  In this study, we analyse "magneto-Stokes" flow, a fundamental
magnetohydrodynamic (MHD) flow that shares the cylindrical-annular geometry of
the Taylor-Couette cell, but uses applied electromagnetic forces to circulate a
free-surface layer of electrolyte at low Reynolds numbers. The first complete,
analytical solution for time-dependent magneto-Stokes flow is presented and
validated with coupled laboratory and numerical experiments. Three regimes are
distinguished (shallow-layer, transitional, and deep-layer flow regimes), and
their influence on the efficiency of microscale mixing is clarified. The
solution in the shallow-layer limit belongs to a newly-identified class of MHD
potential flows, and thus induces mixing without the aid of axial vorticity. We
show that these shallow-layer magneto-Stokes flows can still augment mixing in
distinct Taylor dispersion and advection-dominated mixing regimes. The
existence of enhanced mixing across all three distinguished flow regimes is
predicted by asymptotic scaling laws and supported by three-dimensional
numerical simulations. Mixing enhancement is initiated with the least
electromagnetic forcing in channels with order-unity depth-to-gap-width ratios.
If the strength of the electromagnetic forcing is not a constraint, then
shallow-layer flows can still yield the shortest mixing times in the
advection-dominated limit. Our robust description of momentum evolution and
mixing of passive tracers makes the annular magneto-Stokes system fit for use
as an MHD reference flow.
