Hydrodynamic Approach to Macroscopic Gravity: Thermodynamic Equilibrium via Viscous Dissipation and Quaternion Tensors

Abstract

This paper presents a novel hydrodynamic framework for macroscopic gravity, bridging the gap between Einstein's tensor geometry and Navier-Stokes fluid dynamics. By modeling the gravitational field as a continuous flow of Vacuum Fluctuation Energy, this study introduces the quaternion tensor (q_rot) to resolve the polar coordinate singularity (gimbal lock) inherent in rotating celestial bodies. Furthermore, to prevent the non-physical gravitational collapse observed in conventional mass-density singularity models, a hydrostatic pressure gradient (-∇P) is combined with a radial density stratification function (ρ(r)). This system reaches a permanent thermodynamic equilibrium state by suppressing turbulent kinetic energy (TKE) through a kinematic viscous dissipation term (-ν∇²v). This hydrodynamic formulation successfully eliminates infinite density singularities without relying on the constraints of quantum gravity, providing a computable macroscopic field theory for planetary geometry.