  A consistent and conservative Phase-Field method, including both the model
and scheme, is developed for multiphase flows with an arbitrary number of
immiscible and incompressible fluid phases. The consistency of mass
conservation and the consistency of mass and momentum transport are implemented
to address the issue of physically coupling the Phase-Field equation, which
locates different phases, to the hydrodynamics. These two consistency
conditions provide the ``optimal'' coupling because (i) the new momentum
equation resulting from them is Galilean invariant and implies the kinetic
energy conservation, regardless of the details of the Phase-Field equation, and
(ii) failures of satisfying the second law of thermodynamics or the consistency
of reduction of the multiphase flow model only result from the same failures of
the Phase-Field equation but are not due to the new momentum equation. Physical
interpretation of the consistency conditions and their formulations are first
provided, and general formulations that are obtained from the consistency
conditions and independent of the interpretation of the velocity are
summarized. Several novel techniques are developed to inherit the physical
properties of the multiphase flows after discretization, including the
gradient-based phase selection procedure, the momentum conservative method for
the surface force, and the general theorems to preserve the consistency
conditions on the discrete level. Equipped with those novel techniques, a
consistent and conservative scheme for the present multiphase flow model is
developed and analyzed. Numerical applications demonstrate that the present
model and scheme are robust and effective in studying complicated multiphase
dynamics, especially for those with large-density ratios.
