Unsteady plumes in planar flow of viscous and inviscid fluids

The planar flow of a light fluid into a surrounding heavier fluid is studied. The light fluid is released from the bottom through a vent, which may be at an angle and rises as a plume. It is removed at an upper permeable boundary, which is considered to be a horizontal surface, for simplicity. In the inviscid case, the plume approaches a steady-state configuration, which is well approximated by a one-dimensional asymptotic solution. The stability of the asymptotic solution is discussed. The viscous situation is modelled using Boussinesq theory and shows that entrainment of both fluids into the plume necessarily occurs, so that a steady-state configuration is not achieved. In both the viscous and the inviscid cases, the unsteady solutions are obtained using novel spectral techniques, and these are presented and discussed.