Nondiffusive Hot Spot in a Confined, Narrow Domain

The spatial structure and temporal evolution of a nondiffusive hot spot is examined in a confined, planar enclosure. The situation is modelled by the reactive Euler equations and single-step Arrhenius kinetics with a large (scaled) activation temperature 1/ . The initial state of the medium is taken to be one of uniform pressure but with a prescribed thermal gradient of order across the domain. The induction equations, which govern order disturbances to the initial state, are analyzed in the limit of small α, where α measures the width of the domain relative to the acoustic length based on a characteristic reaction time at the initial state. The solution displays a sequence of finite-time singularities as the hot spot grows in strength and shrinks in size in a spatially homogeneous pressure environment. Ultimately, in a region of exponentially small size, significant pressure gradients appear as acoustic and reaction times become comparable and the hot-spot structure is found to obey the full, but previously analyzed, Clarke equations.