Parametric sensitivity and self-similarity in thermal explosion theory

We study the relations between thermal runaway (also called parametric sensitivity) and self-similarity, an interesting property of the sensitivity functions that has been numerically verified in many explosion and combustion systems. Both concepts are sensitivity related but independent of the particular parameter being perturbed. This independence is emphasized by proposing a new generalized condition for parametric sensitivity. Critically is defined as the point in the parameter space where the nominal trajectory exhibits maximum sensitivity to arbitrary, unstructured perturbations applied at the maximum temperature. The condition for critically reduces to the analysis of the eigenvalues of the Jacobian matrix. In addition to its conceptual generality, the new condition shows that in certain cases there exists no critical Semenov number. The sensitivity functions are shown to satisfy self-similarity relations if and only if the system exhibits critical or supercritical behavior. The onset of self-similarity is explained in terms of two properties of explosion systems, both related to parametric sensitivity. First, the temperature is a dominant variable, and any perturbation in the system affects the conversion mainly through the changes induced in the temperature. This strong coupling of the variables is shown by decomposing the sensitivity functions into direct and indirect terms. Second, the sensitivity equations are pseudo-homogeneous in a characteristic time window, in which the system becomes relatively insensitive to parameter perturbations applied within the same interval. The two properties are shown to imply self-similarity of the sensitivity functions. Relations to earlier parametric sensitivity and self-similarity conditions are discussed.