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Criticality in reactors under domain or external temperature perturbations. (English) Zbl 0739.35003

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.

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

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