Superconductivity & Particle AcceleratorS 2026

According to Landau’s two fluid description, heat transport in He II results from the relative motion of the entropy carrying normal component and the inviscid superfluid component. At sufficiently high relative velocities, quantised vortex lines form a turbulent tangle that produces mutual friction between the two components. In reduced engineering descriptions, this interaction is commonly represented by the Gorter–Mellink relation.

A simplified CFD formulation can be obtained by determining the relative component velocity algebraically from the local temperature and pressure gradients. The resulting counterflow heat flux is introduced into the energy equation as a nonlinear diffusion term. This approach retains the macroscopic thermal resistance associated with mutual friction without solving separate transient momentum equations for the two components. However, the vortex tangle is represented only implicitly through the Gorter–Mellink coefficient and is assumed to adjust instantaneously to local conditions.

The transient wall heating experiment of Fuzier and Van Sciver provides a useful test of this reduction. A short thermal pulse is transported by forced He II while interacting with counterflow and the temperature variation associated with hydraulic pressure loss. Selected calculations show that the algebraic formulation captures the principal transport mechanisms but does not reproduce all features of the measured pulse evolution and decay, indicating the importance of the finite time response of the vortex tangle.

A possible next level in the modelling hierarchy introduces vortex line density, defined as the total length of quantised vortex lines per unit fluid volume, as an independent state variable. Its evolution can be described using an equation of the Vinen type that accounts for vortex transport, production by the relative motion of the normal and superfluid components, and vortex decay. Coupling the evolving vortex line density with mutual friction introduces the finite response time and flow history absent from the algebraic Gorter–Mellink closure. This provides an intermediate description between the reduced counterflow model and the complete Hall–Vinen–Bekarevich–Khalatnikov continuum formulation, which retains the independent dynamics of the normal and superfluid components together with the coarse grained forces produced by quantised vortices.