Stability of convection in the mush under rocking motion

We implement a linear stability analysis for the mush under rocking motion. Owing to the rocking motion, the shifted gravity induces a parallel flow. Without rocking, all the instability modes propagating in different directions are of equal stability criteria. With rocking, except for the longitudinal mode propagating in the direction perpendicular to the induced flow, all modes are to various extents inhibited by the induced flow. More precisely, all these modes are inhibited by the periodical buoyancy reduction in the direction of density gradient, which essentially stabilizes the system without preferred direction.