The tumbling behavior of LCPs at low shear rates is investigated with the help of the Leslie-Ericksen theory restricted to the planar case and to the one-constant approximation. It is shown that the director distortions induced by the tumbling nature of the nematic may easily reach down to the molecular level, where they get saturated. A rotational motion of the director becomes unavoidable in such a case. The time-dependent structure of the director field is then investigated, thus obtaining definite predictions about the shear dependence of the viscosity in the low shear rate range