Hydrodynamic Electron Transport in Regularized Nanocapillary Channels: A Framework for Ultrafast Dissipationless Charge Storage

We formulate a continuum theory in which the dense degenerate electron gas confined in a regularized nanocapillary channel is treated as a viscous, incompressible, charged fluid whose dynamics is governed by the Navier-Stokes momentum balance closed self-consistently by the full Maxwell system through the Lorentz force density and the Maxwell stress tensor. The central object is the coupled fluid-field boundary operator, obtained by linearizing the coupled system about a uniform drift state and imposing a Navier partial-slip condition on a spectrally regularized channel wall. We prove that viscous dissipation vanishes identically if and only if the rate-of-strain tensor vanishes, which in a straight incompressible channel selects rigid plug flow. We derive the Poiseuille-Gurzhi flux with finite slip length, and show that the geometric resistance factor decays as the reciprocal of one plus six times the ratio of slip length to channel width. We then advance the resonant slip conjecture: for a discrete set of capillary widths the Robin-type transverse eigenproblem acquires a marginal mode at which the effective slip length diverges, so that the momentum-relaxing dissipation channel of the hydrodynamic regime closes and the transport becomes dissipationless within the idealized model. We specify a reproducible numerical protocol based on FFT boundary regularization and level-spacing statistics, with the Gaussian Unitary Ensemble Wigner surmise as the acceptance criterion. Finally, we give a formal Lean 4 skeleton of the coupled system and an order-of-magnitude engineering analysis of the power, thermal, and energy-density constraints of charge storage on a seconds time scale.