  Recent observations of supermassive black holes have brought us new
information on their magnetospheres. In this study we attempt a theoretical
modelling of the coupling of black holes with their jets and discs, via three
innovations. First, we propose a semi-analytical MHD description of a steady
relativistic inflow-outflow structure characteristic to the extraction of the
hole rotational energy. The mass-loading is ensured in a thin layer, the
stagnation surface, by a two-photon pair production originating to a gamma-ray
emission from the surrounding disc. The double flow is described near the polar
axis by an axisymmetric meridionally self-similar MHD model. Second, the inflow
and outflow solutions are crossing the MHD critical points and are matched at
the stagnation surface. Knowledge of the MHD field on the horizon give us the
angular momentum and energy extracted from the black hole. Finally, we
illustrate the model with three specific examples of double-flow solutions by
varying the energetic interaction between the MHD field and the rotating black
hole. When the isorotation frequency is half of the black hole one, the
extracted Poynting flux is comparable to the one obtained using the force-free
assumption. In two of the presented solutions, the Penrose process dominates at
large colatitudes, while the third is Poynting flux dominated at mid
colatitudes. Mass injection rate estimations, from disk luminosity and inner
radius, give an upper limit just above the values obtained for two solutions.
This model is pertinent to describe the flows near the polar axis where pair
production is more efficient.
