Thermal stabilities of slabs and hollow spheres with internal heat generation and variable thermal conductivity II. Thermal stability of hollow spheres
type: Item
instance of: scholarly article
title: Thermal stabilities of slabs and hollow spheres with internal heat generation and variable thermal conductivity II. Thermal stability of hollow spheres
zbMATH Open document ID: 0619.73003
DOI: 10.1016/0016-0032(87)90039-1
author: B. George
published in: Journal of the Franklin Institute
publication date: 1987-01-01T00:00:00Z
review text: [For part I, see the article reviewed above (Zbl 0619.73002).] This paper deals with the investigation of the thermal stability of hollow spheres with internal heat generation. The variation of thermal conductivity is inversely as the fourth power of the radius. Five combinations of the boundary conditions are analyzed. In all cases, the critical solutions and parameters are determined. All five chosen critical solutions are listed in a table. There exist many critical solutions for each combination of the boundary conditions. The best critical parameter is chosen by exercising common sense, correspondence principle and naturalness considerations. In this paper, the investigation of the thermal stability problem of hollow spheres leads to a second order nonlinear differential equation: \[ d^ 2\beta \theta /d\xi^ 2+(\Gamma /9)e^{\beta \theta}=0. \] Here \(\beta\) and \(\Gamma\) are known constants. The explanation of the solution of the above mentioned equation can be found in an earlier paper of the author, ibid. 320, 303-322 (1955; Zbl 0581.73119).
Mathematics Subject Classification ID: 74A15
Mathematics Subject Classification ID: 80A20
zbMATH DE Number: 4005681
zbMATH Keywords: thermal stability
zbMATH Keywords: hollow spheres
zbMATH Keywords: internal heat generation
zbMATH Keywords: thermal conductivity
zbMATH Keywords: critical solutions
zbMATH Keywords: common sense
zbMATH Keywords: correspondence principle
zbMATH Keywords: naturalness considerations
zbMATH Keywords: second order nonlinear differential equation
reviewed by: István Ecsedi
MaRDI profile type: Publication
full work available at URL: 0016-0032(87)90039-1
OpenAlex ID: W2068899077
cites work: Thermal stability of solid and hollow squire cylinders
cites work: The role of boundary conditions on the thermal stability of a Landau slab
description: scientific article; zbMATH DE number 4005681