Hydrodynamical model of anisotropic, polarized turbulent superfluids. I: Constraints for the fluxes
The paper is devoted to building up a hydrodynamical model of turbulence in liquid helium II, taking into account the polarization and anisotropy of the vortex tangle according with test results and numerical simulation. As fundamental variables in the one-fluid model of liquid helium II (in absence of vortices) density, velocity and absolute temperature are selected. At the same time, the heat flux is used as an internal variable describing the dynamics of the excitations. Three macroscopic quantities of the model are: (i) the average length of the vortex line per unit volume (vortex line density), which describes only the vortex tangle in a rotating sample and in well-developed counterflow superfluid turbulence; (ii) a pseudovector (axial vector), which describes the polarization of the vortex tangle; (iii) an asymmetric tensor, which models the anisotropy of the vortex tangle inside the considered particle. The goal of the paper is to formulate evolution equations for the three internal variables describing scalar, deviatoric and antisymmetric parts of a vorticity tensor and its acting behavior. With this aim, in the framework of extended thermodynamics with internal variables, the authors develop a (coarse grained) hydrodynamical model of liquid helium II, taking into account the presence of a polarized anisotropic tangle of quantized vortices. The evolution equations for the state variables are obtained starting from the balance equations of mass momentum and energy complemented in further evolution equations for the internal variables describing the dynamics of elementary excitations and the vortex tangle. Then the constraints on the constitutive quantities are deduced from the second law of thermodynamics, using the Liu method of Lagrange multipliers. It is deduced that the Lagrange multipliers are the partial derivatives of the entropy density in terms of their respective conjugated thermodynamic variables. Then, it is shown that the stress tensor is independent on the anisotropy and polarization of the vortex tangle in the first-order approximation. In order to single out the physical meaning and relevance of the constitutive quantities and the Lagrange multipliers, the authors analyze in detail the relations obtained with aim of their physical interpretation. It is shown that in anisotropic situations the description of the vortex tangle needs the whole vorticity tensor. Finally, by using the obtained constitutive expressions, the linearized system of field equations is obtained.
scientific article; zbMATH DE number 6858243
scientific article