A Fully Implicit Model of Compressible Capillary Flows

Small-scale two-phase flows are subject to intense capillary accelerations that must be treated with care in order to avoid artifacts often associated with the numerical methodologies used, such as excessive fragmentation of structures. This analysis proposes a formulation of capillary actions for compressible viscous two-phase flows within the framework of discrete mechanics, where the concept of mass is abandoned in favor of a law of motion that describes the conservation of accelerations, one related to inertia and the other to external actions. With the introduction of the capillary term, the sum of a capillary potential gradient and the dual curl of a vector potential is consistent with the other terms of the law of motion, a formal Helmholtz–Hodge decomposition. This fully compressible formulation reproduces the capillary waves generated by the source terms and the contact and shock discontinuities in the two immiscible fluids. This methodology completely eliminates parasitic currents due mainly to the presence of residual curl in the capillary source terms. Several classic examples demonstrate the validity of this approach.