Blowup Dynamics in a Cylindrical Tube

Abstract

We carry out a numerical investigation of the nonlinear heat equation with a volume heat source that describes combustion with a nonzero background temperature. The problem is considered in cylindrical coordinates assuming radial symmetry. Combustion processes have been studied in both isotropic and anisotropic media. Various types of combustion processes have been identified in the form of thermal dissipative structure, in particular, when combustion is localized in one direction and the combustion region expands in the other direction. Structures form on a nonzero background under certain supercritical perturbations of the background temperature, while subcritical initial disturbances decay and relax to background temperatures. The observed blowups depend on the relations among the parameters describing the power dependence on temperature, thermal conductivity, and the highest order expansion term in the volume heat source.