On heat conductors with a stationary hot spot

We consider a heat conductor having initial constant temperature and zero boundary temperature at every time.The hot spot is the point at which temperature attains its maximum at each given time. For convex conductors, if the hot spot does not move in time, we prove symmetry results for planar triangular and quadrangular conductors.Then, we examine the case of a general conductor and, by an asymptotic formula, we prove that, if there is a stationary critical point, not necessarily the hot spot, then the conductor must satisfy a geometric condition. In particular, we show that there is no stationary critical point inside planar non-convex quadrangular conductors. Mathematics Subject Classification (2000) Primary 35K05, 35K20, 35J05; Secondary 35J25, 35B38, 35B40     

Convex domains with stationary hot spots

Consider the problem of heat flow in a convex domain in ℝⁿ with Dirichlet boundary condition and constant initial temperature. We show that the solution has a fixed hot spot if the domain is invariant under the action of an essential symmetry group. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On characterizing rigid heat conductors

Zusammenfassung Diese kurze Mitteilung charakterisiert isotrope homogene starre Körper, in denen der Wärmestrom nur vom Temperaturgradienten abhängt. Der Begriff des kontrollierbaren Zustandes wird dabei nicht benutzt.     

Controllable states of rigid heat conductors

Zusammenfassung Es werden stationäre Temperaturfelder bestimmt, welche in allen isotropen homogenen starren Körpern ohne äussere Wärmezufuhr möglich sind. Wenn der Wärmestrom nur von der Temperatur und vom Temperaturgradienten abhängt, muss das Temperaturfeld gleichförmig sein. Hängt der Wärmestrom nur vom Temperaturgradienten ab, dann muss das Temperaturfeld schraubenförmig sein. Für den letzten Fall werden Beispiele gegeben.     

Melting of a long cylinder due to an axisymmetric hot spot

By employing a new embedding technique, a short-time analytical solution for the melting of a long cylinder due to an axisymmetric hot spot is presented in this paper. The melt is removed as soon as it is formed. The initial propagation of the melt along the interior of the cylinder exhibits a well-known behaviour. However, the melt propagation along the surface of the cylinder is of a new type and depending on the applied flux whether it is the continuation of the pre-melting flux or diferent, the initial propagation of the melt along the surface is proportional tot ^1/2 ort ^1/4, wheret is the dimensionless time. The melt propagation is much faster along the surface of the cylinder than towards the interior.     

On stationary flows of asymmetric fluids with heat convection

Existence, uniqueness and regularity of solutions of equations describing stationary flows of viscous incompressible isotropic fluids with an asymmetric stress tensor have been considered recently.⁵ In this paper we extend the results of Reference 5 to include heat convection in the hydrodynamic model. We show that the boundary value problem (1.1)–(1.6) below has solutions in appropriate Sobolev spaces, provided the viscosities v and c_a + c_d are sufficiently large. The proof is based on a fixed point argument. Moreover, we show that the solutions are unique if the heat conductivity κ is large enough.     

Dynamics of rotating thermoelastic disks with stationary heat source

Rotating disks are important components of car brakes or sawing units. In both cases, heat effects to be induced via stationary local contacts between pads and disk or workpiece and saw blade, respectively, influence the dynamic behavior and raise interesting problems in theory and practice. Therefore, the discussion of dynamic thermoelasticity in rotating disks with stationary heat sources is of basic interest to understand the interaction of temperature and displacement or stress fields in such structural elements. It will be analyzed here in detail for the case of an elastic disk for which there is a full (but weak) coupling of strain and temperature within the two applications mentioned. As a relatively general case, the combined excitation by a mechanical load and a simultaneously acting heat source will be examined.     

Equipresence and constitutive equations for rigid heat conductors

Zusammenfassung Die Autoren diskutieren neuere Anwendungen des Äquipräsenzprinzips auf Konstitutionsgleichungen ruhender Wärmeleiter und entwickeln sodann eine Theorie, in welcher die Beeinflussung der freine Energie, der Entropie und des Wärmeflusses durch die Geschichte nicht nur der Temperatur, sondern auch des Temperaturgradienten zugelassen ist. Sie formulieren für starre Körper dieser Art notwendige und hinreichende Bedingungen für die Gültigkeit der Clausius-Duhemschen Ungleichung in allen glatten Wärmeprozessen.     

Direct solution of an optimal layout problem for isotropic heat conductors with a volume fraction constraint

This paper considers the problem of the optimal layout of two isotropic heat conductors on a plane subject to a constraint on the overall volume fraction of the better conductor. The problem is solved initially using the traditionalG _m -closure technique. Later, the utility of the direct relaxation technique introduced in Ref. 1 is illustrated as we apply it to verify the original results.This paper is dedicated with gratitude by the author to his advisor K. A. Lurie. The author wishes to acknowledge discussions with D. Clomenil.     

Large time behaviors of hot spots for the heat equation with a potential

We consider the Cauchy problem of the heat equation with a potential which behaves like the inverse square at infinity. In this paper we study the large time behavior of hot spots of the solutions for the Cauchy problem, by using the asymptotic behavior of the potential at the space infinity.     

Solvability of the stationary heat convection problem for a high-viscosity fluid

We study the stationary heat convection problem in the Boussinesq approximation. We derive a priori estimates for its solution. We prove existence and uniqueness theorems for a weak solution of the problem and analyze the smoothness of a weak solution for raised smoothness of the problem data. We consider the two- and three-dimensional cases.     

On the approximation of the thermal conductivity of rigid heat conductors as a Cauchy Problem

Summary It is shown that the material response function of a nonlinear rigid heat conductor may be viewed as the solution to a Cauchy Problem. The measurements necessary to determine sufficient initial and characteristic data are indicated.

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Zusammenfassung Der Näherungswert von nichtlinearem Wärmeleitungsvermögen eines starren Körpers wird äquivalent mit der Lösung eines Cauchy-Problems bewiesen.

     

Computing Canonical Polygonal Schemata with Generalized Maps

This paper shows that a well-known algorithm proposed to compute the canonical polygonal schema of a surface can be transferred onto a 2-dimensional generalized map. We show that transformation rules on polygonal schemata can be achieved in O(1) with generalized maps, which can help optimizing existing algorithms.     

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Consider the nonlinear heat equation $$v_t -\Delta v=|v|^{p-1}v \qquad \qquad \qquad (NLH)$$ in the unit ball of \({\mathbb{R}^2}\) , with Dirichlet boundary condition. Let \({u_{p,\mathcal{K}}}\) be a radially symmetric, sign-changing stationary solution having a fixed number \({\mathcal{K}}\) of nodal regions. We prove that the solution of (NLH) with initial value \({\lambda u_{p,\mathcal{K}}}\) blows up in finite time if |λ ?1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of \({u_{p,\mathcal{K}}}\) and of the linearized operator \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) .     

Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects

We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded in a larger domain, relevant for the global temperature- and magnetic field- distributions. Material inhomogeneities in this larger region lead to transmission relations for the electromagnetic fields and the heat flux on inner boundaries. In the presence of transparent materials, the radiative heat transfer is important and leads to a nonlocal and nonlinear jump relation for the heat flux. We prove the existence of weak solutions, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small.     

Direct solution of an optimal layout problem for isotropic and anisotropic heat conductors on a plane

This paper suggests a procedure of direct construction of the minimal extension of a constrained optimization problem for a heat equation containing controls in the coefficients. For a somewhat simpler problem, this approach has been initiated in Ref. 1.This research has been supported by AFOSR Grant No. 90-0268. The author acknowledges discussions with Dr. Janos Turi.