Density pattern in supercritical flow of liquid ${}^4\mathrm{He}$

A density-functional theory is used to investigate the instability arising in superfluid ${}^4\mathrm{He}$ as it flows at velocity $u$ just above the Landau critical velocity of rotons $v_c$. Confirming an early theoretical prediction by one of us [JETP Lett. 39, 511 (1984)], we find that a stationary periodic modulation of the density occurs, with amplitude proportional to ${(u-v_c)}^{1∕2}$ and wave vector equal to the roton wave vector. This density pattern is studied for supercritical flow both in bulk helium and in a channel of nanometer cross section.

Figure 1

Amplitude of the density modulation along the direction of ${}^4\mathrm{He}$ motion, computed during the functional minimization. Solid line: $u=1.29v_c$, dotted line: $u=1.14v_c$, dashed line: $u=0.99v_c$, dash-dot line: $u=0.84v_c$.

Figure 2

Density profiles along the direction of ${}^4\mathrm{He}$ flow $x$ axis. The three profiles have been calculated, in order of increasing amplitude, with $u=1.07$, 1.22, and 1.37 $v_c$, respectively.

Figure 3

Velocity profile along the direction of ${}^4\mathrm{He}$ flow $x$ axis. Same values of $u$ as in Fig. 2.

Figure 4

Amplitude of the density modulation as a function of the fluid velocity. Points: DF results. Line: fitting function $1.01{[(u-v_c)∕v_c]}^{1∕2}$.

Figure 5

Density profile across the channel section.

Figure 6

Contour plots of the density along the channel.