  We study a mixture of two superfluids with density-density and
current-current (Andreev-Bashkin) interspecies interactions. The
Andreev-Bashkin coupling gives rise to a dissipationless drag (or entrainment)
between the two superfluids. Within the quantum hydrodynamics approximation, we
study the relations between speeds of sound, susceptibilities and static
structure factors, in a generic model in which the density and spin dynamics
decouple. Due to translational invariance, the density channel does not feel
the drag. The spin channel, instead, does not satisfy the usual Bijl-Feynman
relation, since the f-sum rule is not exhausted by the spin phonons. The very
same effect on one dimensional Bose mixtures and their Luttinger liquid
description is analysed within perturbation theory. Using diffusion quantum
Monte Carlo simulations of a system of dipolar gases in a double layer
configuration, we confirm the general results. Given the recent advances in
measuring the counterflow instability, we also study the effect of the
entrainment on the dynamical stability of a superfluid mixture with non-zero
relative velocity.
