Analytical criteria for nonlinear instability of thermal structures

Abstract

Analytical criteria for supercritical and asymptotic stability and for subcritical and superexponential instability are obtained for slab-like, spherical, and cylindrical thermal structures. It is assumed that both, the thermal conductivity κ(T) and the gain/loss function Q(T), can be written as continuous functions of the temperature and they have continuous derivatives. Conditions on κ and Q under which the symmetry of the structure determines the kind of instability (or stability) are obtained. Additionally, it is found that the response of the structure not only depends on the amplitude of the disturbance, but also on whether the disturbance increases or decreases the initial steady temperature. In particular, the threshold value for the amplitude of the disturbances beyond which a linearly stable configuration destabilizes, and explicit conditions for catastrophic heating or cooling are obtained. Applications to structures constituted by atomic and molecular gas are outlined.