The flow of a radiation-dominated ideal fluid through a standing, radiative shock is an important aspect of accretion onto high-luminosity X-ray pulsars. A complete understanding of the shock structure is required in order to analyze the role of Fermi energization in producing the power-law spectra observed from typical X-ray pulsars. The detailed structure of the shock in turn depends on the boundary conditions imposed at the stellar surface, which are dictated by the strong magnetic field. Close to the neutron star, the accreting plasma is constrained to fall along the magnetic field lines, and therefore the flow should stagnate at the stellar surface and the energy flux into the star should vanish. However, previously published models for X-ray pulsar accretion flows generally do not satisfy these conditions. We consider the problem in this paper by obtaining an analytical solution for the velocity profile based on an approximate set of hydrodynamical equations describing the steady, plane-parallel flow of a radiation-dominated ideal fluid along a magnetically confined accretion column, including the escape of radiation through the column walls. The spatial variation of the velocity is parameterized in terms of the energy flux at the sonic point, or equivalently in terms of the upstream Mach number, and the dynamical solution obtained therefore represents a generalization of previous results for radiation-dominated accretion flows. The requirement of downstream stagnation yields an interesting eigenvalue relation involving the accretion rate, the radius of the column, and the energy flux at the sonic point. Flows failing to satisfy this relation cannot be steady and may display a variety of temporal behaviors. The simplicity of the analytical solution makes it a very convenient starting point for calculations of the emergent spectrum, which will be presented in a separate paper.
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