One method and two results are contributed to the complete understanding about MHD laminar flow in annular channel with transverse magnetic field in this paper. In terms of the method, a computationally cheap semi-analytic algorithm is developed based on spectral method and perturbation expansion of Reynolds number $Re$. By virtue of the fast computation, numerous calculating examples with almost continuous varying Hartmann number $M$ and cross-section ratio $eta$ are performed to explore the flow patterns that are missed in previous research. In terms of the results of inertialess regime, we establish the average velocity map and electric-flow coupling demarcation in $eta$-$M$ space. Six phenomenological flow patterns and their analytical approaches are identified according to the boundary layers and electrically coupling modes. In terms of the results of inertial regime, we examine the law of decreasing order-of-magnitude of inertial perturbation on primary flow with increasing Hartmann number. It is identified the proposed semi-analytic solution coincides with the $Re^2/M^{4}$ suppression theory of Baylis & Hunt (J. Fluid Mech., vol. 43, 1971, pp. 423-428) in the case of $M<40$. When $M>40$, the pair of trapezoid vortices of secondary flow begins to crack and there is therefore a faster drop in inertial perturbation as $Re^2/M^{5}$, which is a new suppression theory. When $M>80$, the anomalous reverse vortices are fully developed near Shercliff layers resulting in the slower suppression mode of $Re^2/M^{2.5}$, which confirms the prediction of Tabeling & Chabrerie (J. Fluid Mech., vol. 103, 1981, pp. 225-239).
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