Title: Variational Approach in Studying the Mixture of the Fluids:
Transport and Induced Elastic Stress

Abstract:  From the energetic point of view,
most complicated hydrodynamical and rheological properties
of the non-Newtonian complex fluids arise from the
coupling and competing between the kinetic energy and different
types of internal "elastic" energy. The examples include liquid crystal
materials where the alignment of the molecule director contributes
to the elastic energy; the Magneto-hydrodynamics (MHD) and Electro-
hydrodynamics (EHD) where the magnetical and electrical fields
are the source of the elasticity; different polymerical fluids;
viscoelastical fluids; mixtures of different materials (where the
the elasticity is due to the heterogeneity) and fluids involving
different surfactant materials. The coupling between the transport
of these elastic effects by the flow field and the induced elastic
stresses in the momentum equations assure the Hamiltonian (or dissipative)
nature of the whole system. On the other hand, such coupling also
reflect the influence of the micro-structure of the material to the
hydrodynamical properties of the fluid and the vice versa.

The hydrodynamic theory of mixtures is a good example for
these theories. In this talk, I will discuss a energetic variational
approach involving phase field methods to model the
dynamics of mixtures with free interfaces. The method
can be generalized to the cases of more complicated cases, such as
Marangoni effects, surface viscosity or more general surfactant
situations. When the mixture involves viscoelastic materials, we 
employed a formulation of the system in Eulerian coordinates.
Some analytical,  numerical results as well as open problems 
will be presented.