We explore the possibility that matter bulk flows could generate the required vorticity in the electron-proton-photon plasma to source cosmic magnetic fields through the Harrison mechanism. We analyze the coupled set of perturbed Maxwell and Boltzmann equations for a plasma in which the matter and radiation components exhibit relative bulk motions at the background level. We find that, to first order in cosmological perturbations, bulk flows with velocities compatible with current Planck limits ($beta<8.5times 10^{-4}$ at $95%$ CL) could generate magnetic fields with an amplitude $10^{-21}$ G on 10 kpc comoving scales at the time of completed galaxy formation which could be sufficient to seed a galactic dynamo mechanism.
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