(353g) Faraday Wave Dynamics of Immiscible Systems in Finite Cells

The motivation of this work is to produce an
immiscible Faraday wave system with minimized sidewall stresses, permitting a
comparison to available theory for parameter spaces where wavenumber selection
is governed by sidewall boundary conditions.  The invsicid model of Benjamin
and Ursell¹ provides substantial insight into the fundamental
physics of the instability, but is insufficient for making predictions in
physical systems, in part due to its offering of instabilities arising from
perfect resonance for infinitesimal forcing amplitudes.  Linear damping has
been incorporated into this model to aid matching with experiment, but this
requires a phenomenological parameter and can't be used to make predictions a
priori.  The viscous model of Kumar and Tuckerman² treats the
viscous effects of the system rigorously, and its predictions have been
validated for conditions where sidewalls do not effect wavenumber selection. 
Prediction of threshold amplitudes where the excited wavelength is of the order
of the lateral cell dimensions is much more challenging due to wetting and
contact non-idealities at the sidewalls.

In
these experiments the sidewall contact line is pinned, but the formation of a
sidewall film of the upper fluid allows for an apparent free motion of the bulk
phases and interface.  Minimization of the upper fluid viscosity shows the
thickness of this film decrease and in turn the no-stress limit is approached. 
Making the stress-free assumption, threshold amplitudes for a cell mode and its
bounding co-dimension 2 points are well predicted by the Kumar and Tuckerman
model, with slight deviation due to residual sidewall effects.  Experiments
near the threshold suggest a forcing frequency-dependent transition from
subcritical to supercritical bifurcations.  In the subcritical parameter space,
unbounded growth and wavebreaking is common, while saturation to standing waves
is more common in the supercritical space.  In the saturation to standing
waves, it is seen that the linear spatial form of the mode is well preserved
for forcing amplitudes near the instability threshold, and sufficient increase
promotes the development of secondary instabilities that serve as higher order
system damping.  This behavior helps explain the differences of what can be
expected in a two liquid system versus a single liquid system, a central
question to this work.

[1]
Benjamin, T. and Ursell, F., Proc. R. Soc. London, Ser. A 225, pp.
505-515, 1954.

[2]
Kumar, K. and Tuckerman, L., J. Fluid Mech., 279, pp.
49-68, 1994.