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Talk Abstract

Fingering instability of fluid flow down an inclined plane: an intrinsic linear stability analysis

Fingering instability of fluid flow down an inclined plane: an intrinsic linear stability analysis

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, *L ^{2}*.

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 *L ^{2}*--norm of
solutions to the linearized equation. This amounts to analyzing
the spectrum of the symmetric part