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.
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