Editing Quantum turbulence (section)

=== Two fluid nature ===
[[File:Helium-II_component_densities.png|thumb|375x375px|Fig 7. Component fractions plotted against the temperature, displaying the mixture of normal fluid and superfluid in helium II, where <math>\rho_s/\rho </math> is the superfluid fraction, and <math>\rho_n/\rho </math> is the normal fluid fraction. For temperatures above the critical temperature, the normal fluid makes up the whole fluid.]]
At non-zero temperature <math>T</math>, thermal effects must be taken into account. For atomic gases at non-zero temperatures, a fraction of the atoms are not part of the condensate, but rather form a rarefied (large free mean path) thermal cloud that co-exist with the condensate (which, in the first approximation, can be identified with the superfluid component). Since helium is a liquid, not a dilute gas like atomic condensates,  there is a much stronger interaction between atoms, and the condensate is only a part of the superfluid component. Thermal excitations (consisting of [[phonon]]s and rotons) form a viscous fluid component (very short free mean path, analogous to classical viscous fluid governed by the [[Navier–Stokes equations|Navier-Stokes equation]]), called the normal fluid which coexists with the superfluid component. This forms the basis of Tisza's and Landau's ''two-fluid theory'' describing helium II as the mixture of co-penetrating superfluid and normal fluid components, with a total density dictated by the equation <math>\rho = \rho_n + \rho_s </math>. The table displays the key properties of the superfluid and normal fluid components:.
{| class="wikitable"
!Component
!velocity
!density
![[entropy]]
!viscosity
|-
|superfluid
|<math>\mathbf{v}_s</math>
|<math>\rho_s</math>
|zero
|zero
|-
|classical fluid
|<math>\mathbf{v}_n</math>
|<math>\rho_n</math>
|<math>S</math>
|<math>\mu</math>
|}
The relative proportions of the two components change with temperature, from an all normal fluid flow at the transition temperature <math>T_c </math> (<math>\rho_n/\rho \rightarrow 1 </math> and <math>\rho_s/\rho \rightarrow 0 </math>), to a complete superfluid flow in the zero temperature limit (<math>\rho_s/\rho \rightarrow 1 </math> and <math>\rho_n/\rho \rightarrow 0 </math>). At small velocities, the two-fluid equations are

<math>  \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho_s \mathbf{v}_s + \rho_n \mathbf{v}_n) = 0</math>

<math>  \frac{\partial(\rho S)}{\partial t} + \nabla \cdot (\rho S \mathbf{v}_n) = 0</math>

<math> \rho_s\left[ \frac{\partial \mathbf{v}_s}{\partial t} + (\mathbf{v}_s \cdot \nabla)\mathbf{v}_s\right]  = -\frac{\rho_s}{\rho}\nabla P + \rho_s S \nabla T</math>

<math> \rho_n\left[ \frac{\partial \mathbf{v}_n}{\partial t} + (\mathbf{v}_n \cdot \nabla)\mathbf{v}_n\right]  = -\frac{\rho_n}{\rho}\nabla P - \rho_s S \nabla T + \mu \nabla^2 \mathbf{v}_n</math>

where here <math>P </math> is the pressure, <math>S </math> is the entropy per unit mass and <math>\mu </math> is the viscosity of the normal fluid component as given by the table above. The first of these equations can be identified as being the [[conservation of mass]] equation, while the second equation can be identified as the conservation of entropy. The results of these equations give rise to the phenomena of second sound and thermal counterflow. At large velocities the superfluid becomes turbulent and vortex lines appear; at even larger velocities both normal fluid and superfluid become turbulent.