Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer
- PDF / 891,162 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 59 Downloads / 197 Views
(0123456789().,-volV) (0123456789().,-volV)
ORIGINAL PAPER
Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer Niko Sauer1 Received: 14 May 2020 / Accepted: 7 October 2020 Ó Springer Nature Switzerland AG 2020
Abstract We study a dynamic boundary condition problem in heat transfer which represents the interaction between a conducting solid enclosed by a conducting shell. Both the solid and the shell are thermally inhomogeneous and anisotropic. Interaction is modelled by considering the solid as a source of thermal energy to the shell. A constitutive equation proposed by Carslaw and Jaeger establishes a relation between temperature in the shell and the boundary value of temperature in the solid. This gives rise to a dynamic boundary condition problem that has not been studied in the recent literature. The system of equations so obtained is presented as an implicit evolution equation which involves a pair of unbounded linear operators that map between two different spaces. We extend the operators to a jointly closed pair for which the implicit equation makes sense. The solution of the initial value problem is constructed by means of a holomorphic family of solution operators. The class of admissible initial states is surprisingly large. Keywords Dynamic boundary condition Heat transfer Carslaw-Jaeger relation Mathematics Subject Classification 34G10 35K15 58J35
1 Introduction In the 1947-edition of their scholarly book, Carslaw and Jaeger laid down a fundamental constitutive relation for thermal contact between heat-conducting materials. It states that at a point of contact between two bodies [the normal component of ] thermal flux is proportional to the difference in temperature at the given point and directed toward the lower temperature. This constitutive equation introduces the notion of perfect/imperfect contact by means of a function that is zero at points of perfect contact and positive when contact is imperfect. This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira. & Niko Sauer [email protected] 1
Centre for the Advancement of Scholarship, University of Pretoria, Pretoria, South Africa SN Partial Differential Equations and Applications
48 Page 2 of 20
SN Partial Differ. Equ. Appl. (2020)1:48
This paper is devoted to the study of heat transfer in a solid, represented by a bounded open set X R3 , enclosed by a thin shell modelled as the boundary C ¼ oX. It is assumed that the shell internally conducts thermal energy in a tangential direction and that, according to the Carslaw-Jaeger relation, contact is everywhere imperfect. If u(x, t) denotes the temperature at x 2 X and Uðx0 ; tÞ at x0 2 C at time t [ 0, the following equations arise after scaling to dimensionless variables: ut þ Lu ¼ 0 in X;
ð1Þ
Ut þ KU þ cL u ¼ 0 in C:
ð2Þ
Here the the operators L and K are defined by the differential expressions Lu ¼ r ½aðxÞru and KU ¼ rS ½bðx0 ÞrS U with a and b suitable symmetric matrix functions. The operator L represents
Data Loading...
