Asymptotic solution for the flow due to an infinite rotating disk in the case of small magnetic field (Q1976776)

To the classical hydrodynamical problem on an infinite rotating disk as proposed by von Kármán in 1921, the authors add here the electromagnetic effect. In order to keep the problem simple, it is supposed that the magnetic field is parallel to the axis of rotation, and the fields caused by induction are negligible. The electromagnetic effect retains the motion in azimuthal and radial directions by imposing resisting force components that are proportional to the corresponding velocity components and to the magnetic interaction number \(\beta\). The magnetohydrodynamic problem is studied in the asymptotic limit as \(\beta\) tends to zero. The leading problem goes back to von Kármán's hydrodynamical problem, whereas higher-order problems (those with higher powers of \(\beta)\) are solved following \textit{E. R. Benton} [J. Fluid Mech. 24, 781-800 (1966; Zbl 0141.43702)]. This leads to secular terms which can be determined by stretching the corresponding coordinate. The asymptotic solution and the exact numerical solution are compared, and the validity of the asymptotic solution is checked.