  Recent years have seen a surge of interest in studies of hydrodynamic
transport in electronic systems. We investigate the electron viscosity of
metals and find a new component that is closely related to Coulomb drag. Using
the linear response theory, viscosity, a transport coefficient for momentum,
can be extracted from the retarded correlation function of the momentum flux,
i.e., the stress tensor. There exists a previously overlooked contribution to
the shear viscosity from the interacting part of the stress tensor which
accounts for the momentum flow induced by interactions. This contribution,
which we dub drag viscosity, is caused by the frictional drag force due to
long-range interactions. It is therefore linked to Coulomb drag which also
originates from the interaction induced drag force. Starting from the Kubo
formula and using the Keldysh technique, we compute the drag viscosity of 2D
and 3D metals along with the drag resistivity of double-layer 2D electronic
systems. Both the drag resistivity and drag viscosity exhibit a crossover from
quadratic-in-T behavior at low temperatures to a linear one at higher
temperatures. Although the drag viscosity appears relatively small compared
with the normal Drude component for the clean metals, it may dominate
hydrodynamic transport in some systems, which are discussed in the conclusion.
