= Solution
The <two-fluid model of superfluidity> separates mass current into a <superfluid fraction> and a <normal component of a superfluid>:
$$
\mathbf j=\rho_s\mathbf v_s+\rho_n\mathbf v_n,\qquad \rho=\rho_s+\rho_n.
$$
The normal component carries entropy and responds to shear viscosity; an ideal superfluid flow is entropy-free. These are interpenetrating hydrodynamic responses, not two permanently distinguishable sets of atoms. Thermal <elementary excitations> carry momentum relative to the condensate and generate the normal response.
The <Andronikashvili experiment> gives an operational picture. A closely spaced stack of disks oscillates torsionally in He-II. Viscous coupling entrains the normal response between disks, changing the oscillator's inertia, while vortex-free superfluid need not follow the rotation. The temperature-dependent inertia measures the normal component under the apparatus's coupling conditions. Counterflow also permits <second sound>, primarily an entropy/temperature wave rather than the ordinary density wave of first sound.
<Condensate depletion> is a different quantity. The condensate fraction counts occupation of a single-particle mode, while $\rho_s$ measures the response to phase gradients or motion. In a weak three-dimensional homogeneous Bose gas at zero temperature,
$$
\frac{n-n_0}{n}\simeq\frac8{3\sqrt\pi}\sqrt{na_s^3},
$$
so interactions produce quantum depletion even without thermal excitations. In a translation-invariant Galilean system at zero temperature, the superfluid mass density can nevertheless equal the total density. He-II has strong interaction-induced depletion but a nearly complete zero-temperature superfluid response. Lattices, disorder and finite temperature require their own response calculation; condensate and superfluid fractions should not be equated in general.
The <Landau criterion> follows by considering an excitation of momentum $\mathbf p$ in a moving fluid. Its laboratory-frame creation cost can be $\epsilon(\mathbf p)-\mathbf v\cdot\mathbf p$. A necessary energetic condition against this decay channel is
$$
\boxed{v<v_L,\qquad v_L=\min_{p>0}\frac{\epsilon(p)}p.}
$$
For the low-momentum <Bogoliubov spectrum> of a weakly interacting Bose gas, $\epsilon\simeq c_sp$, giving the sound-speed threshold for that channel. The phonon–roton spectrum of He-II can instead have its minimum ratio near the roton region. Vortex nucleation, boundaries and a moving obstacle can lower a measured critical velocity; the ideal excitation criterion is not a guarantee of dissipation-free flow at every smaller speed in every apparatus. An ideal noninteracting Bose gas has a condensate but quadratic excitations with $v_L=0$, another distinction between condensation and robust superfluidity.
A cold-gas test moved a blue-detuned laser beam through a trapped condensate and measured heating as its speed changed. The repulsive optical obstacle supplies a controlled perturbation; strong heating appeared above a threshold. Its interpretation must account for the inhomogeneous cloud and excitation/nucleation mechanism, rather than identify every observed threshold with the central sound speed.<Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas by C. Raman and others>[1999 abstract: moving blue-detuned laser beam and onset of strong heating]{ref}
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