  We present theoretical study and numerical simulation of Marangoni convection
within ellipsoidal isotropic droplets embedded in free standing smectic films
(FSSF). The thermocapillary flows are analyzed for both isotropic droplets
spontaneously formed in FSSF overheated above the bulk smectic-isotropic
transition, and oil lenses deposited on the surface of the smectic film. The
realistic model, for which the upper drop interface is free from the smectic
layers, while at the lower drop surface the smectic layering still persists is
considered in detail. For isotropic droplets and oil lenses this leads
effectively to a sticking of fluid motion at the border with a smectic shell.
The above mentioned asymmetric configuration is realized experimentally when
the temperature of the upper side of the film is higher than at the lower one.
The full set of stationary solutions for Stokes stream functions describing the
Marangoni convection flows within the ellipsoidal drops were derived
analytically. The temperature distribution in the ellipsoidal drop and the
surrounding air was determined in the frames of the perturbation theory. As a
result the analytical solutions for the stationary thermocapillary convection
were derived for different droplet ellipticity ratios and the heat conductivity
of the liquid crystal and air. In parallel, the numerical hydrodynamic
calculations of the thermocapillary motion in the drops were performed. Both
the analytical and numerical simulations predict the axially-symmetric
circulatory convection motion determined by the Marangoni effect at the droplet
free surface. Due to a curvature of the drop interface a temperature gradient
along its free surface always persists. Thus, the thermocapillary convection
within the ellipsoidal droplets in overheated FSSF is possible for the
arbitrarily small Marangoni numbers.
