  We discuss the ground state and the small-amplitude excitations of a
degenerate vapour of fermionic atoms placed in two hyperfine states inside a
spherical harmonic trap. An equations-of-motion approach is set up to discuss
the hydrodynamic dissipation processes from the interactions between the two
components of the fluid beyond mean-field theory and to emphasize analogies
with spin dynamics and spin diffusion in a homogeneous Fermi liquid. The
conditions for the establishment of a collisional regime via scattering against
cold-atom impurities are analyzed. The equilibrium density profiles are then
calculated for a two-component vapour of 40K atoms: they are little modified by
the interactions for presently relevant values of the system parameters, but
spatial separation of the two components will spontaneously arise as the number
of atoms in the trap is increased. The eigenmodes of collective oscillation in
both the total particle number density and the concentration density are
evaluated analytically in the special case of a symmetric two-component vapour
in the collisional regime. The dispersion relation of the surface modes for the
total particle density reduces in this case to that of a one-component Fermi
vapour, whereas the frequencies of all other modes are shifted by the
interactions.
