We theoretically investigate a spin-mediated conversion from fluid dynamics to voltage, known as spin hydrodynamic generation (SHDG), in oscillatory and transient unsteady flows. We consider unsteady flows of liquid metal between two parallel infinite planes and then calculate its vorticity fields based on the Navier–Stokes equation for an incompressible viscous fluid. The spin accumulation and spin current generated by unsteady flows are derived using a spin-diffusion equation, including spin-vorticity coupling, which is a couple of angular momentum between electron spin and vorticity field in unsteady flows. The estimation of SHDG in liquid mercury flow suggests that an observable magnitude of voltage can be induced in unsteady flows. Our results are expected to enable the realization of high-speed spin devices with unsteady flows and broaden the range of fluid spintronics applicability. •Theoretical investigation of spin-current generation by unsteady fluid dynamic.•The characteristic scales of unsteady flows crucial to spin-current generation are identified.•Unsteady flows can induce observable inverse spin Hall voltage.