A multi-phase SPH method for macroscopic and mesoscopic flows

A multi-phase smoothed particle hydrodynamics (SPH) method for both macroscopic and mesoscopic flows is proposed. Since the particle-averaged spatial derivative approximations are derived from a particle smoothing function in which the neighboring particles only contribute to the specific volume, while maintaining mass conservation, the new method handles density discontinuities across phase interfaces naturally. Accordingly, several aspects of multi-phase interactions are addressed. First, the newly formulated viscous terms allow for a discontinuous viscosity and ensure continuity of velocity and shear stress across the phase interface. Based on this formulation thermal fluctuations are introduced in a straightforward way. Second, a new simple algorithm capable for three or more immiscible phases is developed. Mesocopic interface slippage is included based on the apparent slip assumption which ensures continuity at the phase interface. To show the validity of the present method numerical examples on capillary waves, three-phase interactions, drop deformation in a shear flow, and mesoscopic channel flows are considered.