A derivation is given of the amplitude equations governing pattern
formation in
surface tension gradient-driven Bénard–Marangoni
convection. The amplitude equations
are obtained from the continuity, the Navier–Stokes and the Fourier
equations in the
Boussinesq approximation neglecting surface deformation and buoyancy. The
system
is a shallow liquid layer heated from below, confined below by a rigid
plane and
above with a free surface whose surface tension linearly depends on temperature.
The
amplitude equations of the convective modes are equations of the Ginzburg–Landau
type with resonant advective non-variational terms. Generally, and in agreement
with
experiment, above threshold solutions of the equations correspond to an
hexagonal
convective structure in which the fluid rises in the centre of the cells.
We also
analytically study the dynamics of pattern formation leading not only to
hexagons
but also to squares or rolls depending on the various dimensionless parameters
like
Prandtl number, and the Marangoni and Biot numbers at the boundaries. We
show
that a transition from an hexagonal structure to a square pattern is possible.
We
also determine conditions for alternating, oscillatory transition between
hexagons and
rolls. Moreover, we also show that as the system of these amplitude equations
is
non-variational the asymptotic behaviour (t→∞)
may not correspond to a steady
convective pattern. Finally, we have determined the Eckhaus band for hexagonal
patterns
and we show that the non-variational terms in the amplitude equations enlarge
this band of allowable modes. The analytical results have been checked
by numerical
integration of the amplitude equations in a square container. Like in experiments,
numerics shows the emergence of different hexagons, squares and rolls according
to
values given to the parameters of the system.
Patterns and Their Large-Scale Distortions in Marangoni Convection with Insoluble Surfactant