Abstract

This paper presents a purely continuum mechanics approach to localized wave-packet stability by investigating the hydrodynamic limits of the Madelung formulation of the Schrödinger equation. By treating the quantum wave function via a standard hydro-kinetic change of variables (\(\psi = \sqrt{\rho}e^{iS/\hbar}\)), the system is mapped identically onto a compressible Euler fluid governed by an internal, non-linear Quantum Pressure Tensor (\(Q = -\frac{\hbar^2}{2m} \frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}\)). We derive the exact non-linear constitutive equations for a single-node, three-dimensional localized vortex knot carrying a quantized circulation condition (\(\Gamma = \oint \mathbf{v} \cdot d\mathbf{r} = \frac{n\hbar}{m}\)). Utilizing Navier-Stokes boundary layer theory extended via effective macroscopic viscosity coefficients, we analyze the interfacial shear stress (\(\tau = \mu_{\text{eff}} \frac{dv}{dy}\)) generated when independent vortex topologies overlap. We demonstrate that the localized gradients of the quantum potential Q act as an intrinsic geometric stabilizer, generating an isotropic stagnation pressure field that counteracts dispersion without requiring external boundary walls. Finally, we establish the mathematical thresholds where localized shear stress forces a rapid, deterministic phase-locking transition, providing a classical, hydrodynamic mechanism for environmental decoherence. This numerical solution offers a rigorous continuum framework for analyzing localized topological structures in non-linear fluid substrates.

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.

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