A two‐dimensional electrostatic particle‐in‐cell code is used to simulate a finite‐width plasma streaming across a uniform magnetic field. The simulations show that the plasma polarizes, and non‐neutral charge layers develop along its edges. In an electron–ion plasma, the charge layers are asymmetric and the electron layer is unstable to the diocotron mode. The simulations show that this instability has smaller growth rate for plasma streams that are relatively less dense and wider. For a positive/negative ion plasma with equal mass ions the charge layers are symmetric and the plasma is stable to the diocotron mode. The results show that the diocotron instability leads to vortex structure when the plasma width is greater than the ion gyroradius, but this instability disrupts the entire plasma when the plasma width is of the order of or smaller than the ion gyroradius.