Nematic liquid crystals between antagonistic cylinders: Spirals with bend-splay director undulations

We reexamine the problem of a nematic fluid trapped between two concentric cylinders with different anchoring conditions on each of the cylinders. If the splay constant ${\mathit{K}}_{\mathit{s}}$ is less than the bend constant ${\mathit{K}}_{\mathit{b}}$, the director varies monotonically between the plates. If, however, ${\mathit{K}}_{\mathit{s}}$>${\mathit{K}}_{\mathit{b}}$, the director can undergo a series of spatial oscillations. Observation of these oscillations may allow measurement of ${\mathit{K}}_{\mathit{s}}$/${\mathit{K}}_{\mathit{b}}$. For semiflexible polymer liquid crystals the oscillations should be readily observable and very sensitive to temperature changes.