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Soft modulated phases have been shown to undergo complex morphologicaltransitions, in which layer remodeling induced by mean and Gaussian curvaturesplays a major role. This is the case in smectic films under thermal treatment,where focal conics can be reshaped into conical pyramids and concentric ringstructures. We build on earlier research on a smectic-isotropic, two phaseconfiguration in which diffusive evolution of the interface was driven bycurvature, while mass transport was neglected. Here, we explicitly considerevaporation-condensation processes in a smectic phase with mass transportthrough a coexisting isotropic fluid phase, as well as the hydrodynamicstresses at the interface and the resulting flows. By employing theColeman-Noll procedure, we derive a phase-field model that accounts for avarying density field coupled to smectic layering of the order parameter. Theresulting equations govern the evolution of an interface between a modulatedphase and an isotropic fluid phase with distinct densities, and they capturecompressibility effects in the interfacial region and topological transitions.We first verify a numerical implementation of the governing equations byexamining the dispersion relation for interfacial transverse modes. The inversedecay rate is shown to scale as $Q^{2}$ ($Q$ is the wavenumber of theperturbation) due to hydrodynamic effects, instead of the $Q^{4}$ expected fordiffusive decay. Then, by integrating the equations forward in time, weinvestigate fluid flow on distorted layers and focal conics, and show howinterfacial stresses and density contrast significantly determine the structureof the flow and the evolution of the configuration.

Model of the dynamics of an interface between a smectic phase and an isotropic phase of different density