  In the present work, a general formulation is proposed to implement the
contact angle boundary conditions for the second-order Phase-Field models,
which is applicable to $N$-phase $(N \geqslant 2)$ moving contact line
problems. To remedy the issue of mass change due to the contact angle boundary
condition, a Lagrange multiplier is added to the original second-order
Phase-Field models, which is determined by the consistent and conservative
volume distribution algorithm so that the summation of the order parameters and
the \textit{consistency of reduction} are not influenced. To physically couple
the proposed formulation to the hydrodynamics, especially for
large-density-ratio problems, the consistent formulation is employed. The
reduction-consistent conservative Allen-Cahn models are chosen as examples to
illustrate the application of the proposed formulation. The numerical scheme
that preserves the consistency and conservation of the proposed formulation is
employed to demonstrate its effectiveness. Results produced by the proposed
formulation are in good agreement with the exact and/or asymptotic solutions.
The proposed method captures complex dynamics of moving contact line problems
having large density ratios.
