An electronic current driven through a conductor can induce a current in
another conductor through the famous Coulomb drag effect. Similar phenomena
have been reported at the interface between a moving fluid and a conductor, but
their interpretation has remained elusive. Here, we develop a
quantum-mechanical theory of the intertwined fluid and electronic flows, taking
advantage of the non-equilibrium Keldysh framework. We predict that a globally
neutral liquid can generate an electronic current in the solid wall along which
it flows. This hydrodynamic Coulomb drag originates from both the Coulomb
interactions between the liquid's charge fluctuations and the solid's charge
carriers, and the liquid-electron interaction mediated by the solid's phonons.
We derive explicitly the Coulomb drag current in terms of the solid's
electronic and phononic properties, as well as the liquid's dielectric
response, a result which quantitatively agrees with recent experiments at the
liquid-graphene interface. Furthermore, we show that the current generation
counteracts momentum transfer from the liquid to the solid, leading to a
reduction of the hydrodynamic friction coefficient through a quantum feedback
mechanism. Our results provide a roadmap for controlling nanoscale liquid flows
at the quantum level, and suggest strategies for designing materials with low
hydrodynamic friction