Thermal Convection and Nonlinear Effects of a Superfluid HELIUM-3-HELIUM-4 Mixture in a Porous Medium.

The convective instability of one-component classical fluids in a porous medium confined between two unbounded slabs is studied. This system behaves like a high Prandtl number bulk fluid. It has boundary conditions similar to the stress-free boundary conditions of bulk one-component classical fluids. Both the amplitude expansion method and the Galerkin method are used to investigate the nonlinear steady convection. Two-dimensional rolls are the only stable motion at the onset of convection. Beyond threshold, the steady convection rolls become unstable to formation of cross-roll and zigzag instabilities. Applying the phase -dynamics approach for the zigzag instability, we obtain the diffusion coefficient D, which can signal the onset of instability. We also investigate the convective instability of superfluid ('3)He-('4)He mixtures in porous media. Assuming no interaction between the average superflow and the porous medium and treating the normal flow in the equation of motion like a classical fluid in a porous medium, we find that the superfluid mixtures in a porous medium behave like a classical fluid in a porous medium. Additional two-fluid effects omitted above are calculated at the onset of convection and the instability boundary. The shift of critical Rayleigh number is about 1% or less at the onset of convection and can be as large as 20% or more at the instability boundary for some regions around T (TURN) 1(DEGREES)K. This shift is quite large compared to the corresponding 0.001% shift at the onset of convection for bulk superfluid ('3)He-('4)He mixtures. To investigate the effects of a lateral boundary, the convective instability of classical one-component fluids in porous media inside a box is studied. The zigzag instability does not exist because of the boundary conditions at the sides of the box. The shift of the instability boundary from the infinte layer is obtained. Similar behavior is obtained for a superfluid ('3)He-('4)He mixture in a porous medium, and the two-fluid shift at the instability boundary is of the same order as the two-fluid shift for the infinite layer since the two-fluid effects only affect the z dependence of the rolls.