  We present a discrete exterior calculus (DEC) based discretization scheme for
incompressible two-phase flows. Our physically-compatible exterior calculus
discretization of single phase flow is extended to simulate immiscible
two-phase flows with discontinuous changes in fluid properties such as density
and viscosity across the interface. The two-phase incompressible Navier-Stokes
equations and conservative phase field equation for interface capturing are
first transformed into the exterior calculus framework. The discrete counter
part of these smooth equations is obtained by substituting with discrete
differential forms and discrete exterior operators. We prove the boundedness of
the method for the first order Euler forward and predictor-corrector time
integration schemes in the DEC framework. With a proper choice of two free
parameters, the scheme remains phase field bounded without the requirement of
any ad hoc mass redistribution. We verify the scheme against several standard
test cases (for interface capturing) comprising not only the flat domains but
also the curved domains, leveraging the advantage that DEC operators are
independent of the coordinate system. The results show excellent properties of
boundedness, mass conservation and convergence. Moreover, we demonstrate the
ability of the scheme towards handling large density and viscosity ratios as
well as surface tension in the simulation of various two phase flow physical
phenomena on flat or curved surfaces.
