The transpiration boundary condition (2.59) associated with the semipermeable interfaces significantly increases the difficulty of the hydrodynamical problem as it introduces an extra coupling between the momentum (2.54) and species (2.53) equations. Unless the net influx to the cavity is zero, the problem for the velocity field is ill-posed and has no solution (Gresho 1988). Thus the species equation is coupled to the momentum equation both by the Dirichlet condition and the continuity constraint, as well as the buoyancy force term. This requires special treatment, as described in the following section.