  In this paper the phase behavior and pattern formation in a sheared non-ideal
fluid under a periodic potential is studied. An isothermal two-dimensional
formulation of a lattice Boltzmann scheme for a liquid-vapor system with the
van der Waals equation of state is presented and validated. Shear is applied by
moving walls and the periodic potential will vary along the flow direction. A
region of the parameter space, where in absence of flow a striped phase with
oscillating density is stable, will be considered. At low shear rates the
periodic patterns are preserved and slightly distorted by the flow. At high
shear rates the striped phase looses its stability and a new phase with
interface waves between the liquid and vapor regions will appear. Velocity
field patterns will be also shown.
