  Motivated by problems arising in tear film dynamics, we present a model for
the extensional flow of thin sheets of nematic liquid crystal. The rod-like
molecules of these substances impart an elastic contribution to its response.
We rescale a weakly elastic model due to Cummings et al. [European Journal of
Applied Mathematics 25 (2014): 397-423] to describe a case of moderate
elasticity. The resulting system of two nonlinear partial differential
equations for sheet thickness and axial velocity is nonlinear and fourth order
in space, but still represents a significant reduction of the full system. We
analyze solutions arising from several different boundary conditions, motivated
by the underlying application, with particular focus on dynamics and underlying
mechanisms under stretching. We solve the system numerically, via collocation
with either finite difference or Chebyshev spectral discretization in space,
together with implicit time stepping. At early times, depending on the initial
film shape, pressure either aids or opposes extensional flow, which changes the
shape of the sheet and may result in the loss of a minimum or maximum at the
moving end. We contrast this finding with the cases of weak elasticity and
Newtonian flow, where the sheet retains all extrema from the initial condition
throughout time.
