Thermal stabilities of slabs and hollow spheres with internal heat generation and variable thermal conductivity II. Thermal stability of hollow spheres
[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).
second order nonlinear differential equation
scientific article; zbMATH DE number 4005681