We use the Leslie–Ericksen theory to simulate the shear flow of tumbling nematic polymers. The objectives are to explore the onset and evolution of the roll-cell instability and to uncover the flow scenario leading to the nucleation of disclinations. With increasing shear rate, four flow regimes are observed: stable simple shear, steady roll cells, oscillating roll cells and irregular patterns with disclinations. In the last regime, roll cells break up into an irregular and uctuating pattern of eddies. The director is swept into the flow direction in formations called ‘ridges’, which under favourable flow conditions split to form pairs of ± 1 disclinations with non-singular cores. The four regimes are generally consistent with experimental observations, but the mechanism for defect nucleation remains to be verified by more detailed measurements.