  Temperature and velocity-dependent $^1$S$_0$ pairing gaps, chemical
potentials and entrainment matrix in dense homogeneous neutron-proton
superfluid mixtures constituting the outer core of neutron stars, are
determined fully self-consistently by solving numerically the time-dependent
Hartree-Fock-Bogoliubov equations over the whole range of temperatures and flow
velocities for which superfluidity can exist. Calculations have been made for
$npe\mu$ in beta-equilibrium using the Brussels-Montreal functional BSk24. The
accuracy of various approximations is assessed and the physical meaning of the
different velocities and momentum densities appearing in the theory is
clarified. Together with the unified equation of state published earlier, the
present results provide consistent microscopic inputs for modeling superfluid
neutron-star cores.
