  In the shear flow of liquid crystalline polymers (LCPs) the nematic director
orientation can align with the flow direction for some materials, but
continuously tumble in others. The nematic dumbbell (ND) model was originally
developed to describe the rheology of flow-aligning semi-flexible LCPs, and
flow-aligning LCPs are the focus in this paper. In the shear flow of monodomain
LCPs it is usually assumed that the spatial distribution of the velocity is
uniform. This is in contrast to polymer solutions, where highly non-uniform
spatial velocity profiles have been observed in experiments. We analyse the ND
model, with an additional gradient term in the constitutive model, using a
linear stability analysis. We investigate the separate cases of constant
applied shear stress, and constant applied shear rate. We find that the ND
model has a transient flow instability to the formation of a spatially
inhomogeneous flow velocity for certain starting orientations of the director.
We calculate the spatially resolved flow profile in both constant applied
stress and constant applied shear rate in start up from rest, using a model
with one spatial dimension to illustrate the flow behaviour of the fluid. For
low shear rates flow reversal can be seen as the director realigns with the
flow direction, whereas for high shear rates the director reorientation occurs
simultaneously across the gap. Experimentally, this inhomogeneous flow is
predicted to be observed in flow reversal experiments in LCPs.
