Van de Velde, Eric F.; Ward, Michael J. Criticality in reactors under domain or external temperature perturbations. (English) Zbl 0739.35003 Proc. R. Soc. Lond., Ser. A 434, No. 1891, 341-367 (1991). Summary: The conditions for the onset of thermal runaway in reactors with small non-uniformities is investigated. The reaction is modelled by an Arrhenius heat generation term with a finite activation energy and the dimensionless temperature, \(u_ 0\), is taken to satisfy a nonlinear equation of the form \[ \Delta u_ 0+\lambda_ 0F(u_ 0)=0,\quad x\in D;\quad \partial_ vu_ 0+bu_ 0=0,\quad x\in\partial D. \] We investigate three classes of perturbations of this problem. First, we treat a small temperature variation maintained on the boundary of the domain. Secondly, we consider a small distortion of the boundary of a circular cylindrical domain, and thirdly, we analyse the effect of a small hole in the domain. In each case we derive asymptotic expansions for the critical Frank-Kamenetskij parameter, \(\lambda_ c(\varepsilon)\), where \(\varepsilon\) is a measure of the size of the perturbation.A numerical scheme is then used to determine numerical values for the coefficients in the asymptotic expansion of \(\lambda_ c\). Finally, some of the asymptotic results are compared with corresponding numerical results obtained from a full numerical solution of the perturbed problem. Cited in 2 Documents MSC: 35B20 Perturbations in context of PDEs 35C20 Asymptotic expansions of solutions to PDEs 35K57 Reaction-diffusion equations 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations Keywords:thermal runaway; small non-uniformities; Arrhenius heat generation; finite activation energy; small temperature variation maintained on the boundary; small distortion of the boundary of a circular cylindrical domain; small hole in the domain; asymptotic expansions for the critical Frank-Kamenetskij parameter Software:COLSYS PDFBibTeX XMLCite \textit{E. F. Van de Velde} and \textit{M. J. Ward}, Proc. R. Soc. Lond., Ser. A 434, No. 1891, 341--367 (1991; Zbl 0739.35003) Full Text: DOI