A Deterministic Limit for Acoustic Cavitation: Deriving the Critical Void Radius via Hydrodynamic Pressure Relaxation

The prediction of structural erosion in marine propellers and hydroelectric turbines due to acoustic cavitation relies heavily on the Rayleigh-Plesset equation and empirical Cavitation Numbers (σ). While these thermodynamic and hydrodynamic frameworks effectively estimate the bulk probability of vapor bubble formation under low pressure, they fail to deterministically define the exact spatial boundary where localized inertial collapse triggers a supersonic micro-jet and plasma formation. This paper introduces a strict continuum framework for hydrodynamics. By modeling the fluid-void interface as a dynamic kinematic balance between the spatial capacity for pressure relaxation (fluid damping) and the localized rate of inertial implosion (void collapse), we derive a universal critical cavitation radius (Rcavitation). We demonstrate that cavitation erosion is not a statistical fluid inevitability, but an exact deterministic limit where localized inertial collapse strictly overpowers the advective pressure-diffusing capacity of the surrounding water. We propose a blueprint for Cavitation-Immune Hydrodynamics, allowing for maximum theoretical rotational velocities while guaranteeing absolute structural integrity and acoustic stealth.