We present 3D hydrodynamical simulations of ram pressure stripping of a disc galaxy orbiting in a galaxy cluster. In this paper, we focus on the properties of the galaxies tails of stripped gas. The galactic wakes show a flaring width, where the flaring angle depends on the gas discs cross-section with respect to the galaxys direction of motion. The velocity in the wakes shows a significant turbulent component of a few 100 km/s. The stripped gas is deposited in the cluster rather locally, i.e. within ~150 kpc from where it was stripped. We demonstrate that the most important quantity governing the tail density, length and gas mass distribution along the orbit is the galaxys mass loss per orbital length. This in turn depends on the ram pressure as well as the galaxys orbital velocity. For a sensitivity limit of ~10^19 cm^-2 in projected gas density, we find typical tail lengths of 40 kpc. Such long tails are seen even at large distances (0.5 to 1 Mpc) from the cluster centre. At this sensitivity limit, the tails show little flaring, but a width similar to the gas discs size. Morphologically, we find good agreement with the HI tails observed in the Virgo cluster by Chung et al. (2007). However, the observed tails show a much smaller velocity width than predicted from the simulation. The few known X-ray and H$alpha$ tails are generally much narrower and much straighter than the tails in our simulations. Thus, additional physics like a viscous ICM, the influence of cooling and tidal effects may be needed to explain the details of the observations. We discuss the hydrodynamical drag as a heat source for the ICM but conclude that it is not likely to play an important role, especially not in stopping cooling flows.
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