A model of arch stability in a conical hopper section has been derived, based upon the approaches of Walters, Walker and Li. A simple flow rule has resulted from this analysis. The model has been applied to vibrationally activated conical hoppers. Two limiting conditions were considered: the push mode, in which the vibrated wall section compresses the bulk material at plastic yield; and the pull mode, in which the wall releases stress upon the contained material and, with in the limit, exerts no hoop stress over the vibrated section of the hopper wall. The pull mode is assumed to induce a switch in stress orientation from a passive to an active stress state. Theoretically, this is very effective at causing flow within the contained materials. The model has been applied to a number of scenarios involving typical bulk materials, and the following conclusions have been deduced: (1) Vibration can be very effective at inducing flow in hopper systems and can generate flows at shallower wall-slope angles and much smaller outlet diameters than conventional analysis will permit. (2) The flow regime from a vibrationally induced system will probably be stick-slip. (3) The most effective strategy is to induce vibration over the complete height of the hopper. This will permit flow through very small outlets (theoretically down to zero diameter).However, there are mechanical problems associated with this approach. (4) When vibration is applied to a section of the cone, the location of the vibration source is acritical design parameter. The proposed model may provide the beginnings of a rational basis for design in this respect. (5) The ’one-third cone height’ vibration mount rule is not generally applicable.