Counter-flow instability of a quantum mixture of two superfluids

We study the instability of a mixture of two interacting counter-flowing superfluids. For a homogeneous system, we show that superfluid hydrodynamics leads to the existence of a dynamical instability at a critical value of the relative velocity v(cr). When the interspecies coupling is small the critical value approaches the value v(cr) = c(1) + c(2), given by the sum of the sound velocities of the two uncoupled superfluids, in agreement with the recent prediction of [Y. Castin, I. Ferrier-Barbut, C. Salomon, arXiv:1408.1326 (2014)] based on Landau's argument. The crucial dependence of the critical velocity on the interspecies coupling is explicitly discussed. Our results agree with previous predictions for weakly interacting Bose-Bose mixtures and applies to Bose-Fermi superfluid mixtures as well. Results for the stability of transversally trapped mixtures are also presented.