Some gas flows which obey Charles’ Law

Abstract

The effects of substantial temperature and hence density changes on a low-speed ‘incompressible’ flow can be modelled by adopting Charles’ Law as one of the equations of state. It is found that planar radial inflows or outflows constitute a group of solutions which are self-similar for arbitrary temperature variations. When the temperature is written as a separable function of radius and polar angle, ordinary differential equations result. Permissible solutions include some with discontinuities in the temperature gradient across a radial line (streamline); this is a rough model of a diffusion flame and it is used to illustrate some features of a variable-density flow in a channel with radial walls in the presence of such a ‘flame’.

Exact analytical solutions are given for the situation in which temperature increases linearly with radius; no boundary layers appear for either outflow or inflow. Approximate analytical solutions are presented for the case of a relatively rapid inflow with temperature independent of the radius; a velocity boundary layer exists at the walls and in the neigh bourhood of the ‘flame’, although the latter is of small velocity amplitude.