On Shapes and Forms: Population balance dynamics of corrugated stirred fronts

We introduce a unified framework to discuss the emergence of corrugations on material interfaces transported by random media. Relating the shape of these interfaces to the stirring field giving birth to it, we formalize a population balance dynamics for the $r-$elements (segments of length $r$) needed to cover the interface contour in the course of its deformation. As long as corrugations grow kinematically, shapes change continuously, their fractal dimension $d_f(r,t)$ is a non-monotonous function of the scale $r$, and increases in time $t$ with no bounds. Interface creation and destruction balance however in self-propagating fronts like flames, and in fronts smearing by molecular diffusion, through a mixing induced overlap mechanism, leading to a stationary shape. These findings, which help reexamining old observations in a new perspective, also reconcile kinetics with geometry.