  We consider the sharp interface limit of a convective Allen-Cahn equation,
which can be part of a Navier-Stokes/Allen-Cahn system, for different scalings
of the mobility $m_\varepsilon=m_0\varepsilon^\theta$ as $\varepsilon\to 0$. In
the case $\theta>2$ we show a (non-)convergence result in the sense that the
concentrations converge to the solution of a transport equation, but they do
not behave like a rescaled optimal profile in normal direction to the interface
as in the case $\theta=0$. Moreover, we show that an associated mean curvature
functional does not converge the corresponding functional for the sharp
interface. Finally, we discuss the convergence in the case $\theta=0,1$ by the
method of formally matched asymptotics.
