Confined flows under a strong vertical magnetic field and rotating flows have in common at least two striking features : first, far from walls, the velocity field is quasi-two-dimensional and then, a simple structured boundary layer arises along walls perpendicular to the magnetic field direction (Hartmann layer) or to the axis of rotation (Ekman layer). We present a study of these striking features in cases where other forces such as inertia or viscous friction are not negligible in comparison with either the Lorentz or the Coriolis force. It is shown that interaction between vertical and horizontal coupling leads, on the one hand, to a quadratic velocity field in function of the vertical coordinate (barrel effect), and on the other hand, to jets in the Hartmann and Ekman layers. A wall-model for several configurations of the Hartmann layer is then built (inertial effects, horizontal component of the magnetic field, conducting walls...), in order to avoid heavy numerical computation of these layers. Striking properties of quasi-2D flows are then used to build three different 2D models, using vertical averaging of the motion equations. Theses simple models only rely on physical assumptions and do not need any numerical tuning. The PSM2000 model for MHD flow with inertia under strong vertical field provides simulations that are very close to experimental results and points out the fact that local Ekman pumping in Hartmann layers acts approximately as a diffusion process of the vorticity field along the average current lines. An analogous model without inertial effects but with non-stationary and non-homogenous field first allows to find out cases where the Hartmann layers master the global flow, and then provides a quick tool to study the possibility of controlling liquid steel flows thanks to a sliding magnetic field (in continuous casting processes). Finally, a simple quasi-geostrophic model provides a theoretical starting point for the experimental study of parallel layers in spin-up and spin-down problems.
