  Soft modulated phases have been shown to undergo complex morphological
transitions, in which layer remodeling induced by mean and Gaussian curvatures
plays a major role. This is the case in smectic films under thermal treatment,
where focal conics can be reshaped into conical pyramids and concentric ring
structures. We build on earlier research on a smectic-isotropic, two phase
configuration in which diffusive evolution of the interface was driven by
curvature, while mass transport was neglected. Here, we explicitly consider
evaporation-condensation processes in a smectic phase with mass transport
through a coexisting isotropic fluid phase, as well as the hydrodynamic
stresses at the interface and the resulting flows. By employing the
Coleman-Noll procedure, we derive a phase-field model that accounts for a
varying density field coupled to smectic layering of the order parameter. The
resulting equations govern the evolution of an interface between a modulated
phase and an isotropic fluid phase with distinct densities, and they capture
compressibility effects in the interfacial region and topological transitions.
We first verify a numerical implementation of the governing equations by
examining the dispersion relation for interfacial transverse modes. The inverse
decay rate is shown to scale as $Q^{2}$ ($Q$ is the wavenumber of the
perturbation) due to hydrodynamic effects, instead of the $Q^{4}$ expected for
diffusive decay. Then, by integrating the equations forward in time, we
investigate fluid flow on distorted layers and focal conics, and show how
interfacial stresses and density contrast significantly determine the structure
of the flow and the evolution of the configuration.
