Onset of convection in dilute superfluidHe3-He4mixtures. I. Unbounded slab

The thermohydrodynamics of a dilute superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixture confined between parallel horizontal planes is studied with the two-fluid equations of Landau and Khalatnikov. When heated from above, downward normal fluid flow concentrates ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ impurities near the cooler bottom plate, creating a potentially unstable density gradient. The linearized equations for the convective amplitudes contain nonclassical two-fluid terms proportional to ${\overrightarrow{\mathrm{v}}}_n^0·\overrightarrow{\nabla }$ and to $\overrightarrow{\nabla }·{\overrightarrow{\mathrm{v}}}_n^{\prime }$, which alter the critical Rayleigh number from that for a classical one-component fluid. This shift is expected to be smaller near $T\approx 1$ K.