  We consider a multiphysics model for the flow of Newtonian fluid coupled with
Biot consolidation equations through an interface, and incorporating total
pressure as an unknown in the poroelastic region. A new mixed-primal finite
element scheme is proposed solving for the pairs fluid velocity - pressure and
displacement - total poroelastic pressure using Stokes-stable elements, and
where the formulation does not require Lagrange multipliers to set up the usual
transmission conditions on the interface. The stability and well-posedness of
the continuous and semi-discrete problems are analysed in detail. Our numerical
study is framed in the context of applicative problems pertaining to
heterogeneous geophysical flows and to eye poromechanics. For the latter, we
investigate different interfacial flow regimes in Cartesian and axisymmetric
coordinates that could eventually help describe early morphologic changes
associated with glaucoma development in canine species.
