  We discuss the dynamics of the accretion flow onto a black hole driven by
Poynting flux in a simplified model of a two-dimensional accretion disk. In
this simplified model, the condition of the stationary accretion flow is found
to impose a nontrivial constraint on the magnetic field configuration. The
effect of the magnetic field on the accretion flow is discussed in detail using
the paraboloidal and hyperboloidal type configuration for the poloidal
structure suggested by Blandford in 1976. It is demonstrated explicitly that
the angular velocity of the disk, $\Omega_D$, deviates from the Keplerian
angular velocity. The angular velocity of the rigidly-rotating magnetic
surface, $\Omega_F$, does not have to be the same as the angular velocity of
the disk for the paraboloidal type configuration. But for the hyperboloidal
type configuration, it is found that we can set $\Omega_F = \Omega_D$, which
corresponds to an accretion disk of perfect conductor. We discuss the numerical
solutions of the stream equation for stationary accretion flow in the
Schwarzschild background using a paraboloidal type configuration. The dynamics
of the accretion disk is found to depend strongly on the ratio of the accretion
rate to the magnetic field strength.
