Magnetohydrodynamic secondary flows at high Hartmann numbers

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Abstract
Laminar flows in annular ducts of rectangular cross-section subjected to a constant axial magnetic field B0 are considered. The equations of flow are treated by a perturbation method involving infinite series expansions in ascending powers of the ratio Λ = a/R0 (where a and R0 are respectively the height and the mean radius of the annular duct).The leading terms of the series are calculated in the range of high values of the Hartmann number M by means of a boundary-layer technique. When M is large, the secondary flow pattern exhibits two profoundly distinct features. Firstly, in the Hartmann and interior regions, secondary flows have a one-dimensional structure and involve no inertial effects if the curvature of the duct is small enough; secondly, in thin layers near the walls parallel to the magnetic field, the secondary flow pattern is dramatically influenced by the conductivities of the walls: varying these conductivities gives rise to either one or several counter-rotating eddies. When the Reynolds number of the flow increases, inertial effects emanating from these layers penetrate the core of the duct by convective transport. Order-of-magnitude arguments show that the mean velocity is affected by secondary flow effects when becomes large, where K is the Dean number of the flow.

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