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In this paper, we construct a framework for investigatingmagnetohydrodynamical jet structure of spinning black holes (BHs), whereelectromagnetic fields and fluid motion are governed by the Grad-Shafranovequation and the Bernoulli equation, respectively. Assuming steady andaxisymmetric jet structure, we can self-consistently obtain electromagneticfields, fluid energy density and velocity within the jet, given proper plasmaloading and boundary conditions. Specifically, we structure the two coupledgoverning equations as two eigenvalue problems, and develop full numericaltechniques for solving them. As an example, we explicitly solve the governingequations for the split monopole magnetic field configuration and simplifiedplasma loading on the stagnation surface where the poloidal fluid velocityvanishes. As expected, we find the rotation of magnetic field lines is draggeddown by fluid inertia, and the fluid as a whole does not contribute to energyextraction from the central BH, i.e., the magnetic Penrose process is notworking. However, if we decompose the charged fluid as two oppositely chargedcomponents, we find the magnetic Penrose process does work for one of the twocomponents when the plasma loading is low enough.

Towards a Full MHD Jet Model of Spinning Black Holes--I: Framework and a split monopole example