The `hump' effect in solid propellant combustion

Abstract

An eikonal equation modelling the propagation of combustion fronts in striated media is studied via a level set formulation. Travelling fronts are obtained, and their speeds turn out to be monotone with respect to the angle of the striations. An effective equation for thin meniscus striations is derived from a homogenization process, thus explaining the so-called 'hump' effect. Finally, the time-asymptotic stabilization of unsteady fronts propagating in straight striations is proved.

Large time behavior of fronts governed by eikonal equations

“…Proof of Theorem 4.1. It is a variant of [25]. Following [25], one shows easily that u t ∞ and Du ∞ are uniformly bounded and by a comparison result we also have…”

“…The H ε (p) can also be viewed as the velocities of travelling waves and it is shown in [25] that these velocities are given by the following formulas where α denotes the slope between the y-axis and the planar front, i.e. α = p 2 /p 1 :…”

“…The main underlying motivation is a model of solid combustion in heterogeneous media where the flame front is assumed to propagate in R N with a periodic, space dependent normal velocity. We refer to Namah and Roquejoffre [25] and reference therein for a more detailed presentation of the model. This type of models was proposed by Landau in the 40's, in particular a simpler example when R is a constant: even in this case, it is known that fronts which are smooth for all time do not exist in general, and in Barles [3], a weak formulation to study the simple Landau model was proposed; it was based on the idea that the moving front can be identified as the 0-level set of the unique viscosity solution of an eikonal equation.…”

“…In two space dimensions, and when the function R is constant along parallel lines, the problem is treated in [25] where two results are obtained.…”

“…In Section 2, we provide the general approach to the asymptotic behaviour of the front. Section 3 is devoted to explicit computations, and in particular to the connections with the approach in [25]. Section 4 is devoted to the case when there exist lines of maximal value for R, and Section 5 treats the 2-dimensional case.…”