  The morphogenesis of cells and tissues involves an interplay between chemical
signals and active forces on their surrounding surface layers. The complex
interaction of hydrodynamics and material flows on such active surfaces leads
to pattern formation and shape dynamics which can involve topological
transitions, for example during cell division. To better understand such
processes requires novel numerical tools. Here, we present a phase-field model
for an active deformable surface interacting with the surrounding fluids. The
model couples hydrodynamics in the bulk to viscous flow along the diffuse
surface, driven by active contraction of a surface species. As a new feature in
phase-field modeling, we include the viscosity of a diffuse interface and
stabilize the interface profile in the Stokes-Cahn-Hilliard equation by an
auxiliary advection velocity, which is constant normal to the interface. The
method is numerically validated with previous results based on linear stability
analysis. Further, we highlight some distinct features of the new method, like
the avoidance of re-meshing and the inclusion of contact mechanics, as we
simulate the self-organized polarization and migration of a cell through a
narrow channel. Finally, we study the formation of a contractile ring on the
surface and illustrate the capability of the method to resolve topological
transitions by a first simulation of a full cell division.
