Surface-tension-driven Bénard (Marangoni) convection
in liquid layers heated from
below can exhibit a long-wavelength primary instability that differs from
the more
familiar hexagonal instability associated with Bénard. This
long-wavelength instability
is predicted to be significant in microgravity and for thin liquid
layers. The instability
is studied experimentally in terrestrial gravity for silicone oil layers
0.007 to 0.027 cm
thick on a conducting plate. For shallow liquid depths (<.017 cm for
0.102 cm2 s−1 viscosity liquid), the
system evolves to a strongly deformed long-wavelength
state which can take the form of a localized depression (‘dry spot’)
or a localized elevation (‘high spot’), depending on the thickness
and thermal conductivity of the
gas layer above the liquid. For slightly thicker liquid depths
(0.017–0.024 cm),
the formation of a dry spot induces the formation of hexagons. For even
thicker
liquid depths (>0.024 cm), the system forms only the hexagonal convection
cells.
A two-layer nonlinear theory is developed to account properly for the effect
of
deformation on the interface temperature profile. Experimental results
for the
long-wavelength instability are compared to our two-layer theory and to
a one-layer
theory that accounts for the upper gas layer solely with a heat transfer
coefficient.
The two-layer model better describes the onset of instability and also
predicts the
formation of localized elevations, which the one-layer model does not predict.
A
weakly nonlinear analysis shows that the bifurcation is subcritical.
Solving for steady states of the system shows that the subcritical pitchfork
bifurcation curve never turns
over to a stable branch. Numerical simulations also predict a subcritical
instability
and yield long-wavelength states that qualitatively agree with the experiments.
The
observations agree with the onset prediction of the two-layer model, except
for very
thin liquid layers; this deviation from theory may arise from small non-uniformities
in the experiment. Theoretical analysis shows that a small non-uniformity
in heating
produces a large steady-state deformation (seen in the experiment) that
becomes
more pronounced with increasing temperature difference across the liquid.
This
steady-state deformation becomes unstable to the long-wavelength instability
at a
smaller temperature difference than that at which the undeformed state
becomes
unstable in the absence of non-uniformity.
