half-angles, =/6 and /2, and with several combinations of parameters relating to the boundary conditions.

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Résumé Une transformation de Naylor est employée afin de déterminer les distributions axisymétriques stationnaires de la température dans les cônes sphériques finis où les conditions aux limites comprennent une variété des conditions à la surface sphérique (isolation ou perte de chaleur conformément à la loi de Newton) et une variation de la température assignée à la surface conique (en particulierf(r)=r ^m ,m0). Les solutions sont obtenues sous forme d'intégrales d'inversion et de développements en série de Fourier-Legendre. Les calculs fondés sur les intégrales d'inversion sont accomplis pour les cônes de demi-angles, -/6 et /2, et pour différentes combinaisons de paramètres qui apparaissent dans les conditions aux limites.

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Acknowledgement. This research was supported in part by Grant AFOSR-69-1779 at North Carolina State University at Raleigh, Raleigh, North Carolina. 27607.     

Hyperbolic heat conduction in a circular plate from green’s function

We show the existence and uniqueness of Green’s function of the Neumann problem for the axisymmetric hyperbolic heat conduction equation in a circular plate and present its explicit and rigorous computation. As an application, we use this function in order to compute the temperature profile in a circular plate irradiated by a continuous Gaussian laser source.     

Inverse heat conduction problem in a thin circular plate and its thermal deflection

An inverse problem of transient heat conduction in a thin finite circular plate with the given temperature distribution on the interior surface of a thin circular plate being a function of both time and position has been solved with the help of integral transform technique and also determine the thermal deflection on the outer curved surface of a thin circular plate defined as 0 ? r ? a, 0 ? z ? h. The results, obtained in the series form in terms of Bessel’s functions, are illustrated numerically.     

Steady-state thermal stresses in an elastic cone

Zusammenfassung Diese Arbeit befasst sich mit der Bestimmung der stationären Wärmespannungen, die in einem elastischen Kreiskegel durch eine unstetige Verteilung der Oberflächentemperatur hervorgerufen werden. Die Temperatur der Mantelfläche wird oberhalb und unterhalb einer festen Entfernung von der Kegelspitze konstant angenommen. Das zugehörige Wärmeleitungsproblem wurde in einer früheren Veröffentlichung unabhängig gelöst. In der vorliegenden Arbeit wird eine strenge Lösung für die gesuchten Wärmespannungen mit Hilfe der Mellin-Transformation hergeleitet. Das hier benutzte Verfahren ist auch anwendbar auf das gewöhnliche Randwertproblem für einen elastischen Kegel, der durch eine entsprechende unstetige Oberflächenbelastung beansprucht wird.

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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.     

Variational analysis of circular cone programs

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/co-derivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.     

Steady-state motions of a massif with a circular cavity under an antiplane deformation

We give a solution of the problem of antiplane deformation of an isotropic massif with a cavity of circular cross-section, reinforced by a multilayer cylinder under steady-state motions. We give the results of numerical studies that characterize the influence of reinforcement on the stress distribution in a neighborhood of the cavity. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 78–81.     

Torsion of a circular cone with static and dynamic loading

Integral transformation methods—the Mellin transform for statics and the Lebedev-Kontorovich transform for dynamics—are used to construct analytic solutions of the problem of the torsion of an elastic circular cone. Assuming that external forces are concentrated in the neighbourhood of the vertex of the cone, the asymptotic behaviour of the far field is investigated. It is shown that the leading term of the asymptotic expansion is governed by the magnitude of the moment of the external forces, so that the St Venant principle is satisfied in the cases under consideration.     

Steady-state problem of complex heat transfer

The problem of radiative-conductive-convective heat transfer in a three-dimensional domain is studied. The existence of a weak solution of the problem is proved, and sufficient conditions for the uniqueness of a solution are found. The temperature distribution in a three-dimensional channel is determined in numerical experiments.     

On a proposed model for heat conduction

^** Corresponding author. Email: jcsong{at}hanyang.ac.kr A system of partial differential equation for modelling theconduction of heat was proposed by Ghaleb & El-Deen Mohamedein(1989). According to their theory, the initial-value problemfor the temperature is ill-posed. In this paper, two well-posedproblems for the temperature are introduced and investigated.     

On a nonstandard problem for heat conduction in a cylinder

Decay bounds are derived for the solution of a heat conduction problem in a semi-infinite cylinder when the lateral surface is held at zero temperature, a nonzero temperature is prescribed on the finite base, and the temperature at time T is prescribed to be a constant multiple of the temperature at initial time. Both energy and pointwise decay bounds are computed for a range of values of the constant multiple. Such problems were originally introduced as a means of stabilizing the backward-in-time problem for the heat equation.     

Half-line solutions of a nonlinear heat conduction problem

We solve a half-line problem for a nonlinear diffusion equation with a given time-dependent thermal conductivity at the origin. The problem reduces to a linear Volterra integral equation, which is solvable by Picard’s process of successive approximations. We analyze some explicit examples numerically. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 58–65, July, 2007.     

On a radially symmetric inverse heat conduction problem

In this paper, we treat an inverse problem for a radially symmetric heat equation, which arises from non-destructive evaluation by thermal imaging. The problem can also be considered as an inverse heat conduction problem. Based on a weighted energy method, we give a conditional stability estimate. A feasible regularization method is provided for numerical simulation. The reconstruction experiment is done for verifying the efficiency of the regularization method.     

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.     

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.     

Reconstruction of spatial heat sources in heat conduction problems

In this short article, we consider the problem of recovering unknown spatial heat sources in heat equations. Applying Tikhonov's regularization approach, we define and obtain stable solutions to approximate the unknown sources from overspecified non-smooth data. We will also conduct numerical computations to demonstrate the applicability of our approximation.