Many multiphysics problems involve time-dependent free boundaries (or
interfaces) where the interface motion is controlled by interactions
among microscopic forces (e.g. surface tension) and external
macroscopic driving forces (e.g. far-field temperature or heat flux
for solidification problems, external flow for biomembranes). During
the evolution, the moving interface experiences instability due to the
imbalance of forces. In the first part of this talk, we consider the
Hele-Shaw problem (Saffman-Taylor instability) and crystal growth
problem (Mullins-Sekerka instability). We will show how these
instabilities can be suppressed using appropriate boundary
conditions. In particular, we demonstrate the existence of
self-similar evolving, symmetric universal patterns. In the second
part of this talk, we consider a bio-membrane problem. We will discuss
the model and show how surface tension can be used to control the
phase decomposition along the surface of the membrane. In particular,
we demonstrate how flow, morphology and phase decomposition are
coupled during the dynamics.