We study the ability of a plasmonic structure under illumination to release heat and induce fluid convection at the nanoscale. We first introduce the unified formalism associated with this multidisciplinary problem combining optics, thermodynamics, and hydrodynamics. On this basis, numerical simulations were performed to compute the temperature field and velocity field evolutions of the surrounding fluid for a gold disk on glass while illuminated at its plasmon resonance. We show that the velocity amplitude of the surrounding fluid has a linear dependence on the structure temperature and a quadratic dependence on the structure size (for a given temperature). The fluid velocity remains negligible for single nanometer-sized plasmonic structures (<1 nm/s) due to a very low Reynolds number. However thermal-induced fluid convection can play a significant role when considering either micrometer-size structures or an assembly of nanostructures.