Flame eaters are not stirlings.
A flame eater is an internal combustion engine.
A stirling is an external combustion engine.
The difference being that;
A flame eater is an open system, the head into which cold air is drawn is heated, the cold air expands due to the heat and the force provided by the expansion moves the piston. One can take a model aero engine, heat the head with sunlight thru a magnifying glass and it will run after a helping hand to start it up... this is the equivalent to a flame eater.
A stirling is a sealed system, the working fluid (or gas) is contained within the engine. A power piston and displacer are moved due to the expansion and (mostly) contraction of the fluid within the system by to heat applied externally to one part of the engine.
Normally both internal and external combustion engines require an outside energy source to get them going... i.e. to kick over a flywheel... & once running they will continue to run until the heat source is removed.
paul