Multiphase Flows

Abstract

In this chapter we derive the equations of motion of multiphase fluids. In the classical theory of multiphase flow, each phase is associated with its own conservation equations (of mass, momentum, energy and chemical species), assuming that it is at local equilibrium and separated from the other phases by zero-thickness interfaces, with appropriate boundary conditions. Instead, here we describe the so-called diffuse interface, or phase field, model, assuming that interfaces have a non-zero thickness, i.e. they are “diffuse”, as it is more fundamental than the classical, sharp interface theory and is therefore more suitable to be coupled to all non-equilibrium thermodynamics results. After describing van der Waals’ theory of coexisting phases at equilibrium (Sect. 9.2), in Sect. 9.3 we illustrate the main idea of the diffuse interface model, leading to the definition of generalized chemical potentials, where the non uniformity of the composition field is accounted for. Then, in Sect. 9.4, the equations of motion are derived by applying the principle of minimum action, showing that an additional, so called, Korteweg, reversible force appears in the momentum conservation equation. This force is proportional (with a minus sign) to the gradient of the generalized chemical potential and therefore tends to restore the equilibrium conditions (where chemical potentials are uniform). Finally, in Sect. 9.5, we show that for incompressible and symmetric binary mixtures the governing equations simplify considerably.