The behavior of the Doi model for liquid crystalline polymers, using a closure approximation due to Hinch and Leal (HL1), is studied and compared to the behavior using the simple quadratic closure. It is found that the HL1 closure gives some aphysical results for simple shear flow, but describes a much richer solution structure for this flow than the quadratic closure, including tumbling and wagging phenomena that are found both experimentally, and in solutions of the Doi theory without closure approximations. However, there is a very sharp transition in the predicted behavior as the flow is changed very slightly to one in which the strain rate exceeds the vorticity: the orientation distribution becomes flow aligning for all values of the nematic potential, and the phenomena of tumbling and wagging are suppressed completely. Although the quadratic closure model does not exhibit tumbling in simple shear flow, and thus has been sharply criticized as an approximation of the Doi model, it does exhibit tumbling for flows that are only very slightly more rotational than simple shear flow, and the sharp transition to flow aligning solutions with added strain rate occurs just prior to simple shear, instead of just after as in the HL1 model. Finally, we find that the transition from flow‐aligning to periodic solutions occurs via two distinct mechanisms, one occurring for Pe≳20 where there is a strong coupling between the director orientation and the degree of alignment and the other for smaller Pe where the degree of orientation is unchanged by the flow.