  We investigate the sharp interface limit of a diffuse interface system that
couples the Allen--Cahn equation with the instationary Navier--Stokes system in
a bounded domain in $\mathbb{R}^d$ with $d \in \{2,3\}$. This model is used to
describe a propagating front in a viscous incompressible flow with the width of
the transition layer being characterized by a small parameter $\varepsilon>0$.
We show that the solutions converge to a limit two-phase fluid system with
surface tension that couples the mean curvature flow and the Navier--Stokes
system. The main assumptions are that the evolution of the limit system is
sufficiently regular and that the associated evolving interface does not
intersect the boundary of the container. For quantitatively well-prepared
initial data, we even establish an optimal convergence rate. This is the first
rigorous result of this kind which is valid in all physically relevant ambient
dimensions.
