A two‐dimensional electrostatic particle‐in‐cell code is used to simulate the convection of a finite stream of initially neutral plasma across a uniform magnetic field. The simulations show that the stream loses momentum with distance as a result of two erosion mechanisms that have greater effects for denser plasmas: (1) erosion of the charge layers at the sides of the stream as a result of velocity shear and (2) erosion of the head of the stream as a result of charge separation where ions travel ahead of the electrons. The electron charge layer exhibits a velocity shear that excites the diocotron instability. This instability occurs earlier for denser plasmas but it does not appear when the length of the stream is shorter than four wavelengths. The charge separation at the head of the stream causes the eroded plasma to drift with a sheared velocity. A flutelike instability develops at the head of the plasma stream for sufficiently dense plasmas. The simulations show that the plasma is eroded faster by the head erosion mechanism. Electric field fringe effects cause the plasma head to broaden and the tail of the plasma to narrow. The simulations show that although the plasma configuration is changed a great deal by erosion and fringe effects, the stream is convected across the magnetic field with a constant velocity for sufficiently dense plasmas and with a velocity that approaches the injection for denser plasmas. The simulations also show that the convection velocity for a partially and for a completely eroded plasma is the same.