Context. Magnetic batteries are potential sources that may drive the generation of a seed magnetic field, even if this field is initially zero. These batteries can be the result of nonaligned thermodynamic gradients in plasmas, as well as of special and general-relativistic effects. So far, magnetic batteries have only been studied in ideal magnetized fluids.Aims. We studied the nonideal fluid effects introduced by the energy flux in the vortical dynamics of a magnetized plasma in curved space-time. We propose a novel mechanism for generating a heat-flux-driven magnetic seed within a simple accretion disk model around a Schwarzschild black hole.Methods. We used the 3 + 1 formalism for the splitting of the space-time metric into space-like and time-like components. We studied the vortical dynamics of a magnetized fluid with a heat flux in the Schwarzschild geometry in which thermodynamic and hydrodynamic quantities are only dependent on the radial coordinate. Assuming that the magnetic field is initially zero, we estimated the linear time evolution of the magnetic field due to the inclusion of nonideal fluid effects.Results. When the thermodynamic and hydrodynamic quantities vary only radially, the effect of the coupling between the heat flux, the space-time curvature, and the fluid velocity acts as the primary driver for an initial linearly time-growing magnetic field. The plasma heat flux completely dominates the magnetic field generation at a specific distance from the black hole, where the fluid vorticity vanishes. This distance depends on the thermodynamical properties of the Keplerian plasma accretion disk. These properties control the strength of the nonideal effects in the generation of seed magnetic fields.Conclusions. We find that heat flux is the main driver of a seed magnetic field in black hole accretion disks if the geometry, plasma dynamics, and thermodynamics share the same axial symmetry. This suggests that nonideal fluid effects may play a major role in the magnetization of astrophysical plasmas.