This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat,...

An entire solution of the Allen-Cahn equation $\Delta u=f\left(u\right)$, where $f$ is an odd function and has exactly three zeros at $\pm 1$ and $0$, e.g. $f\left(u\right)=u(u^2-1)$, is called a $2k$ end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks (up to a multiplication by $-1$) like the one dimensional, odd, heteroclinic solution $H$, of $H^{\text{'}\text{'}}=f\left(H\right)$. In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space of such solutions....

The hot-wire method, based on the recording of the temperature development in time in a testing sample, supplied by a probe with its own thermal source, is useful to evaluate the thermal conductivity of materials under extremal loads, in particular in refractory brickworks. The formulae in the technical standards come from the analytical solution of the non-stationary equation of heat conduction in cylindric (finally only polar) coordinates for a simplified formulation of boundary conditions, neglecting...

The aim of this paper is to prove some properties of the solution to the Cauchy problem for the system of partial differential equations describing thermoelasticity of nonsimple materials proposed by D. Iesan. Explicit formulas for the Fourier transform and some estimates in Sobolev spaces for the solution of the Cauchy problem are proved.

This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient $\kappa \left(\lambda \right)$ to be piecewise constant and null for small values of the wavelength $\lambda$ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important...

In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model supplemented with the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce...

In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (shortly SBL) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature (${\lambda }^0\left(\vartheta \right),K^A\left(\vartheta \right),A=1,2,3$). The requirements of the II law of thermodynamics...