The Coupling of Motion and Conductive Heating of a Gas by Localized Energy Sources

This paper investigates the time evolution of the near-isobaric flow field produced in a gas after the sudden application of a constant heat flux from a localized energy source. The problems of plane, line, and point heat sources are all investigated, with a power law for the temperature dependence of the thermal conductivity, after reduction to a quasi-linear heat equation for the temperature. In the planar and spherical cases, the constant heat flux defines scales for the length and time, which are used to nondimensionalize the problem. Numerical integration is used to provide the evolution of the temperature and velocity, and limiting solutions corresponding to small and large rescaled times are obtained. In the axisymmetric case, due to the absence of characteristic length and time scales, the solution is seen to admit a self-similar description in terms of the nondimensional heat flux. Profiles of temperature and radial velocity are provided for different values of this parameter, and the asymptotic limits of both small and large heating rates are addressed separately. The analysis reveals, in particular, the existence of front solutions when the resulting temperatures become much larger than the initial temperature, as occurs for sufficiently large times for the planar source, for sufficiently small times for the point source, and for sufficiently large heating rates for the line source.