  We study theoretically internal flows in a small oblate droplet suspended on
the circular frame. Marangoni convection arises due to a vertical temperature
gradient across the drop and is driven by the surface tension variations at the
free drop interface. Using the analytical basis for the solutions of Stokes
equation in coordinates of oblate spheroid we have derived the linearly
independent stationary solutions for Marangoni convection in terms of Stokes
stream functions. The numerical simulations of the thermocapillary motion in
the drops are used to study the onset of the stationary regime. Both analytical
and numerical calculations predict the axially-symmetric circulatory convection
motion in the drop, the dynamics of which is determined by the magnitude of the
temperature gradient across the drop. The analytical solutions for the critical
temperature distribution and velocity fields are obtained for the large
temperature gradients across the oblate drop. These solutions reveal the
lateral separation of the critical and stationary motions within the drops. The
critical vortices are localized near the central part of a drop, while the
intensive stationary flow is located closer to its butt end. A crossover to the
limit of the plane film is studied within the formalism of the stream functions
by reducing the droplet ellipticity ratio to zero value. The initial stationary
regime for the strongly oblate drops becomes unstable relative to the
many-vortex perturbations in analogy with the plane fluid films with free
boundaries.
