A 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, a dimensionally calibrated quaternion tensor is introduced to resolve the topological coordinate singularity (gimbal lock phenomenon) inherent in rotating celestial bodies. Furthermore, to prevent the non-physical gravitational collapse observed in conventional mass-density singularity models, an isotropic hydrostatic pressure tensor is coupled with a radial density stratification function. This system reaches a permanent thermodynamic equilibrium state by suppressing the exponential divergence of turbulent kinetic energy (TKE) through a kinematic viscous dissipation term opposing the non-linear convective vortex acceleration. This hydrodynamic formulation successfully mitigates infinite density singularities without relying on the constraints of quantum gravity, providing a computable closed-form macroscopic field theory for planetary geometry.