Abstract
We develop a mathematical theory of flame propagation and analyze the
stability of a flame front.
We consider a premixed, combustible fluid and model the front between
the burnt and unburnt regions as an infinitely thin curve propagating in a
direction normal to itself at a constant speed.
We assume that the specific volume of a fluid particle increases by a fixed
amount when it burns.
Our results show a deep analogy between the equations of flame
propagation and hyperbolic systems of conservation laws.
We introduce the notion of ignition curves and an entropy condition
which enable us to solve the equations of flame propagation in the absence of
fluid motion.
We prove that any initial front asymptotically approaches a circle as it burns,
and that if two fronts start close to each other, they remain so.
As the front moves, it may form cusps, which are the result of colliding
ignition curves and form in the same way that shocks develop in the
solution of hyperbolic systems.
These cusps absorb sections of the flame front, destroying
information about the initial shape of the front: once
a cusp forms, it is impossible to retrieve the initial data by solving
the equations of motion backwards in time.
We use our theory to discuss the difficulties involved in a numerical
approximation to the equations of flame propagation.
Finally, we analyze a numerical technique, developed by Chorin, that does not
rely on a discrete parameterization of the initial front, and use it to
illustrate the results of our theorems.
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