We study the ability of a plasmonic structure under illumination to release heat and induce fluid convection at the nanoscale. We first present a unified formalism to treat this multidisciplinary problem combining optics, thermodynamics and hydrodynamics and work out the orders of magnitude for each of these contributions. On this basis, numerical simulations were performed to compute the temperature profile and the velocity field of the surrounding fluid for a gold disk on a glass while illuminated at its plasmon resonance. The influence of the disk size and the fluid height is investigated. We found that while the fluid velocity remains negligible for single nanometer-sized plasmonic structures due to very low Reynolds number, thermal-induced fluid convection can play a significant role when considering either an ensemble of nanostructures or micrometre-size plasmonic structures.