The paper presents a theory of nonlinear evolution and
secondary instabilities in
Marangoni (surface-tension-driven) convection in a two-layer
liquid–gas system with
a deformable interface, heated from below. The theory takes into account the motion
and convective heat transfer both in the liquid and in the gas layers. A system of
nonlinear evolution equations is derived that describes a general case of slow
long-scale evolution of a short-scale hexagonal Marangoni convection pattern near the
onset of convection, coupled with a long-scale deformational Marangoni instability.
Two cases are considered: (i) when interfacial deformations are negligible; and (ii)
when they lead to a specific secondary instability of the hexagonal convection.
In case (i), the extent of the subcritical region of the hexagonal Marangoni
convection, the type of the hexagonal convection cells, selection of
convection patterns – hexagons, rolls and squares – and
transitions between them are studied, and the
effect of convection in the gas phase is also investigated.
Theoretical predictions are compared with experimental observations.
In case (ii), the interaction between the short-scale hexagonal
convection and the
long-scale deformational instability, when both modes of Marangoni convection are
excited, is studied. It is shown that the short-scale convection
suppresses the deformational instability. The latter can appear as a
secondary long-scale instability of
the short-scale hexagonal convection pattern. This secondary instability is shown to
be either monotonic or oscillatory, the latter leading to the excitation of
deformational waves, propagating along the short-scale hexagonal
convection pattern and modulating its amplitude.