A detailed understanding of how the infalling matter in accretion columns is decelerated is essential for the calculation of the emitted radiation. On neutron stars, the deceleration takes place mainly by the interaction of the plasma with radiation, at least for the high-luminosity sources. We report on our two-dimensional calculations of the hydrodynamic flow in such accretion columns. The radiation transport is treated in the diffusion approximation, and we are looking for a stationary solution for the velocity field. The dependence of the results on physical parameters, especially on the accretion rate is discussed. Due to the non-linearity of the problem it turns out that only in certain parameter ranges stationary solutions seem to exist. For accretion rates higher than a critical value there are no stationary accretion flows. This leads us to the conclusion that a time-dependent picture for the accretion is unavoidable.