PH YS ICAL RE VIE% A VOLUME 24, NUMBER 2 AUGUST 1981 Stationary convective instability in a superfluity 'He-'He mixture V. Steinberg Department of Physics, University of California, Santa Barbara, Cahfornia 93106 IReceived 30 October 1980) The stationary convection instability of a superfluid mixture is considered. The criteria for the instability onset in a superfluid region of the 'He-'He phase diagram and in different limiting cases are obtained. It is shown that in difFerent regions of the phase diagram, depending on the magnitude of the kinetic coefficients, the stationary- instability criterion is similar to that of either the regular binary mixture with abnormal thermal diffusion, to the pure liquid, or to the compressible pure liquid. The stability is also strongly dependent on concentration and temperature, and the critical temperature gradient rises sharply with decreasing temperature and concentration. At very low temperature (bek)w 0.5 K) and concentration (dilute solution) the superfluid solution becomes stable with respect to stationary convection. Asymptotic behavior of the criterion in the vicinity of the A, line, the tricritical point and for infinitely dilute solutions is also estimated. I. INTRODUCTION k'z &Co = -K& 0. T - O'T (2) 'The Rayleigh-Benard problem has been the sub- ject of considerable interest in recent years both theoretically" and experimentally. " From an experimental point of view, as shown by Ahlers, ' cryogenic fluids have an advantage for investiga- tion of hydrodynamic instabilities due to a high- temperature resolution and low extraneous heat transport. The He'-He4 mixture in a superfluid region between the ~ line and coexistence curve of separation represents an unusual two-component system due to a very wide variation (several or- ders of magnitude) of the thermodynamic and ki- netic properties. ' 'The superfluid nature of this mixture also renders it a unique Benard system. It is well known that in an He'-He' superfluid mixture a temperature gradient can exist in equil- ibrium in contrast to pure He if (Ref. 7). This temperature gradient leads to a corresponding concentration gradient 8 S Sc pt. " p, z where p, T, and C are the density, the tempera- ture, and the weight concentration respectively, S is the entropy per 1 cm' of solution and Z = p(p, - p, ), p is the chemical potential, and sub- script "0'* corresponds to mechanical-equilibrium conditions. Thus, the superfluid He' component moves to the warm boundary and causes the light He' atoms to be concentrated near the cold boundary. A sim- ilar concentration distribution occux s in a regular binary mixture with a large abnormal thermodif- fusion effect kr & 0 (Ref. 8); in our case we also have Such systems are unstable with respect to sta- tionary convection when heated from above and with respect to oscillatory convection when heated from below, '" and differ significantly from the Hayleigh-Benard instability in a pure liquid. In the latter case the instability occurs due to a den- sity gradient which becomes unstable in the gra- vitational field. But in a binary mixture it also may be caused by the separation of the time scales in relaxation of temperature and concen- tration fluctuations. " The physical reason for this instability is clear; in the system heated from above the concentra- tion perturbations are destabilizing whereas the temperature perturbations tend to stabilize the system. In the case where the relaxation time of the concentration fluctuations is much larger than the relaxation time of the temperature fluctuat- ionss the system becomes unstable with respect to stationary convection at the certain critical value of the temperature gradient. A similar explana- tion holds for the oscillatory instability (oversta- bility) onset in the system heated from below. As for a regular binary mixture with abnormal thexmal diffusion, it is natural to expect the onset of a stationary instability in the superfluid mixture when heated from above and the onset of an oscil- latory instability when heated from below. In this paper I will discuss just the stationary stability of a horizontal layer of a superfluid He'-He' mix- ture. The oscillatory instability of this system will be the subject of the following paper. The hydrodynamic equations in the Boussinesq approximation are discussed in Sec. II. In Sec. III the two limiting cases of stationary instability are considered: for one the dissipation of super- 1981 The American Physical Society