  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.

Fluid Dynamics;Mesoscale and Nanoscale Physics