Title: Mass Conservative and Energy Stable Finite Difference Methods for the Quasi-incompressible Navier-Stokes-Cahn-Hilliard system: Primitive Variable and Projection-Type Schemes,
Abstract: In this paper we describe two fully mass conservative, energy stable, finite
difference methods on a staggered grid for the quasi-incompressible
Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible
fluid flow with variable density and viscosity. Both methods, namely the
primitive method (finite difference method in the primitive variable
formulation) and the projection method (finite difference method in a
projection-type formulation), are so designed that the mass of the binary fluid
is preserved, and the energy of the system equations is always non-increasing
in time at the fully discrete level. We also present an efficient, practical
nonlinear multigrid method - comprised of a standard FAS method for the
Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy
for the Navier-Stokes equation - for solving these equations. We test the
scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor
instability. Quantitative comparisons are made with existing analytical
solutions or previous numerical results that validate the accuracy of our
numerical schemes. Moreover, in all cases, mass of the single component and the
binary fluid was conserved up to 10 to -8 and energy decreases in time.