Multi-Scale Perturbation Analysis in Hydrodynamics of the Superfluid Turbulence. Derivation of the Dresner Equation

TL;DR

This paper develops a multi-scale perturbation method to derive a simplified nonlinear heat conduction equation for superfluid turbulence, clarifying the validity of the Dresner equation in modeling slow heat transfer processes in HeII.

Contribution

The paper introduces a multi-scale perturbation analysis to derive the Dresner equation from hydrodynamic equations of superfluid turbulence, linking fast and slow processes.

Findings

Derived the Dresner equation from first principles.

Identified the range of validity for the Dresner phenomenological model.

Provided a systematic approach to eliminate fast sound processes.

Abstract

The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or counterflows) of HeII in the presence of a chaotic set of vortex filaments, so called superfluid turbulence. The HST equations govern both a slow variation of the hydrodynamic variables due to dissipation related to the vortex tangle and fast processes of the first and second sound propagation. This circumstance prevents an effective numerical simulations of the problems of unsteady heat transfer in HeII. By virtue of a pertinent multi-scale perturbation analysis we show how one can eliminate the fast processes to derive the evolution equation for the slow processes only. We then demonstrate that the long-term evolution of a transient heat load of moderate intensity obeys the nonlinear heat conductivity equation, often referred to as the Dresner equation. We also compare our approach against the Dresner phenomenological derivation and establish a range of validity of the latter.