Abstract We present a three-parameter, self-similar family of steady, axisymmetric, nonrelativistic solutions that unifies the morphology, kinematics, and viscous transport of accretion–ejection flows. The triplet (, , ) governs the radial power-law indices of angular velocity, density, and kinematic viscosity, respectively. In the inviscid limit, the geometric index continuously organizes the flow topology—from flared, toroidal envelopes (2) —while the stratification index controls mass loading and helical pitch. Introducing a scale-free viscosity (r) r^ preserves separability and yields an analytic viscous correction r^ -1 to the meridional velocities, with amplitude set by a coupling V. This framework provides closed-form expressions for velocity fields, streamlines, and stream surfaces, enabling quantitative morphology diagnostics such as the opening-angle profile () and contour-based RMSE for direct comparison with simulations or observations. The resulting (, , ) atlas defines a transparent analytic baseline for global HD/GRMHD models, clarifies how viscosity tilts self-similar stream surfaces, and offers benchmark solutions for reduced or physics-informed neural network surrogates.
Bernal et al. (Thu,) studied this question.