Fully developed, gravity-driven flow in an open channel of arbitrary electrical conductance and orientation to an applied magnetic field is investigated. The formulation of the model equations and the numerical solution methodology are described in detail. Numerical solutions of the model equations for the flow velocity profile, induced magnetic field profile, and the uniform film height as a function of Hartmann number, field angle, flow rate, and channel conductivity are presented and discussed. The parameter ranges explored are those most representative of tokamak divertor surface protection schemes, where the field is predominantly coplanar in orientation. The formation of jets in velocity and the occurrence of abrupt jumps in uniform film height are seen as the wall conductance increases. Regimes where the flow is dominated by the smaller transverse field component instead of the larger coplanar field are also observed. Simple analytic relations predicting the film height are given for the different flow regimes.