  The main goal of this paper is to establish the nonlocal-to-local convergence
of strong solutions to a Navier--Stokes--Cahn--Hilliard model with singular
potential describing immiscible, viscous two-phase flows with matched
densities, which is referred to as the Model H. This means that we show that
the strong solutions to the nonlocal Model H converge to the strong solution to
the local Model H as the weight function in the nonlocal interaction kernel
approaches the delta distribution. Compared to previous results in the
literature, our main novelty is to further establish corresponding convergence
rates. Before investigating the nonlocal-to-local convergence, we first need to
ensure the strong well-posedness of the nonlocal Model H. In two dimensions,
this result can already be found in the literature, whereas in three
dimensions, it will be shown in the present paper. Moreover, in both two and
three dimensions, we establish suitable uniform bounds on the strong solutions
of the nonlocal Model H, which are essential to prove the nonlocal-to-local
convergence results.
