Institute for Mathematics and Its Applications
Felix Otto, University of California-Santa Barbara
We consider the flow of a thin film of a viscous fluid down an inclined plane. Experiments show that the initially horizontal front of the thin film breaks into fingers. The emerging finger pattern has a characteristic wave length. Our goal is to recover the instability with characteristic wave length within the standard model of this flow problem.
The standard model is the lubrication approximation of the quasi stationary Stokes flow within the thin flow domain limited by the free surface, driven by gravity and surface tension. Mathematically speaking, it comes in form of a degenerate parabolic (second and forth order) equation for the height h of the free surface. The goal is to establish the instability of its 1-dimensional traveling wave solution and to show that the wave length of the fastest growing perturbation is finite. The standard proceeding is to analyze the linearization of the equation around the traveling wave solution in the moving frame.
The conventional way to analyze the linearization is to consider the eigenvalue problem related to the differential operator T defining the linearization, as done by Troian et al. But Bertozzi and Brenner pointed out that there is a parameter regime where all eigenvalues of T have negative real part, although instabilities are observed experimentally. Bertozzi and Brenner also observed transient growth in numerical simulations of the linearized equation. This is not unconsistent with the theoretical result since T is not normal with respect to, say, L2.
We propose an analysis of the linearized equation which recovers the instability with characteristic wave number. It is based on the ``energy method'' applied by Spaid and Homsy. They consider the transient decay or increase of the L2--norm of solutions to the linearized equation. This amounts to analyzing the spectrum of the symmetric part SymL2(T) of (T) w. r. t. L2.