Multiphase flows are generally modeled assuming that the different phases are separated by an interface, that is a surface of zero thickness. Imposing that the condition of local equilibrium is satisfied, all physical properties are allowed to change discontinuously across the interface. Naturally, that results in a free boundary problem, which means that one of the main problems of this approach is to determine the position of the interface. In microdevices, an additional difficulty is encountered, as most lengthscales of the systems are comparable to the real interface thickness; this problem arises also in modeling drop coalescence and break up and moving contact angles. In these cases, it is more reasonable to use a different approach, proposed at the end of the 19th century by Rayleigh and Van der Waals, where interfaces have a non-zero thickness, i.e. they are “diffuse”, so that all quantities, such as density or composition, vary continuously.
The equations of motion are derived, showing how additional stresses, so called Korteweg stresses, arise naturally as reversible body forces, that tends to minimize the free energy of the system. In addition, constitutive relations for the diffusive fluxes are proposed, that insure that the problem is well posed, that is the rate of change of the total energy of the system equals its energy dissipation.
Several case studies will be presented to demonstrate the advantages of the diffuse interface method in modeling multiphase flows in microdevices, as compared to the classical two-phase flow approach. Among the examples that will be presented, here we mention the following problems:
a) mixing, spinodal decomposition and nucleation of macroscopically quiescent regular mixtures;
b) deformation, coalescence and break-up of fluid volumes under shear flows;
c) drop movement through an interface and from a moving nozzle;
d) film ruptures and coarsening in dewetting;
e) heat transfer enhancement due to phase change;
f) chaotic mixing in microfluidics;
g) spontaneous emergence of complex structures during growth far from equilibrium.
The aim of this course is to formulate and apply the diffuse interface model for one-component, two-phase fluids and for liquid binary mixtures, to model multiphase flows in confined geometries.
A considerable emphasis will be devoted to the advanced numerical modeling schemes that have be developed so far, stressing the computational difficulties encountered in implementing the diffuse interface method. In particular, stability problems will be analyzed, showing how they can be overcome.

Coordinator: Roberto Mauri (Università di Pisa, Italy)