In　(Comm　Pure　Appl　Math　62(4):502-564,　2009),　Hou　and　Lei　proposed　a　3D　model　for　the　axisymmetric　incompressible　Euler　and　Navier-Stokes　equations　with　swirl.　This　model　shares　a　number　of　properties　of　the　3D　incompressible　Euler　and　Navier-Stokes　equations.　In　this　paper,　we　prove　that　the　3D　inviscid　model　with　an　appropriate　Neumann-Robin　or　Dirichlet-Robin　boundary　condition　will　develop　a　finite　time　singularity　in　an　axisymmetric　domain.　We　also　provide　numerical　confirmation　for　our　finite　time　blowup　results.　We　further　demonstrate　that　the　energy　of　the　blowup　solution　is　bounded　up　to　the　singularity　time,　and　the　blowup　mechanism　for　the　mixed　Dirichlet-Robin　boundary　condition　is　essentially　the　same　as　that　for　the　energy　conserving　homogeneous　Dirichlet　boundary　condition.　Finally,　we　prove　that　the　3D　inviscid　model　has　globally　smooth　solutions　for　a　class　of　large　smooth　initial　data　with　some　appropriate　boundary　condition.　Both　the　analysis　and　the　results　we　obtain　here　improve　the　previous　work　in　a　rectangular　domain　by　Hou　et　al.　(Adv　Math　230:607-641,　2012)　in　several　respects.