  A finite difference numerical scheme is proposed and analyzed for the
Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex
splitting is applied to the chemical potential, which in turns leads to the
implicit treatment for the singular logarithmic terms and the surface diffusion
term, and an explicit update for the expansive concave term. The convective
term for the phase variable, as well as the coupled term in the Stokes
equation, are approximated in a semi-implicit manner. In the spatial
discretization, the marker and cell (MAC) difference method is applied, which
evaluates the velocity components, the pressure and the phase variable at
different cell locations. Such an approach ensures the divergence-free feature
of the discrete velocity, and this property plays an important role in the
analysis. The positivity-preserving property and the unique solvability of the
proposed numerical scheme are theoretically justified, utilizing the singular
nature of the logarithmic term as the phase variable approaches the singular
limit values. An unconditional energy stability analysis is standard, as an
outcome of the convex-concave decomposition technique. A convergence analysis
with accompanying error estimate is provided for the proposed numerical scheme.
In particular, a higher order consistency analysis, accomplished by
supplementary functions, is performed to ensure the separation properties of
numerical solution. In turn, using the approach of rough and refined error
(RRE) estimates, we are able to derive an optimal rate convergence. To
conclude, several numerical experiments are presented to validate the
theoretical analysis.
