  A coupled phase-field and hydrodynamic model is introduced to describe a
two-phase, weakly compressible smectic (layered phase) in contact with an
isotropic fluid of different density. A non-conserved smectic order parameter
is coupled to a conserved mass density in order to accommodate non-solenoidal
flows near the smectic-isotropic boundary arising from density contrast between
the two phases. The model aims to describe morphological transitions in smectic
thin films under heat treatment, in which arrays of focal conic defects lead to
conical pyramids and concentric rings through curvature dependent evaporation
of smectic layers. The model leads to an extended thermodynamic relation at a
curved surface that includes its Gaussian curvature, non-classical stresses at
the boundary and flows arising from density gradients. The temporal evolution
given by the model conserves the overall mass of the liquid crystal while still
allowing for the modulated smectic structure to grow or shrink. A numerical
solution of the governing equations reveals that pyramidal domains are sculpted
at the center of focal conics upon a temperature increase, which display
tangential flows at their surface. Other cases investigated include the
possible coalescence of two cylindrical stacks of smectic layers, formation of
droplets, and the interactions between focal conic domains through flow.
