We present a numerical method to evolve the flow dynamics of a model for nematic liquid crystalline polymers (LCPs) in a recirculating cavity flow, coupling the Doi model for the microstructure of the fluid with the incompressible Navier-Stokes equations of motion. In particular, we model the flow dynamics of a LCP confined to a square channel with a lid moving at a constant speed under the simplifying assumption that there are no spatial gradients in the direction of the channel's length, i.e., the only relevant flow dynamics occur in a 2-dimensional, square flow domain. Flow patterns typical of this system include rotation near the center of the domain, extensional flow near the boundaries, and counter-rotating vortices in the corners. We use the spherical harmonics transform to develop a Fokker-Planck solver to evolve the microstructure and utilize the Projection Method to evolve the macroscopic flow.

At increasing Deborah numbers, the model captures the formation of +1/2 and +1 disclinations in the microstructure and suggests four regimes of dynamics of the microstructure: tumbling between two anisotropic layers, stabilization of the tumbling with formation of a +1/2 disclination, destabilization with the formation of a second +1/2 disclination, and re-stabilization. This progression of the dynamics demonstrates a wagging-to-flow-aligning transition at De ≈ 3 rather than at De ≈ 5, as observed experimentally by Larson and Mead [Liquid Crystals, 15 (1993), pp. 151-169]. Furthermore, instead of simply a tumbling-to-wagging transition at De ≈ 2, this model predicts a transition from tumbling to equilibrating at De ≈ 1 and then to wagging at De ≈ 2.