  In this paper, we construct a framework for investigating
magnetohydrodynamical jet structure of spinning black holes (BHs), where
electromagnetic fields and fluid motion are governed by the Grad-Shafranov
equation and the Bernoulli equation, respectively. Assuming steady and
axisymmetric jet structure, we can self-consistently obtain electromagnetic
fields, fluid energy density and velocity within the jet, given proper plasma
loading and boundary conditions. Specifically, we structure the two coupled
governing equations as two eigenvalue problems, and develop full numerical
techniques for solving them. As an example, we explicitly solve the governing
equations for the split monopole magnetic field configuration and simplified
plasma loading on the stagnation surface where the poloidal fluid velocity
vanishes. As expected, we find the rotation of magnetic field lines is dragged
down by fluid inertia, and the fluid as a whole does not contribute to energy
extraction from the central BH, i.e., the magnetic Penrose process is not
working. However, if we decompose the charged fluid as two oppositely charged
components, we find the magnetic Penrose process does work for one of the two
components when the plasma loading is low enough.
