Hot Spots in the Weak Detonation Problem and Special Relativity

Abstract

Problem statement: The initiation of a detonation in an explosive gaseous mixture in the high activation energy regime, in three space dimensions, typically leads to the formation of a singularity at one point, the “hot spot”. It would be suitable to have a description of the physical quantities in a full neighborhood of the hot spot. Results of this paper: (1) To achieve this, it is necessary to replace the blow-up time, or time when the hot spot first occurs, by the blow-up surface in four dimensions, which is the set of all hot spots for a class of observers related to one another by a Lorentz transformation. (2) A local general solution of the nonlinear system of PDE modeling fluid flow and chemistry, with a given blow-up surface, is obtained by the method of Fuchsian reduction. Advantages of this solution: (i) Earlier approximate solutions are contained in it, but the domain of validity of the present solution is larger; (ii) it provides a signature for this type of ignition mechanism; (iii) quantities that remain bounded at the hot spot may be determined, so that, in principle, this model may be tested against measurements; (iv) solutions with any number of hot spots may be constructed. The impact on numerical computation is also discussed.

Keywords

• weak detonation
• high activation regime
• nonlinear PDEs
• Fuchsian reduction analysis
• Lorentz transformation
• blow-up