  In this article, we propose a novel scalar-transport model for the simulation
of scalar quantities in two-phase flows with a phase-field method
(diffuse-interface method). In a two-phase flow, the scalar quantities
typically have disparate properties in two phases, which results in effective
confinement of the scalar quantities in one of the phases, in the time scales
of interest. This confinement of the scalars lead to the formation of sharp
gradients of the scalar concentration values at the interface, presenting a
serious challenge for its numerical simulations.
  To overcome this challenge, we propose a model for the transport of scalars.
The model is discretized using a central-difference scheme, which leads to a
non-dissipative implementation that is crucial for the simulation of turbulent
flows. Furthermore, the provable strengths of the proposed model are: (a) the
model maintains the positivity property of the scalar concentration field, a
physical realizability requirement for the simulation of scalars, when the
proposed criterion is satisfied, (b) the proposed model is such that the
transport of the scalar concentration field is consistent with the transport of
the volume fraction field, which results in the enforcement of the effective
zero-flux boundary condition for the scalar at the interface; and therefore,
prevents the artificial numerical diffusion of the scalar across the interface.
  Finally, we present numerical simulations using the proposed model in a wide
range of two-phase flow regimes, spanning laminar to turbulent flows; and
assess: the accuracy and robustness of the model, the validity of the
positivity property of the scalar concentration field, and the enforcement of
the zero-flux boundary condition for the scalar at the interface.
