  This work considers flows from an accretion disk corotating with the aligned
dipole magnetic field of a rotating star. Ideal magnetohydrodynamics (MHD) is
assumed with the pressure and density related as $p \propto \rho^\gamma$ and
with $\rho {\bf v}^2 \ll {\bf B}^2/4\pi$, where ${\bf v}$ is the flow velocity.
This limit corresponds to the Alfv\'en radius for the disk accretion larger
than the corotation radius. Transonic flows, which go from subsonic motion near
the disk to supersonic inflow near the star, are shown to be possible only for
a narrow range of $R_d \sim r_c \equiv (GM/\Omega^2)^{1/3}$, where $R_d$ is the
radius at which the dipole field line intersects the disk, $r_c$ the corotation
radius, $M$ the mass of the star, and $\Omega$ its angular rotation rate. The
transonic flows have very different behaviors for $\gamma > 7/5$ and $< 7/5$.
In both cases, the plasma flow velocity ${\bf v}$ (which is parallel to ${\bf
B}$) increases with decreasing distance $R$ from the star. However, for $\gamma
>7/5$, the Mach number ${\cal M} \equiv |{\bf v}|/c_s$ (with $c_s$ the sound
speed) initially increases with $R$ decreasing from $R_d$, but for $R$
decreasing from $\approx 0.22 R_d$ (for $\gamma=5/3$)the Mach number
surprisingly {\it decreases}. In the other limit, $\gamma < 7/5$, ${\cal M}$
increases monotonically with decreasing $R$. Application of these results is
made to funnel flows to rotating magnetized neutron stars and young stellar
objects.
