= Two-fluid model of superfluidity
{title2=$\rho=\rho_s+\rho_n$}
The <two-fluid model of superfluidity> gives the <normal component of a superfluid> and the <superfluid> independent <velocities>.
An elementary uniform-flow example makes the distinction operational. Write the <mass> flux per unit <area> and the ideal <entropy> flux as
$$
\mathbf j=\rho_s\mathbf v_s+\rho_n\mathbf v_n,\qquad
\mathbf j_S=\rho s\mathbf v_n,\qquad \rho=\rho_s+\rho_n.
$$
Here $s$ is <specific entropy>, in $\mathrm{J\,kg^{-1}\,K^{-1}}$, and $\rho_s,\rho_n$ are response <mass densities>. The <mass density> $\rho_s$ used here gives the three-dimensional <phase stiffness> coefficient $\rho_s(\hbar/m)^2$ when $\mathbf v_s=(\hbar/m)\nabla\theta$ is substituted into the <kinetic energy density> $\rho_s v_s^2/2$. For a stipulated $\rho_s=3\rho/4$, $\rho_n=\rho/4$ and $v_n=3u$ along a channel, zero net <mass> flux requires $v_s=-u$. Yet $j_S=3\rho su$ is positive. Opposing <velocities> can therefore transport <entropy> while transporting no net <mass>.
This example explains why excellent apparent <heat> transport need not be ordinary diffusion of <heat> through a stationary <liquid>. In the ideal zero-net-<mass>-flux limit, the <heat> flux associated with this <entropy> transport is $Tj_S$. Viscous losses, <quantum vortices> and external <heat> exchange require additional terms. The model also supports <second sound>.
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