Deuterium NMR spectroscopy has been used to study the director dynamics of the nematic liquid-crystal system cetyl trimethylammonium bromide/D2O under the action of applied viscous torques. Shear forces were applied using a custom-built Couette cell that was introduced into an NMR superconducting magnet, so that its rotational axis was parallel to the magnetic field direction, along which the liquid-crystal director originally aligned. Subsequently, the inner cylinder of the cell was rotated continuously at different rates using a stepper motor. The resulting time evolution and ultimate steady-state orientation of the director, governed by the competition between the applied viscous torque with elastic and magnetic terms, was measured via observed changes in the deuterium spectrum. Using a simple gearbox allowed unprecedented access to a low-shear-rate regime in which, above a threshold shear rate, the director of part of the sample was observed to reorient, while the remaining part still aligned with the magnetic field. Subsequent increases in the applied rotational rate were found to increase the relative proportion of the orienting fraction. Spatially resolved NMR spectra showed that the orienting and field-aligned fractions formed separated bands across the gap of the Couette cell, with director reorientation being initiated at the moving inner wall. The behavior was found to be consistent with the often ignored variation in velocity gradient manifest across the gap of a cylindrical cell, so that as the angular frequency of the inner cylinder was increased the radial location of the critical shear rate required for reorientation traversed the gap. Once the applied rotational rate was sufficient to reorient the director of the entire sample, the dependence of the exhibited steady-state orientation on the average applied shear rate was measured. These results could be fitted to an analytical solution of the force-balance equation, made tractable by the assumption that the elasticity term was of minor significance and could be ignored. Additionally, the use of a numerical solution of the full force-balance equation, which explicitly includes elasticity and secondary flow and additionally allows the time evolution of the director orientation to be calculated, was investigated.