Laplace's law and the interfacial momentum source in two-phase models.

A two-phase flow model with liquid-solid transformation [M. Hütter, Phys. Rev. E 64, 011209 (2001)] is discussed, focusing on two elements: (1) the driving force for nucleation and growth and (2) the contribution of phase interfaces to the momentum balance. According to the model, nucleation and growth are partly driven by deviations from the equilibrium pressure difference between the phases, obtained as the surface tension times the ratio of the rates of change of two structural variables: the interfacial area per unit volume and the solid volume fraction. This is shown to be the proper extension of Laplace's law to nondilute conditions. Contrary to the classical result, the equilibrium pressure difference changes sign at a volume fraction around 50% because the amount of interfacial area lost due to impingement starts to outweigh the amount gained by growth. Hütter did not notice this and consequently misinterpreted a source term in his evolution equation for the momentum density. This term involves the surface tension times the interfacial area per unit volume, which is always nonnegative and hence not related to Laplace's law, as assumed in earlier two-phase models [M. Ishii, (Eyrolles, Paris, 1975); J. Ni and C. Beckermann, Metall. Trans. B 22, 349 (1991)]. An alternative derivation of the interfacial momentum source is presented here, which shows that Hütter's result correctly expresses the balance of forces on a representative volume element and should have been presented as a correction, rather than a corroboration, of the previous works mentioned.