Flow in an insulating rectangular duct at the entry of a magnet

The main assumptions of the previous theories on duct flows in non-uniform magnetic fields are revisited in the case of a slotted duct with a small aspect ratio ε = b/a. In particular, a div-andcurl-free two-dimensional magnetic field is selected, and inertia is taken into account. A new set of equations is proposed and solved numerically for weak, moderate and strong inertial effects, which are characterized by the number R = ε²Re/Ha, where Re and Ha are the Reynolds and Hartmann numbers. Results are in fair agreement with available experimental data and with a newly developed asymptotic analysis, which provides scaling laws for typical velocities, length scales, pressure distribution and head losses. In the inertialess regime, the flow exhibits an upstream region, where the side jets form on both sides of an almost motionless core, whose typical length scales as (εHa)^1/3, and then a rapid transition toward the Shercliff fully-developed flow across a transverse layer, whose thickness scales as ε and is Ha-independent; most of the pressure variation, scaling as Ha^1/2/R, takes place within this transverse layer. When inertia increases, the upstream region, where the M-shaped velocity profile forms, becomes shorter and shorter, with a typical length of the order of (ε²Ha/R)^1/3. In turn, the length scale of the transition toward the downstream Shercliff flow becomes very long, of the order of R, as R increases. The flow predicted in this transition region, made of a core with a universal M-shaped profile, between the classic Shercliff layers along the side walls, is in fair agreement with experimental measurements. In the regime with strong inertia, the pressure variation is found to scale as Ha/εR.